Geometric and Functional Analysis Seminar
Geometric and Functional Analysis Seminar
Organizers: Kari Astala, Ilkka Holopainen, Pekka Pankka, Eero Saksman, Hans-Olav Tylli, Xiao Zhong
The Geometric and Functional Analysis Seminar combines (from September 2018) the earlier Geometric Analysis Seminar and the Functional Analysis Seminar at the department. The combined seminar is devoted to research in analysis in a wide sense, including Functional analysis and Geometric analysis, as well as its applications. The talks will be held in room C124 (Exactum building, Kumpula Science campus) on Thursday 12-14 o'clock (the main time slot), and occasionally on Tuesday 14-16 o'clock (reserve slot).
TALKS AUTUMN 2024
Thursday 12.12.2024 12.15-14 o'clock in room C124
Gaetan Leclerc (Helsinki): Fourier dimension and dynamical fractals
Abstract:
Consider the triadic Cantor set equipped with the Cantor law. It happens that its cumulative distribution function, the devil staircase, is Hölder regular, and its best exponent of regularity is ln(2)/ ln(3), which is exactly the Hausdorff dimension of the Cantor set. Moreover, one can show that the Fourier transform of the Cantor law decay like |xi|^{− ln 2/ ln 3} “on average”. This is no coincidence, and hint for a deeper link between Fractal Geometry and Fourier Analysis. In this talk we will detail and explore this link through the notion of Fourier Dimension. We will introduce the Fourier dimension, compute it on some easy examples, quote some natural questions that arise, and then discuss a (partial) state of the art on the topic. We will in particular discuss why we expect dynamical fractals to have positive Fourier dimension when the underlying dynamical system is nonlinear.
----
Thursday 5.12.2024 12.15-14 o'clock in room C124
Sören Mikkelsen (Helsinki): Sharp semiclassical spectral asymptotics for Schrödinger operators without full regularity.
Abstract:
In this talk we will discuss some recent results on sharp semiclassical spectral asymptotics for Schrödinger operators obtained without full regularity. We will in the talk present the results and discuss what challenges arise when full regularity is not assumed.
----
Thursday 21.11.2024 12.15-14 o'clock in room C124
Teri Soultanis (University of Jyväskylä): Curve fragments and differentiability of Lipschitz functions.
Abstract:
The classical Rademacher's theorem on the a.e. differentiability of Lipschitz functions on Euclidean space has a famous generalization (due to Cheeger) to metric measure spaces (mms) supporting a Poincare inequality. This result has inspired a large body of research on the differentiability of Lipschitz functions on mms, in which the role of curve fragments as a substitute for directional derivatives has become apparent.
In this talk I discuss the connection between differentiability of Lipschitz functions and metric Sobolev spaces. In articular I'll describe how, using tehniques from metric Sobolev spaces, one can establish some weak (i.e. directional) differentiability of Lipschitz functions with essentially no assumptions on the mms.
----
Thursday 14.11.2024 12.15-14 o'clock in room C124
Jarmo Jääskeläinen (Helsinki): Homeomorphic Sobolev extensions of parametrizations of Jordan curves.
Abstract
Each homeomorphic paramatrization of a Jordan curve via the unit circle extends to a homeomorphism of the entire plane. It is a natural question to ask if such a homeomorphism can have some Sobolev regularity.
This prompts the simplified question: for a homeomorphic embedding of the unit circle into the plane, when can we find a homeomorphism from the unit disk that has the same boundary values and integrable first-order distributional derivatives?
We give the optimal geometric criterion for the interior Jordan domain so that there exists a Sobolev homeomorphic extension for any homeomorhic parametrization of the Jordan curve. The problem is partially motivated by trying to understand which boundary values can correspond to deformations of finite energy. This is a joint work with Bouchala, Koskela, Xu and Zhou.
----
Thursday 7.11.2024 12.15-14 o'clock in room C124
Ikshu Neithalath (Helsinki): Sheaf-theoretic SL(2,C) Floer homology and Knot Surgeries
Abstract:
SL(2,C) Floer homology is a 3-manifold invariant defined by Abouzaid-Manolescu that applies objects from algebraic geometry such as character varieties and perverse sheaves of vanishing cycles to the study of low-dimensional topology. We will describe the construction of this invariant as well as independent work on computing it for surgeries on some knots
-----
Thursday 10.10.2024 12.15-14 o'clock in room C124
Marcin Gryszowka (IMPAN, Warsaw): Carleson measures on domains in Heisenberg groups
Abstract:
Carleson measures are a useful tool in harmonic analysis. They are connected with nontangential maximal function and so called square function. However, they were mostly studied in the Euclidean setting. We prove characterization of Carleson measures in Heisenberg group. Moreover, for a subelliptic harmonic function we give a bound on a norm of its square function in terms of norm of its boundary data and prove that square of norm of gradient of a subelliptic harmonic function gives rise to a Carleson measure. We mostly work with nontangentially accessible domains. The talk is based on joint work with Tomasz Adamowicz.
----
Thursday 3.10.2024 12.15-14 o'clock in room C124
Kari Vilonen (University of Melbourne): Unitary representations of Lie groups and Hodge theory.
Abstract:
The determination of the unitary dual of a Lie group is a longstanding problem in mathematics. The unitary dual consist of all irreducible unitary representations of a given Lie group. The question is important for the purposes of L^2 harmonic analysis, for example, for the Langlands program. There have been many attempts to solve this problem and to put it in some general conceptual context. For example, in the 1960’s Kostand and Kirillov introduced the orbit method where the idea is to quantize co-adjoint orbits. In this talk I will explain a new point of view to this question. The idea is that representations carry a Hodge structure and the Hodge structure can be used to determine unitarity. The Hodge structure arises from the theory of mixed Hodge modules of Saito. This is joint work with Dougal Davis establishing earlier conjectures made by Schmid and myself several years ago.
----
Tuesday 1.10.2024 14.15-16 o'clock in room B222 (Exactum, 2. floor) [Please note exceptional time and place]
Andre Guerra (ETH Zurich): The Monge-Ampere equation and quasiconformal maps
Abstract:
In the complex plane, there is a surprising correspondence between solutions of the Monge-Ampère equation and solutions of a certain Beltrami equation associated to SO(2). Under this correspondence, the W^{2,p} regularity of solutions to Monge--Ampère corresponds to a precise unique continuation principle for solutions of the Beltrami equation. We will sketch a proof of the latter, which relies on ideas coming from Differential Inclusions. The talk is based on joint work with G. De Philippis and R. Tione.
----
Thursday 26.9.2024 12.15-14 o'clock in room C124
Sauli Lindberg (Helsinki): Weakly continuous nonlinear quantities in magnetohydrodynamics.
Abstract:
In mathematical fluid dynamics, convex integration has been used for the past couple of decades to prove flexibility statements for weak solutions (breakdown of classical conservation laws and non-uniqueness in initial value problems). As a counterpart, for weak solutions with more regularity/integrability, there are rigidity statements showing conservation and uniqueness. In the latter results, a key role is played by weakly continuous nonlinear quantities (compensated compactness quantities). I illustrate this phenomenon in the case of ideal MHD (a combination of Euler and Maxwell equations). The talk is based on joint works with Daniel Faraco, Lauri Hitruhin and László Székelyhidi Jr.
TALKS SPRING 2024
Thursday 23.5.2024 12.15-14 o'clock in room C124
Toni Ikonen (Helsinki): Beppo—Levi and Newtonian Sobolev spaces: equivalence in metric measure spaces.
Abstract:
In this talk, we continue the previous overview (GAFA seminar, 14.3.2024) of the equivalence of four definitions of Sobolev spaces on metric measure spaces for p greater or equal to one. We expand upon the fact that every Sobolev function in the Beppo—Levi sense has a representative in the Newtonian space; as Newtonian representatives are sensitive to changes in sets of positive capacity, this is the most involved part of the full equivalence proof.
We introduce a new curve modulus à la Savaré ('23) and outline its role in the construction of the aforementioned representative. Along the way, we prove that a certain geometrically important curve family having positive and finite modulus in the Savaré sense contains a compact curve subfamily of positive modulus in the same sense. The extraction of the compact subfamily is important for a certain minmax argument. Since modulus is not a Choquet capacity when p = 1, the compact subfamily is constructed with purely elementary techniques.
Based on joint work with L. Ambrosio (Scuola Normale Superiore), D. Lučić and E. Pasqualetto (University of Jyväskylä).
-----
Friday 26.4.2024 14.15-16 o'clock in room C124 [Please note exceptional time]
Håkan Hedenmalm (KTH Stockholm): Hyperbolic Fourier series and the Klein-Gordon equation.
Abstract:
We show that any distribution or ultra distribution on the extended real has an expansion as a hyperbolic Fourier series. The hyperbolic Fourier series were considered in work of Hedenmalm and Montes in 2011 (Annals). We connect this with the interpolation theory of the Klein-Gordon equation.
-----
Thursday 25.4.2024 12.15-14 o'clock in room C124
Peter Lindqvist (NTNU Trondheim): Anomalies in Non-Linear Rayleigh Quotients
Abstract (see pdf-file):
-----
Friday 19.4.2024 14.00-15.00 o'clock in room C124 [Please note the exceptional time and that the talk starts at 14.00 o'clock]
Constanze Liaw (University of Delaware): A Survey of Aleksandrov-Clark Theory and Generalizations
Abstract:
We will begin by explaining Aleksandrov-Clark Theory: First note that Beurling’s Theorem says that any non-trivial shift-invariant subspace of the Hardy space $H^2(\mathbb{D})$ is of the form $\theta H^2(\mathbb{D})$ for an inner function $\theta$. Now, for a fixed inner $\theta$, we form the model space, that is, the orthogonal complement of the corresponding shift-invariant subspace in the Hardy space. Consider the compressed shift, which is the application of the shift to functions from the model space followed by the projection to the model space. Clark observed that all rank one perturbations of the compressed shift that are also unitary have a particular, simple form. Following this discovery, a rich theory was developed connecting the spectral properties of those unitary rank one perturbations with properties of functions from the model space, more precisely, with their non-tangential boundary values. Some intriguing perturbation results were obtained via complex function theory.
Generalizations we will consider in the mini course include the following. Model spaces can be defined but turn out considerably more complicated when $\theta$ is not inner. Finite rank perturbations were investigated. Parts of the theory have been studied for some several variables settings.
-----
Thursday 11.4.2024 12.15-14 o'clock in room C124
Maria Gordina (University of Connecticut): Dirichlet boundary problems in metric measure spaces
Abstract:
We consider Dirichlet boundary problems in metric measure spaces. Results include properties of the spectrum, regularity and $L^{p}$-estimates of eigenfunctions, as well as irreducibility of the corresponding stochastic processes. A number of examples will be given including both local and non-local Dirichlet forms, and applications to limit laws such as small deviations and large time behavior of the heat content.
-----
Thursday 4.4.2024 12.15-14 o'clock in room C124 (joint with the Mathematical Physics Seminar)
Joonas Turunen (Universität Wien): A Wiener-Hopf factorization for planar maps coupled with the O
loop model with mixed boundary conditions.
Abstract:
Random maps coupled with the O loop model encompass many important random geometries coupled with matter fields of statistical physics. From a mathematical point of view, they provide a beautiful interplay between combinatorics, analysis, geometry and probability. In this talk, I discuss a particular case of the model on planar quadrangulations with so-called Dirichlet-Neumann boundary conditions (with an analogy to boundary value problems for PDEs). From a simple combinatorial decomposition, we are able to find a Wiener-Hopf factorization for the generating functions of the model and to solve it analytically. We also compare our results with those obtained for a similar model on triangulations by Kazakov and Kostov in the physics literature in 1992, finding a surprising critical exponent which is different from the one for triangulations. Based on a joint work with Jérémie Bouttier (Sorbonne University) and Grégory Miermont (ENS de Lyon).
-----
Thursday 28.3.2024 12.15-13 o'clock in room C124 [60 min. talk]
Vladimir Gutlyanskii (National Academy of Sciences of Ukraine): On integral modulus estimates for q.c. maps in R^n and cavitations.
-----
Thursday 21.3.2024 12.15-13.15 o'clock in room C124 [60 min. talk]
Alexander Borichev (Aix-Marseille Universite): The Szego minimum problem.
Abstract:
Given a measure mu on the unit circle, we study the rate of approximation of the constant function by polynomials of degree n vanishing at the origin in L^2(mu). We deal mainly with measures outside of the Szego class.
-----
Thursday 14.3.2024 12.15-14 o'clock in room C124 [transferred talk from 15.2.2024]
Toni Ikonen (Helsinki): Sobolev spaces, derivations and divergence measures.
Abstract:
In this talk, we overview four definitions of Sobolev spaces on metric measures spaces for p greater or equal to one: based on approximation by Lipschitz functions (Cheeger), integration by parts relative to derivations (Di Marino), and two generalizations of the ACL property (Shanmugalingam and Ambrosio-Gigli-Savaré, respectively).
The approaches are known to be equivalent when p > 1 (Ambrosio-Gigli-Savaré) and during the talk we explain how to extend the result to p = 1. The approach when p = 1 requires new ideas since the gradient flow methods yield the BV theory instead (Ambrosio-Di Marino) and the 1-modulus is not a Choquet capacity (V. H. Exnerová, O. F.K. Kalenda, J. Malý, O. Martio).
Based on joint work with L. Ambrosio (Scuola Normale Superiore), D. Lučić and E. Pasqualetto (University of Jyväskylä).
-----
Thursday 7.3.2024 12.15-14 o'clock in room C124
Jose Angel Pelaez (Universidad de Malaga): Composition of analytic paraproducts and the radicality property for spaces of symbols of bounded integral operators.
Abstract (see pdf-file):
-----
Thursday 29.2.2024 12.15-14 o'clock in room C124
Alexandru Aleman (University of Lund): Invariant subspaces of generalized differentiation and Volterra operators. [CANCELLED on 29.2; to be rescheduled]
Abstract:
The problem of describing the invariant subspaces of the usual Volterra operator goes back to Gelfand (1938) and this direction contains many ingenious ideas. Invariant subspaces for differentiation on C^\infty were studied much later by Korenblum and myself and subsequent work with Baranov and Belov. I intend to give a presentation of some of these ideas and then continue with a more abstract setting consisting of an unbounded operator D with a compact quasi-nilpotent right inverse V.
It turns out that under certain general conditions one can prove similar results for a large class of examples (for D) containing Schrödinger operators and many other cannonical systems. This is a report about joint work with Alex Bergman.
-----
Thursday 22.2.2024 12.15-14 o'clock in room C124 (joint with the Mathematical Physics Seminar)
Aernout van Enter (University of Groningen): One-sided versus two-sided regularity properties
Abstract:
Stochastic processes can be parametrised by time (such as occurs in Markov chains), in which case conditioning is one-sided (on the past), as naturally occurs in Dynamical Systems, or by one dimensional space (which is the case, for example, for one-dimensional Markov fields), as is natural in Statistical Mechanics, where the conditioning is two-sided (on the right and on the left). I will discuss some examples, in particular generalising this distinction to g-measures versus Gibbs measures, where, instead of a Markovian dependence, the weaker property of continuity (in the product topology) is considered. In particular I will discuss when the two descriptions (one-sided or two-sided) produce the same objects and when they are different. We show moreover the role one-dimensional entropic repulsion plays in this setting.
Based on joint work with R. Bissacot, E. Endo and A. Le Ny, and S. Shlosman.
-----
Thursday 15.2.2024 12.15-14 o'clock in room C124
Toni Ikonen (Helsinki): Sobolev spaces, derivations and divergence measures. Note: the talk has been shifted to Thursday 14.3 (because of strike action on 15.2)
Abstract:
In this talk, we overview four definitions of Sobolev spaces on metric measures spaces for p greater or equal to one: based on approximation by Lipschitz functions (Cheeger), integration by parts relative to derivations (Di Marino), and two generalizations of the ACL property (Shanmugalingam and Ambrosio-Gigli-Savaré, respectively).
The approaches are known to be equivalent when p > 1 (Ambrosio-Gigli-Savaré) and during the talk we explain how to extend the result to p = 1. The approach when p = 1 requires new ideas since the gradient flow methods yield the BV theory instead (Ambrosio-Di Marino) and the 1-modulus is not a Choquet capacity (V. H. Exnerová, O. F.K. Kalenda, J. Malý, O. Martio).
Based on joint work with L. Ambrosio (Scuola Normale Superiore), D. Lučić and E. Pasqualetto (University of Jyväskylä).
-----
Thursday 8.2.2024 12.15-14 o'clock in room C124
Lauritz Streck (University of Cambridge): Entropy methods in Fractal Geometry
Abstract:
For much exciting recent progress on self-similar sets and measures, entropy methods have played a crucial role. We start by defining the entropy associated to a self-similar measure and giving an overview of its role in some recent breakthroughs in fractal geometry. We then show how one can reduce certain questions about the entropy to the algebraic properties of the parameters defining the self-similar measure, and why absolute continuity and power Fourier decay of the measure are present in these cases.
---
Thursday 25.1.2024 12.15-14 o'clock in room C124
Tomasz Cias (Adam Mickiewicz University Poznan): Runge type approximation theorems for kernels of certain partial differential operators
Abstract:
Let $H(\Omega)$ denote the space of holomorphic functions defined on a open subset $\Omega$ of the complex plane $\mathbb{C}$, i.e., the kernel of the Cauchy-Riemann operator on the space $C^\infty(\Omega)$ of smooth functions on $\Omega$.
From Runge's approximation theorem it follows that for open subsets $\Omega_1\subset\Omega_2$ of $\C$ the restriction map $r\colon H(\Omega_2)\to H(\Omega_1)$, $rf:=f_{\mid{\Omega_1}}$, has dense range if and only if no compact connected component of $\mathbb{C}\setminus\Omega_1$ is contained in $\Omega_2$.
In 1955, Lax and Malgrange extended independently this result to kernels of elliptic partial differential operators with constant coefficients. Some other classes of partial differential operators were considered later by several authors.
In this talk we will discuss the problem of characterization of density of the range of restriction maps between the kernels of certain partial differential operators defined on the spaces $C^\infty(\Omega)$ as well as on the spaces $\mathcal E(F)$ of smooth Whitney jets, where $\Omega$ and $F$ are open and closed subsets of $\mathbb{R}^d$, respectively. This is joined work with Thomas Kalmes from Chemnitz University of Technology.
Abstract (see pdf-file):
----
Thursday 18.1.2024 12.15-14 o'clock in room C124
Carsten Peterson (Universität Paderborn): Quantum ergodicity on Bruhat-Tits buildings
Abstract:
Originally, quantum ergodicity concerned equidistribution properties of Laplacian eigenfunctions with large eigenvalue on manifolds for which the geodesic flow is ergodic. More recently, several authors have investigated quantum ergodicity for sequences of spaces which "converge" to their common universal cover and when one restricts to eigenfunctions with eigenvalues in a fixed range. Previous authors have considered this type of quantum ergodicity in the settings of regular graphs, rank one symmetric spaces, and some higher rank symmetric spaces. We prove analogous results in the case when the underlying common universal cover is the Bruhat-Tits building associated to PGL(3, F) where F is a non-archimedean local field. This may be seen as both a higher rank analogue of the regular graphs setting as well as a non-archimedean analogue of the symmetric space setting.
TALKS AUTUMN 2023
Thursday 7.12.2023: no seminar.
There is a related miniconference Quasiconformal Geometry, Analysis on Metric Saces and Minimal Surfaces to honor the career of Ilkka Holopainen on Friday 8.12.2023 (Exactum, room C222). For details of the program see
https://www.mv.helsinki.fi/home/pankka/ilkkafest.html
Thursday 30.11.2023 12.15-14 o'clock in room C124
Oscar Kivinen (Aalto): Harmonic analysis on p-adic groups and braids
Abstract:
In the first part of the talk, I will give an introduction to harmonic analysis on p-adic Lie algebras and groups, focusing on the GL case. I will then describe some recent progress in the field (such as the computation of “Shalika germs”) due to myself and Cheng-Chiang Tsai. The classical braid groups enter the story in a somewhat unexpected way, and I will describe the general expectation about the role of braids in this type of harmonic analysis.
----
Thursday 23.11.2023 12.15-14 o'clock in room C124
Sari Rogovin (Aalto): Remarks about Gehring-Hayman Theorem and Uniformity
Abstract:
We will discuss first shortly about the historical background of the Gehring-Hayman type theorems. Then we look at how this relates to uniformity of a space and talk about some equivalent characterizations for the Gehring-Hayman type theorem in metric space setting.
----
Thursday 9.11.2023 12.15-14 o'clock in room C124
Mario Ullrich (Johannes Kepler Universität Linz): High-dimensional integration and the curse of dimension (part 2 of FiRST-CoE Lecture series)
Abstract:
In Part 2, the focus is on high-dimensional integration and approximation, and the dependence of the error on the dimension. Here, we mainly discuss the "curse of dimension" for classical (isotropic) spaces C^k on domains, and that the (expectedly ineffective) product rules are indeed optimal in high-dimensions.
I will mention several open problems in the field. In both parts, I'll try to introduce all the necessary concepts in detail and therefore think that no expertise is required to follow the talk.
For the abstract of part 1, see the link https://www.helsinki.fi/fi/projektit/first/events
----
Thursday 2.11.2023 12.15-14 o'clock in room C124
Will Hide (Durham University): Spectral gaps for random covers of hyperbolic surfaces
Abstract:
----
Thursday 26.10.2023 12.15-14 o'clock in room C124
Susanna Heikkilä (Helsinki): Quasiregular curves and cohomology
Abstract:
In this talk, we define quasiregular ellipticity in the setting of quasiregular curves. We also discuss the cohomology of quasiregularly elliptic manifolds and give examples of (non-)ellipticity.
----
Thursday 19.10.2023 12.15-14 o'clock in room C124
Kari Vilonen (University of Melbourne): Geometric Satake, part 2.
The lecture is part of a lecture series organised by the Finnish Centre of Excellence in Randomness and Structures (FiRST). Part 1 takes place on Wednesday 18.10 at 14-16 o'clock (room C124) and part 3 on Tuesday 24.10 at 14-16 o'clock (room C124).
Abstract:
I will give a leisurely introduction to a result which is basic to geometric approaches to the Langlands program. In particular, I will explain how the dual group of a reductive group arises naturally from the geometry of the affine Grassmannian.
----
Thursday 12.10.2023 12.15-14 o'clock in room C124
Athanasios Tsantaris (Helsinki): Quasiconformal curves.
Abstract:
Quasiconformal omega-curves, for some smooth, non-vanishing, closed form omega, are embeddings from R^n to R^m satisfying a form of the distortion inequality. They can be considered as analogues of classical quasiconformal mappings in the case where the dimension of the domain and the codomain differ. In this talk we are going to discuss the relationship of these maps with quasiconformal mappings between metric spaces.
----
Thursday 5.10.2023 12.15-14 o'clock in room C124
Santeri Miihkinen (University of Reading and University of Helsinki): The infinite Hilbert matrix on spaces of analytic functions
Abstract:
----
Thursday 28.9.2023 12.15-14 o'clock in room C124
Shusen Yan (Central China Normal University): On the critical points for the Robin functions
Abstract:
The study of blow-up solutions for many nonlinear elliptic problems will lead to the investigation of the non-degenerate critical points of the Robin functions. In this talk, I will present some results on this aspect, with emphasis on the effects of the small holes in the domain on the existence and non-degeneracy of the critical points.
|
-----
Thursday 21.9.2023 12.15-14 o'clock in room C124
Joni Teräväinen (University of Turku): Generation of the multiplicative group of integers by a few small primes
Abstract:
A conjecture of Erdős states that, for any large prime q, every element of the multiplicative group of integers modulo q is represented by a product of two primes where each factor is less than q. I will discuss a proof of a ternary version of Erdős' conjecture, namely that products of three primes less than q represent every element of the group. The proof is based on a Fourier-analytic argument which we call a multiplicative transference principle, some tools from additive combinatorics, and some number theory input on character sums. This is joint work with Kaisa Matomäki.
----
Thursday 7.9.2023 12.15-14 o'clock in room C124
Toni Ikonen (Helsinki): Pushforward of currents under Sobolev maps
Abstract:
Recently, I proved that any Sobolev map from a Riemannian manifold into a complete metric space pushes forward almost every integral current to an Ambrosio-Kirchheim integral current in the metric target, where 'almost every' is understood in a modulus sense. The existence of the pushforward has several consequences. In particular, it implies a positive answer to an open question by Onninen and Pankka on sharp Hölder continuity of quasiregular curves.
TALKS SPRING 2023
The talks in the seminar during the Spring will take place on-site in lecture room C124 (Exactum).
-----
Tuesday 20.6.2023 14.15-16 o'clock in room C124 (Please note exceptional time)
Ilmari Kangasniemi (Cincinnati): Quasiregular values and Rickman's Picard theorem
Abstract: A locally n-Sobolev map f : Ω → ℝⁿ has a (K, Σ)-quasiregular value at a point y ∈ ℝⁿ if it satisfied the generalized distortion inequality |Df
|ⁿ ≤ K det(Df) + Σ|f - y|ⁿ, where Σ is a locally Lᵖ-integrable function for p > 1 and y ∈ ℝⁿ is a fixed point. This condition provides single-value versions of various standard results of quasiregular maps, such as the Liouville and Reshetnyak's theorems. In this talk, we discuss a quasiregular values -version of Rickman's Picard Theorem, which in this generality turns out to be about the number of quasiregular values in the boundary ∂f(ℝⁿ). The proof builds upon the recent new proof of Rickman's Picard theorem by Bonk and Poggi-Corradini. Joint work with Jani Onninen.
-----
Tuesday 13.6.2023 14.15-15 o'clock in room C124 (Please note exceptional time)
Nikolai Kuchumov (LPSM, Sorbonne University Paris): A variational principle for multiply-connected domains
Abstract:
In the talk we will focus on a probabilistic features of random domino tilings (a.k.a. dimer configurations) of a multiply-connected domain. We will compare the classical Arctic circle theorem and its analog for the Aztec diamond with a hole, where the height function obtains a monodromy, non-zero increment going around a loop. A more suitable definition of the height function will be presented, which will give us access to a variational principle for multiply-connected domains.
-----
Thursday 1.6.2023 12.15-14 o'clock in room C124
Oleksiy Dovgoshey (National Academy of Sciences of Ukraine): Rigidity and related properties of semimetric spaces
Abstract:
Slides:
-----
Thursday 25.5.2023 12.15-14 o'clock in room C124
David Fisher (Rice University, Houston): Tukia type theorems for Carnot-by-Carnot groups
Abstract:
Motivated by Mostow's proof of Mostow rigidity, Tukia proved landmark results about when a group of quasi-conformal mappings of R^n where quasiconformally conjugate to a conformal action. Gromov later pointed out that Tukia's theorem had fairly immediate consequences for the quasi-isometric rigidity of fundamental groups of hyperbolic manifolds.
In this talk I will discuss recent generalizations with Dymarz and Xie of Tukia's theorem to a broader class of spaces, namely what we call Carnot-by-Carnot groups. The motivation for studying this class of spaces (and even broader ones) also comes from the study of quasi-isometric rigidity for certain groups and spaces.
-----
Tuesday 16.5.2023 13.15-14 o'clock in room C124 (Please note exceptional time)
Andras Stipsicz (Alfred Renyi Institute of Mathematics, Budapest): Knot and surfaces in dimension four
Abstract:
Continuous and smooth differs most drastically in dimension four, and it seems that properties of surfaces (and knots in four-manifolds with boundary) display this difference in the most transparent way. The constructions rely on topological ideas, while obstructions use global analysis and differential geometry. In the lecture I will recall the basic results of the subject, list the most important problems, and report on some advances in these questions.
-----
Thursday 4.5.2023 12.15-14 o'clock in room C124
Lassi Päivärinta (Tallinn University of Technology): New thoughts on the inverse screen scattering problem
Abstract:
We describe some new ideas in the inverse scattering problem for screens in Euclidean spaces, particularly for the one passive measurement problem. In this problem, the transmitter is in a fixed location, the energy is fixed but the measurement direction varies. In mathematical terms, the far field is known only for a single incoming wave and a fixed wave number. The dimension of our data is half the dimension of the data for the fixed energy inverse scattering problem. Near the end of the talk, we give a simple result for this problem, obtained using a method that seems promising for solving many more such problems. On our journey, Mellin meets Fourier and Hilbert on the half-line.
This is based on joint work with Emilia Blåsten (Lappeenranta University of Technology), Petri Ola (University of Helsinki) and Sadia Sadique (Tallinn University of Technology).
-----
Thursday 27.4.2023 12.15-14 o'clock in room C124
Sean Gomes (Helsinki): Semiclassical scarring in KAM Hamiltonian systems
Abstract:
In this talk we will discuss the phenomenon of semiclassical scarring of eigenfunctions in KAM Hamiltonian systems of dimension 2. The persistence of a large measure family of flow-invariant Lagrangian tori in the classical dynamics is reflected on the quantum side by the scarring of quantum limits of eigenfunctions on almost all of these tori for a generic family of perturbations. A key ingredient in the proof are the quasimodes associated to such tori, constructed by Colin de Verdiere and subsequently Popov in the Gevrey regularity setting. This talk is based on joint work with Andrew Hassell.
-----
Thursday 20.4.2023 12.15-14 o'clock in room C124
Victor Macia (Universidad Autonoma, Madrid): The distance to the border of a random tree
Abstract:
The height of a rooted tree is the maximum distance from the root to the leaves. The asymptotic behaviour of the number of rooted labeled Cayley trees with n nodes and restrictions on this metric quantity was first studied by Renyi and Szekeres. Subsequently, de Bruijn, Knuth, and Rice studied the same problem for the family of rooted plane trees and Flajolet, Gao, Odlyzko, and Richmond for that of the rooted binary plane trees.
The distance to the border of a rooted tree is the minimum distance from the root to the leaves. The height and the distance to the border are two, extreme, complementary cases. We study the asymptotic behaviour of simple varieties of trees with n nodes and distance to the border >= k, as the number of nodes n escapes to infinity. These families of rooted trees include, for instance, the family of rooted labeled Cayley, plane, and binary trees. We will also translate these results into the setting of the rooted random trees.
-----
Tuesday 18.4.2023 14.15-16 o'clock in room C124 (Please note exceptional time)
Jonathan Kirby (University of East Anglia): Complex powers
Abstract:
The Fundamental Theorem of Algebra states that every non-constant complex polynomial in one variable has a complex zero. Equivalently, the complex field is algebraically closed. It follows, essentially from Hilbert's Nullstellensatz, that any system of complex polynomial equations, in several variables, which has a solution in some extension field of the complexes already has a complex solution. What happens if we consider not just polynomials but allow the complex exponential map, or ``polynomials'' with complex exponents, like $z^i + z = 1$?
It turns out that we cannot reduce to just one variable at a time, and we also have to take into account transcendence issues. After doing this, the notions analogous to algebraic closedness are called exponential-algebraic closedness and powers-algebraic closedness. I will discuss recent progress towards proving that the complex field is exponentially-algebraically closed, the recent result of Gallinaro that it is powers-algebraically closed, and obstacles in proving the full conjecture. I will also mention a model-theoretic consequence: a partial solution to Zilber's quasiminimality conjecture.
This is joint work with Aslanyan, Gallinaro, and Mantova.
-----
Thursday 13.4.2023 12.15-14 o'clock in room C124
Jarkko Siltakoski (University of Jyväskylä): On the bounded slope condition and parabolic equations
Abstract:
We consider the equation ∂tu − div(Df(t, Du))=0, where the integrand f(t, ξ) is integrable in time and convex in the second variable. Assuming that the initial and boundary datum satisfies the so called bounded slope condition, we prove the existence of a unique variational solution to the corresponding Cauchy-Dirichlet problem. Moreover, we show that the solution is Lipschitz continuous in space. This talk is based on a joint work with Leah Schätzler.
-----
Thursday 6.4.2023 12.15-14 o'clock in room C124
Efstathios Konstantinos Chrontsios Garitsis (Univ. Illinois at Urbana-Champaign): Holomorphic distortion of the Assouad spectrum
Abstract:
It is known how holomorphic maps change certain dimension notions. For instance, the Hausdorff dimension of compact sets does not change when a holomorphic map is applied, while the Box-counting dimension decreases (due to its Lipschitz stability). Fraser and Yu in 2018 introduced the notion of the Assouad spectrum, which is a collection of dimensions that can change in an uncontrollable way under Lipschitz maps. We prove that this notion also in general decreases under holomorphic maps and we provide examples showing this is sharp. As a corollary, this result implies upper bounds on the distortion of the Assouad spectrum (and dimension) under planar quasiregular maps.
-----
Thursday 30.3.2023 12.15-14 o'clock in room C124
Janne Junnila (Helsinki): Piecewise-geodesic Jordan curves on the sphere
Abstract:
In this talk I will discuss Jordan curves on the Riemann sphere passing through n fixed points that have the property that every arc between two consecutive points is a hyperbolic geodesic in the simply connected region bounded by the rest of the arcs.
Basic properties of such curves have been previously studied by Marshall, Rohde and Wang, and based on their works I will during the first half of the talk present alternative characterisations of these curves, as well as give some ideas on how to show their existence. Time permitting, I will also mention some applications.
The second part of the talk will focus on showing the uniqueness of continuously differentiable piecewise geodesic curves of a given isotopy type. This is part of (unfinished) joint work with Bonk, Marshall, Rohde and Wang.
------
Thursday 23.3.2023 12.15-14 o'clock in room C124
Adrian Llinares (NTNU Trondheim): Contractive inequalities for analytic functions.
Abstract:
Contractive inclusions between spaces of analytic functions have attracted the attention of the experts because of their multiple applications. In this talk, we will show some of these contractive inequalities and discuss their most immediate consequences.
------
Thursday 16.3.2023 12.15-14 o'clock in room C124
Tadeusz Iwaniec (Syracuse University): Complex harmonic capacitators
Abstract:
-----
Thursday 9.3.2023 12.15-14 o'clock in room C124
Susanna Heikkilä (Helsinki): De Rham algebras of closed quasiregularly elliptic manifolds are Euclidean
Abstract:
In this talk, I will discuss a result stating that, if a closed manifold admits a non-constant quasiregular mapping from the Euclidean space, then the de Rham cohomology algebra of the manifold embeds into the Euclidean exterior algebra as a subalgebra. This embedding of algebras yields a homeomorphic classification of closed simply connected orientable 4-manifolds admitting a non-constant quasiregular mapping from R^4. The talk is based on joint work with Pekka Pankka.
-----
Special joint Distinguished Visitor Lecture and GAFA-seminar talk:
Friday 3.3.2023 14.15-15 o'clock in Ahlfors Auditorium A111, Exactum
Karen E. Smith (University of Michigan): Measuring Singularities of algebraic and analytic varieties
Abstract:
Consider a single polynomial or analytic function f in a neighborhood of a point p. Its zero locus is a hypersurface X containing p, which is typically a singular (non-smooth) point of X. In algebraic geometry, there is great interest in understanding such singularities or and quantifying "how singular" a particular singular point may be. For complex hypersurfaces, I will define a numerical measure of the singularity (called the analytic index of singularity) in terms of the integrability of a natural real-valued function in a neighborhood of p, and describe a technique for determining this integrability using Hironaka's famous theorem on resolution of singularities. For varieties defined over more exotic fields, however, a completely different approach is needed: here, we show how to iterate the Frobenius map to produce a numerical measure of singularities called the F-pure threshold, which has beautiful fractal-like properties. Remarkably, these two approaches turn out to be closely related.
-----
Thursday 2.3.2023 12.15-14 o'clock in room C124
Siarhei Finski (Center of Mathematics Laurent Schwartz, École Polytechnique): Submultiplicative norms and filtrations on section rings
Abstract:
A section ring of a polarised projective manifold is the graded ring of holomorphic sections of tensor powers of the polarising line bundle. A graded norm on the section ring is called submultiplicative if the norm of products of holomorphic sections is no bigger than the products of norms.
Submultiplicative norms on section rings arise naturally in complex geometry and functional analysis. In the former context, they appear in the study of holomorphic extension problems, submultiplicative filtrations (related to K-stability) and Narasimhan-Simha pseudonorms. In the latter context, they appear in the study of projective tensor norms on polynomial rings.
We show that submultiplicative norms on section rings of polarised projective manifolds are asymptotically equivalent to sup-norms associated with metrics on the polarising line bundle. We then derive several applications of this result to the aforementioned problems.
The holomorphic extension theorem of Ohsawa and Takegoshi, semiclassical analysis in complex geometry, pluripotential theory and functional-analytic study of projective tensor norms play a prominent role in our work.
-----
Thursday 23.2.2023 12.15-14 o'clock in room C124
Hans-Olav Tylli (Helsinki): Strange properties of closed subideals of the algebra of bounded operators
Abstract:
I will describe recent joint work with Henrik Wirzenius (Helsinki) about surprising properties of closed subideals of the algebra L(X) of bounded operators on Banach spaces X. The closed linear subspace I is a closed J-subideal of L(X) if I is a closed ideal of J and J is a closed ideal of L(X). The J-subideal I is non-trivial if I is not an ideal of L(X).
For instance, there is a Banach space Z and an uncountable family of non-trivial closed K(Z)-subideals that are pairwise isomorphic as Banach algebras, which is not possible for closed ideals of L(X) by a result of Chernoff. (Here K(Z) are the compact operators on Z.) I will also construct non-trivial closed S(X)-subideals of L(X) for classical Banach spaces such as X = L^p (for p different from 2) and X = C(0,1), where S(X) is the closed ideal of the strictly singular operators on X.
-----
Thursday 16.2.2023 12.15-14 o'clock in room C124
Aleksis Koski (Helsinki): Restricted Quasiconvexity of the Burkholder functional
Abstract:
The Burkholder functional B_p is perhaps the most important rank-one convex functional in the plane due to its various connections to singular integrals, martingales, and the vector valued calculus of variations. B_p is therefore a prime candidate for which to consider the validity of the notorious Morrey conjecture of whether rank-one convexity implies quasiconvexity. The Iwaniec conjecture concerning the exact norms of the Beurling transform is also equivalent to the quasiconvexity of the Burkholder functional.
In this talk we discuss recent joint work together with K. Astala, D. Faraco, A. Guerra, and J. Kristensen. We study the Burkholder functional from the context of nonlinear elasticity, where the axiom of non-interpenetration of matter gives natural motivation to consider a restricted class of mappings which satisfy B_p(Df) ≤ 0 pointwise. In particular, we show the quasiconvexity of B_p in this class and provide a version of the area inequality for B_p.
-----
Thursday 9.2.2023 12.15-14 o'clock in room C124
Sauli Lindberg (Helsinki): Convex integration in 3D MHD
Abstract:
Magnetohydrodynamics (MHD) couples Navier-Stokes and Maxwell equations to study the interplay of a plasma and a magnetic field. In ideal MHD, where viscosity and resistivity are set to zero, smooth solutions conserve total energy (sum of kinetic and magnetic energies) and magnetic helicity. Nevertheless, in view of numerical and empirical evidence, ideal MHD should possess weak solutions that 1) arise at the ideal limit of Leray-Hopf solutions, 2) conserve magnetic helicity but 3) dissipate total energy. I will discuss partial and related results and also describe the main mathematical tool, convex integration, in the context of MHD. The talk is based on joint works with Daniel Faraco, Lauri Hitruhin and László Székelyhidi Jr.
-----
Thursday 2.2.2023 12.15-14 o'clock in room C124
Simon Nowak (Universität Bielefeld): Nonlocal gradient potential estimates
Abstract:
We consider nonlocal equations of order larger than one with measure data and present pointwise bounds of the gradient in terms of Riesz potentials. These gradient potential estimates lead to fine regularity results in many commonly used function spaces, in the sense that "passing through potentials" enables us to detect finer scales that are difficult to reach by more traditional methods. The talk is based on joint work with Tuomo Kuusi and Yannick Sire.
-----
Thursday 19.1.2023 12.15-14 o'clock in room C124
Otte Heinävaara (Princeton): Planes in Schatten-3
Abstract:
Schatten-p spaces are non-commutative variants of the usual l_p spaces. While, as normed spaces, they cannot be embedded into L_p spaces, they share various local characteristics with them. For instance, it was proven by Ball, Carlen and Lieb that Schatten-p and L_p have the same moduli of uniform convexity and smoothness.
We prove that in the case of p = 3, in fact any two-dimensional subspace of Schatten_p is linearly isometric to a subspace of L_p. We also conjecture the same is true for any p >= 1.
In the first part of the talk we overview the basic properties of Schatten-p spaces and state and reformulate the main result. In the second part we sketch the proof of the result and, time permitting, discuss and motivate the aforementioned conjecture.
TALKS AUTUMN 2022
The talks in the seminar during the Autumn will mainly take place on-site in lecture room C124 (Exactum).
-----
Tuesday 20.12.2022 14.15-16 o'clock in room C124 (on-site talk; please note exceptional time)
Toni Annala (IAS, Princeton): Topological defects and topologically protected tricolorings
Abstract:
Topological defects are singularities in a continuous order parameter field which can not be removed by a local modification that preserves continuity. They appear in various types of ordered media, such as liquid crystals, Bose--Einstein condensates, and vacuum structures of Yang--Mills theories. Due to the richness of the topological phenomena supported in such systems, the theory of topological defects has attracted persistent scientific fascination since the 70s.
After reviewing background on general topological defects, I will turn my attention to topological vortices, i.e., codimension-one topological defects. I will explain how, under certain homotopical assumptions that are satisfied in many realistic systems, topological vortex configurations admit faithful presentations in terms of colored link diagrams. The most well-known coloring scheme of links is given by tricolorings: each arc of the link diagram is colored by one of three possible colors (red, green, or blue) in such a way that, in each crossing, either all arcs have the same color, or all arcs have a different color. A tricolored link is topologically protected if it cannot be transformed into a disjoint union of unlinked simple loops by a sequence of color-respecting isotopies and color-respecting local cut-and-paste operations. The latter operations are referred to as allowed local surgeries. We use equivariant bordism groups of three-manifolds to construct invariants of colored links that are conserved in allowed local surgeries, and employ the invariant to classify all tricolored links up to local surgeries. The talk is based on joint work with Hermanni Rajamäki, Roberto Zamora Zamora, and Mikko Möttönen.
-----
Thursday 15.12.2022 12.15-14 o'clock in room C124 (on-site talk)
Saara Sarsa (Helsinki): Second order regularity for singular/degenerate parabolic equations
Abstract:
Consider the homogeneous parabolic equation $$ u_t-|Du|^{\gamma}\Delta_p^Nu=0, $$ where $\Delta_p^Nu$ denotes the normalized $p$-Laplacian, $p>1$ and $\gamma>-1$. I will discuss the spatial second order regularity of its viscosity solutions.
The talk is based on joint work with Yawen Feng and Mikko Parviainen.
-----
Thursday 8.12.2022 12.15-14 o'clock in room C124 (on-site talk)
Alexander Shamov (Helsinki): Sobolev globalizations of representations of SL_2(R)
-----
Thursday 1.12.2022 12.15-14 o'clock in room C124 (on-site talk)
Eero Hakavuori (Helsinki): Sub-Riemannian manifolds and their abnormal curves
Abstract:
One of the features distinguishing sub-Riemannian geometry from Riemannian geometry is the existence of so called abnormal curves, which are Lipschitz curves with the property that first order variations are constrained to some lower dimensional space. The abnormal curves are the focus of some important open problems in sub-Riemannian geometry such as the regularity of length-minimizing curves and the Sard problem. In this talk I will give a brief overview of these problems and present some results on how complicated the abnormal curves can be.
-----
Thursday 24.11.2022 12.15-14 o'clock in room C124 (on-site talk)
Toni Ikonen (Helsinki): Coarea inequality for Sobolev functions on nonsmooth 2D manifolds
Abstract:
One of the key tools in the proof of the coarea formula of Sobolev functions on Euclidean spaces is the Eilenberg inequality of Lipschitz functions. A suitable formulation of the inequality remains valid in all metric spaces and Lipschitz functions and the inequality has recently found important applications in the uniformization of nonsmooth 2D manifolds.
In a joint work with B. Esmayli (JYU) and K. Rajala (JYU), we extended the Eilenberg inequality to all nonsmooth 2D manifolds and to the class of all Sobolev functions satisfying a (weak) maximum principle. As an application of the new Eilenberg inequality, we proved that given an element of the class with a p-integrable upper gradient, for p greater or equal to two, then the function is continuous. This has applications to potential analysis on nonsmooth 2D manifolds.
We discuss some of the main ideas of the proofs which involve a surprising amount of planar topology. The talk is based on joint work with Behnam Esmayli and Kai Rajala.
------
Thursday 17.11.2022 12.15-14 o'clock in room C124 (on-site talk)
Pertti Mattila (Helsinki): Parabolic rectifiability
Abstract:
Rectifiable sets are basic concepts of geometric measure theory. In addition to Euclidean spaces, they have been studied in general metric spaces and Carnot groups. I shall discuss definitions and basic properties in the case of parabolic metric.
-----
Thursday 3.11.2022 12.15-14 o'clock in room C124 (on-site talk)
Olli Martio (Helsinki): Capacities from moduli
-----
Thursday 6.10.2022 12.15-14 o'clock in room C124 (on-site talk)
Lauri Hitruhin (Helsinki): Curves with exponentially integrable distortion
Abstract:
I will talk about generalizing mappings with exponentially integrable distortion to a setting where the target space has bigger dimension than the domain of definition. I show that just following the definition of Pankka for Quasiregular curves is not enough for exponentially integrable distortion, as we lose continuity and "Lusin N" condition. The talk is based on joint work with Athanasios Tsantaris.
-----
Thursday 29.9.2022 12.15-14 o'clock in room C124 (on-site talk)
Konrad Kolesko (University of Giessen): Limit theorem for general branching processes.
Abstract:
In 1981, Olle Nerman, in his famous paper, proved a strong law of large numbers for the general branching process N(t), which says that after exponential normalization converges to a certain random variable. In my talk, I will present new results that give an asymptotic extension of N(t) up to Gaussian fluctuations. This result unifies and extends various central limit theorem type results for certain branching processes.
-----
Thursday 22.9.2022 12.15-14 o'clock in room C124 (on-site talk)
Anna Zatorska-Goldstein (University of Warsaw): Potential estimates and local behavior to nonlinear elliptic equations
Abstract:
We provide pointwise estimates in terms of a nonlinear potential of a measure datum, for solutions to elliptic problems involving a second-order operator with Orlicz growth and measurable coefficients. We investigate regularity consequences. The talk is based on joint work with I. Chlebicka and F. Giannetti.
-----
Thursday 15.9.2022 12.15-14 o'clock in room C124 (on-site talk)
Cong Yao (Massey University, New Zealand): Minimization of Exponential Distortions
Abstract:
We consider minimisers of the $p$-exponential conformal energy for homeomorphisms $f:\ID\to\ID$ of finite distortion $\IK(z,f)$ with given boundary data, $f|\partial \ID=f_0$,
\[ E_p(f)=\int_\ID \exp[p\IK(z,f)]\; dz \]
Homeomorphic minimisers always exist. The Euler-Lagrange equations show that the inverses of sufficiently regular minimisers have holomorphic Hopf differential. We take that as a starting point and establish that if $h=f^{-1}:\ID\to\ID$ has holomorphic Hopf differential $\Phi=\exp(p\IK_h)h_w\overline{h_\wbar}$, then $h$, and hence $f$, are diffeomorphisms.
-----
Thursday 8.9.2022 12.15-14 o'clock in room C124 (on-site talk)
Athanasios Tsantaris (Helsinki): Quasiregular dynamics and Zorich maps
Abstract:
One dimensional complex dynamics refers to the study of iteration of holomorphic maps between Riemann surfaces. One of the most well studied families of maps in complex dynamics is the exponential family $E_\lambda(z):=\lambda e^z$, $\lambda\in \mathbb{C}\setminus\{0\}$. Zorich maps are the quasiregular higher dimensional analogues of the exponential map on the plane. In this talk we are going to discuss how one can develop an iteration theory for quasiregular maps and how many of the well known results about the dynamics of the exponential family generalize to the higher dimensional setting of Zorich maps.
------
Thursday 25.8.2022 two on-site talks in room C124
- 12.15-13.00 o'clock Juan Souto (IRMAR, Universite de Rennes):
Effective estimation of the dimension of a manifold from random samples
Abstract:
We give explicit theoretical and heuristic bounds for how big does a data set sampled from a reach-1 submanifold of Euclidean space need to be so that one can estimate its dimension with at least 90% confidence.
- 13.15-14 o'clock Erik Duse (KTH, Stockholm): Generic illposedness of the energy-momentum equations and differential inclusions
TALKS SPRING 2022
The talks in the seminar during the Spring will take place either via the Zoom conference system or on site (in room C124 in Exactum). The length of the talks will be 60-90 minutes. More information about the arrangement of the talks (including the Zoom-links, if applicable) will be sent by e-mail beforehand.
-----
Thursday 9.6.2022 12.15-14 o'clock in room C124 (on-site talk)
David Fisher (Indiana University, Bloomington): Rigidity of low dimensional group actions
Abstract:
I will discuss recent work with Brown and Hurtado proving several cases of Zimmer's conjecture. Zimmer conjectured that certain large groups, namely higher rank lattices, cannot act on compact manifolds of relatively low dimension. My goal is to motivate the conjecture, provide some background and then provide some indication of the larger ideas used in the proofs, which borrow ideas from several different areas of mathematics.
-----
Tuesday 7.6.2022 14.15-16 o'clock in room C124 (on-site talk)
Ilmari Kangasniemi (Syracuse University): Fibers of monotone mappings of finite distortion
Abstract:
In two dimensions, uniform limits of homeomorphisms can be characterized by monotonicity: a map f: ℝ² → ℝ² is monotone if the fibers f⁻¹{y} are connected. In higher dimensions, the most crucial condition is instead that f is cellular, i.e. that the fibers f⁻¹{y} are nested intersections of topological balls. We show that if f : Ω → Ω' is a proper, monotone mapping of finite distortion in n dimensions, with |Df| ∈ Lⁿ(Ω) and K ∈ Lᵖ(Ω) where p = n/2 - 1, then f is at least very close to being cellular: more precisely, the fibers f⁻¹{y} of f are nested intersections of rational homology balls. We also show that for n=3, the given exponent p is sharp. Joint work with Jani Onninen.
-----
Thursday 19.5.2022 12.15-14.00 o'clock in room C124 (on-site talk)
Medet Nursultanov (Helsinki): Diffusion on a manifold with localized trap.
Abstract:
Let us consider a Brownian particle confined to a bounded domain. It is sojourning until it hits the certain part of the domain. This part is called as a window/trap if it is open subset of the boundary/interior. We compute the asymptotic expansion of the mean sojuorning time for Brownian particle on Riemannian manifold as window/trap is shrinking. This problem is also called as a "Narrow escape/ capture problem". (Works with Willim Trad, Justin Tzou and Leo Tzou)
-----
Thursday 31.3.2022 12.15-14.00 o'clock in room C124 (on-site talk)
Xavier Ros-Oton (Universitat de Barcelona): The singular set in the Stefan problem
Abstract:
The Stefan problem, dating back to the XIXth century, is probably the most classical and important free boundary problem. The regularity of free boundaries in the Stefan problem was developed in the groundbreaking paper (Caffarelli, Acta Math. 1977). The main result therein establishes that the free boundary is $C^\infty$ in space and time, outside a certain set of singular points.
The fine understanding of singularities is of central importance in a number of areas related to nonlinear PDEs and Geometric Analysis. In particular, a major question in such a context is to establish estimates for the size of the singular set. The goal of this talk is to present some new results in this direction for the Stefan problem. This is a joint work with A. Figalli and J. Serra.
-----
Thursday 24.2.2022 12.15-13.15 o'clock (talk via Zoom-link)
Emiel Lorist (Helsinki): A discrete framework for the interpolation of Banach spaces
Abstract:
In the study of parabolic boundary value problems (BVPs) from a functional analytic viewpoint, interpolation spaces show up as the spaces of initial and boundary values. For the space of initial values this is well-established and connected to the classical real interpolation method. For the space of boundary values this insight is more recent and connected to the so-called Gaussian- and 
-----
Thursday 13.1.2022 12.15-13.15 o'clock (talk via Zoom-link)
Aleksis Koski (Universidad Autónoma de Madrid): Homeomorphic Extension Problems in Geometric Analysis
Abstract:
One of the most fundamental problems in Geometric Analysis is to understand which properties of a boundary map allow for a homeomorphic extension with specific geometric and analytic properties. Classical results such as the Beurling-Ahlfors extension result or the Radó-Kneser-Choquet theorem constitute some of the basic building blocks needed to solve these problems in 2D space. In this talk, we will review the known planar theory of questions such as the Sobolev Jordan-Schönflies problem of extending a boundary map between Jordan domains as a Sobolev homeomorphism. Moreover, we discuss some first approaches in higher dimensions where many of the techniques crucial to the planar theory simply fail. This talk is based on joint work with Stanislav Hencl and Jani Onninen.
TALKS AUTUMN 2021
The talks in the seminar during the Autumn will take place either via the Zoom conference system or on site (in room C124 in Exactum). The length of the talks will be 60-90 minutes. More information about the arrangement of the talks (including Zoom-links) will be sent by e-mail beforehand.
-----
Thursday 9.12.2021 12.15-13.15 o'clock (talk via Zoom-link)
Toni Ikonen (University of Jyväskylä): Two-dimensional metric spheres from gluing hemispheres
Abstract:
We are interested in metric spheres (Z, d) obtained by gluing two hemispheres of the sphere along an orientation-preserving homeomorphism g: S -> S of the equator, where d is the canonical distance that is locally isometric to the sphere off the seam. We investigate under which assumptions on g the (Z, d) is quasiconformally equivalent to the sphere, in the geometric sense.
As a necessary condition, we mention that g must be a welding homeomorphism with conformally removable welding curves. While investigating sufficient conditions, we observed that g is bi-Lipschitz if and only if (Z, d) has a 1-quasiconformal parametrization whose Jacobian is comparable to the Jacobian of a quasiconformal mapping taking the sphere onto itself. When g is a quasisymmetry and g is absolutely continuous, we show that (Z, d) admits a quasiconformal parametrization; such a parametrization does not exist, for example, for welding homeomorphisms corresponding to the von Koch snowflake. We also discuss some relaxations on the quasisymmetry assumption.
The talk is based on the preprint "Two-dimensional metric spheres from gluing hemispheres".
-----
Thursday 2.12.2021 12.15-14 o'clock (on-site seminar in room C124)
Jani Virtanen (University of Reading): Schatten class Hankel operators on the Fock space
Abstract:
A unique feature in the theory of the Fock space is the property that the Hankel operator H_f is compact if and only if H_{\bar f} is compact when f is bounded. This result of Berger and Coburn is not true for Hankel operators on the Bergman space or the Hardy space. In the first part, I will show that the reason for this is the lack of bounded analytic functions on the complex plane as conjectured by Zhu. A natural question then arises as to whether an analogous phenomenon holds true for Hankel operators in the Schatten classes. This was answered in the affirmative for the Hilbert-Schmidt class by Bauer. In the second part, I will present a complete characterization of Schatten class Hankel operators on the Fock space and show how it can be used to deal with the Berger-Coburn phenomenon on the other Schatten classes. The first part is joint work with Raffael Hagger and the second with Zhangjian Hu.
-----
Thursday 25.11.2021 12.15-14 o'clock (on-site seminar in room C124)
Juan-Carlos Felipe-Navarro (University of Helsinki): Null-Lagrangians and calibrations for nonlocal elliptic functionals
Abstract:
This talk will be devoted to introduce a null-Lagrangian and a calibration for nonlocal elliptic functionals in the presence of an extremal field. First, I will review the classical Weierstrass theory of extremal fields from the Calculus of Variations. Next, I will explain how to extend it to the nonlocal setting, where our model functional is the one associated to the fractional Laplacian. Finally, I will show as applications that monotone solutions are minimizers and that minimizers are viscosity solutions.
This is a joint work with Xavier Cabré (ICREA-UPC) and Iñigo U. Erneta (UPC-BGSMath).
-----
Thursday 18.11.2021 12.15-14 o'clock (on-site seminar in room C124)
Tuomas Sahlsten (Aalto University): Spectral geometry of random Riemann surfaces
Abstract:
There have been several recent advances in spectral graph theory such as Friedman’s proof of Alon’s conjecture, local semicircle law on the spectrum, Quantum Unique Ergodicity for eigenvectors of the Laplacian on random graphs, and recent breakthrough of Bordenave-Collins on Friedman’s conjecture. This provides hope for many of the analogous problems on random Riemann surfaces. The systematic study of random surfaces began in the works of Brooks-Makover and Mirzakhani, where they estimated the spectral gap of the Laplacian. After these there have been some striking results on random surfaces that are even stronger that are known for arithmetic surfaces, where usually extra symmetries are available (e.g. Hecke operators). The core idea is that the randomisation will average out pathological surface geometries causing complications in the periodic orbit structure of the geodesic flow that is pivotal e.g. in the trace formula. In this talk I will describe some of the methods such as the trace formula and Mirzakhani’s integration formula, which we applied to the delocalisation of eigenfunctions of the Laplacian on random surfaces with C. Gilmore, E. Le Masson and J. Thomas.
-----
Thursday 4.11.2021 12.15-14 o'clock (this is a hybrid talk: on-site in room C124 as well as via a Zoom-link)
Titus Hilberdink (University of Reading) : The spectrum of certain arithmetical matrices and Beurling primes
Abstract:
In this talk we discuss a class of arithmetical matrices and their truncations. We investigate spectral asymptotics of some particular group of examples, exploiting their tensor product structure. Finally, we point out a connection to the theory of Beurling's generalised prime systems.
-----
Thursday 7.10.2021 12.15-13.15 o'clock (talk via the Zoom conference system)
Yacin Ameur (University of Lund): An explicit charge-charge correlation function at the edge of a two-dimensional Coulomb droplet
Abstract:
-----
Thursday 16.9.2021 12.15-14 o'clock (this is a hybrid talk: on-site in room C124 as well as via the Zoom-link https://helsinki.zoom.us/j/63162315348 )
Håkan Hedenmalm (KTH Stockholm): Soft Riemann-Hilbert problems and planar orthogonal polynomials
Abstract:
It is known that planar orthogonal polynomials with respect to exponentially varying weights can be characterized in terms of a matrix dbar-problem. In recent work with A. Wennman, an asymptotic expansion for the orthogonal polynomials was found. The proof required scaffolding in terms of the construction of an invariant flow. Here, we find a direct approach in terms of a guess for the Cauchy potential in the Riemann-Hilbert problem. This seems to connect with integrable hierarchies but that aspect remains to explore. The solution allows for better control of the error term, and the main underlying equation is solved by finite Neumann series expansion.
TALKS SPRING 2021
The talks in the seminar during the Spring will exceptionally take place via the Zoom conference system. The length of the talks will be about 60 minutes. More information on the Zoom-links will be sent by e-mail before the talks.
-----
Thursday 20.5.2021 12.15-13.15 o'clock
Daniel Meyer (University of Liverpool): Uniformization of quasiconformal trees.
Abstract:
Quasisymmetric maps are generalizations of conformal maps and may be viewed as a global versions of quasiconformal maps. Originally, they were introduced in the context of geometric function theory, but appear now in geometric group theory and analysis on metric spaces among others. The quasisymmetric uniformization problem asks when a given metric space is quasisymmetric to some model space. Here we consider ``quasiconformal trees''. We show that any such tree is quasisymmetrically equivalent to a geodesic tree. Under additional assumptions it is quasisymmetric to the ``continuum self-similar tree''. This is joint work with Mario Bonk.
-----
Thursday 6.5.2021 12.15-13.15 o'clock
Kristian Seip (NTNU Trondheim): Idempotent Fourier multipliers acting contractively on $H^p$ spaces
Abstract:
We describe the idempotent Fourier multipliers that act contractively on $H^p$ spaces of the $d$-dimensional torus $\mathbb{T}^d$ for $d\geq 1$ and $1\leq p \leq \infty$. When $p$ is not an even integer, such multipliers are just restrictions of contractive idempotent multipliers on $L^p$ spaces, which in turn can be described by suitably combining results of Rudin and And\^{o}. When $p=2(n+1)$, with $n$ a positive integer, contractivity depends in an interesting geometric way on $n$, $d$, and the dimension of the set of frequencies associated with the multiplier. Our results allow us to construct a linear operator that is densely defined on $H^p(\mathbb{T}^\infty)$ for every $1 \leq p \leq \infty$ and that extends to a bounded operator if and only if $p=2,4,\ldots,2(n+1)$.
The talk is based on joint work with Ole Fredrik Brevig and Joaquim Ortega-Cerd\`{a}.
-----
Thursday 29.4.2021 12.15-13.15 o'clock
Hans-Olav Tylli (Helsinki): Closed ideals in the quotient algebra of compact-by-approximable operators
Abstract:
-----
Thursday 22.4.2021 12.15-13.15 o'clock
Ville Tengvall (University of Jyväskylä): About the local injectivity of quasiregular mappings with a small inner dilatation
Abstract:
We discuss the local injectivity of quasiregular mappings with an inner dilatation close to two. The talk is motivated by so-called Martio's conjecture which states that every quasiregular mapping in dimension n > 2 with an inner dilatation less than two is a local homeomorphism. The talk is partly based on a joint work with Aapo Kauranen and Rami Luisto.
-----
Thursday 25.3.2021 12.15-13.15 o'clock
Cliff Gilmore (University College Cork): The Dynamics of weighted composition operators on Fock spaces
Abstract:
The study of weighted composition operators acting on spaces of analytic functions has recently developed into an active area of research. In particular, characterisations of the bounded and compact weighted composition operators acting on Fock spaces were identified by, amongst others, Ueki (2007), Le (2014), and Tien and Khoi (2019).
In this talk I will examine some recent results that give explicit descriptions of bounded and compact weighted composition operators acting on Fock spaces. This allows us to prove that Fock spaces do not support supercyclic weighted composition operators. This is joint work with Tom Carroll (University College Cork).
-----
Thursday 18.3.2021 12.15-13.15 o'clock
Erik Duse (KTH Stockholm): Second order scalar elliptic pde:s, Dirac operators and a generalization of the Beltrami equation
Abstract:
------
Thursday 4.3.2021 12.15-13.15 o'clock
Teri Soultanis (Universiteit Nijmegen): p-Weak differentiable structures
Abstract:
Combining the modulus and plan approaches to Sobolev spaces yields a representation of minimal weak upper gradients as maximal directional derivatives along generic curves. In this talk I explain how this result can be used to extract geometric information of the underlying space and define a p-weak differentiable structure (with an associated differential for Sobolev functions), in the spirit of the seminal work of Cheeger. This structure coincides with Cheeger's differentiable structure on PI-spaces, and is also compatible with Gigli's cotangent module. The talk is based on joint work with S. Eriksson-Bique.
------
Thursday 18.2.2021 12.15-13.15 o'clock
Susanna Heikkilä (Helsinki): Signed quasiregular curves.
Abstract:
In this talk, I will discuss a subclass of quasiregular curves, called signed quasiregular curves, which contains holomorphic curves and quasiregular mappings. Signed quasiregular curves satisfy a weak reverse Hölder inequality that implies a growth result of Bonk-Heinonen type. I will present this growth result and the main ideas of its proof.
------
Thursday 11.2.2021 12.15-13.15 o'clock
Benny Avelin (Uppsala University): Approximation of BV functions using neural networks
Abstract:
In this talk, I will focus on a recent result together with Vesa Julin, concerning the approximation of functions of Bounded Variation (BV) using special neural networks on the unit circle. I will present the motivation for studying these special networks, their properties, and hopefully some proofs. Specifically the results we will cover: the closure of the class of neural networks in $L^2$, a uniform approximation result, and a localization result.
------
Thursday 4.2.2021 12.15-13.15 o'clock
Sylvester Eriksson-Bique (University of Oulu): Infinitesimal Splitting with Positive Modulus
Abstract:
In this talk we study the geometric properties that positive modulus causes for a subset of Euclidean space. On the one hand, many spaces with positive modulus families of curves resist embedding into Euclidean spaces. On the other hand, examples of measures on subsets of Euclidean spaces supporting positive modulus often take the form of product measures. Jointly with Guy C. David we found an explanation for this phenomenon: If a doubling measure on a subset of Euclidean space supports positive modulus, then infinitesimally it splits as a product. Thus, either a space does not embed, or it exhibits some form of product structure. This yields a new way to prevent bi-Lipschitz embeddings, which notably does not depend on a Poincare inequality. We present this result, some of its corollaries and the main scheme of the proof.
------
Thursday 28.1.2021 12.15-13.15 o'clock
Sauli Lindberg (Aalto University): Nonlinear versions of the open mapping theorem
Abstract:
I will discuss nonlinear analogues of the classical Banach-Schauder open mapping theorem. A prototypical one is stated informally as follows. Consider a constant-coefficient system of PDEs with scaling symmetries, posed over $\mathbb{R}^n$ or $\mathbb{R}^n \times [0,\infty)$, which is stable under weak-$*$convergence. The solution-to-datum operator is then surjective if and only if it is open at the origin.
I will also briefly discuss applications to scale-invariant nonlinear PDEs such as the Euler equations and the Jacobian equation. The talk is based on joint work with A. Guerra and L. Koch.
-------
Thursday 21.1.2021 12.15-13.15 o'clock
Alexandru Aleman (University of Lund): Factorizations induced by complete Nevanlinna-Pick factors
Abstract:
TALKS AUTUMN 2020
The talks in the seminar during the Autumn will exceptionally take place via the Zoom conference system. The length of the talks will be about 60 minutes. More information on the Zoom-links will be sent by e-mail before the talks.
-------
Thursday 3.12.2020 12.15-13.15 o'clock
Mitja Nedic (Helsinki): Quasi-Herglotz functions that are identically zero in one half-plane
Abstract:
Herglotz functions are holomorphic functions on the cut-plane satisfying a certain symmetry and positivity condition. Quasi-Herglotz functions are then constructed as complex linear combinations of Herglotz functions. In this talk, we will focus on the subclass of quasi-Herglotz functions that are identically zero in one half-plane and present different characterizaiton results and examples. This talk is based on joint work with Annemarie Luger.
-------
Thursday 26.11.2020 12.15-13.15 o'clock
Olli Järviniemi (Helsinki): Pretentious number theory and Halász's theorem
Abstract:
We give an introduction on pretentious number theory, a subfield of number theory focusing on the behavior of multiplicative functions. In particular, we discuss Halász's theorem, a fundamental result on the "long" averages of multiplicative functions.
------
Thursday 19.11.2020 12.15-13.15 o'clock
Istvan Prause (University of Eastern Finland): The genus-zero five-vertex model
Abstract:
The five-vertex model is a probability measure on monotone nonintersecting lattice path configurations on the square lattice where each corner-turn is penalised by a weight. I’ll introduce an inhomogeneous “genus-zero” version of this non-determinantal model and study its limit shape problem. That is, we are interested in the typical shape of configurations for large system size and fixed boundary conditions. The talk is based on joint work with Rick Kenyon.
-------
Thursday 12.11.2020 12.15-13.15 o'clock
Olli Hirviniemi (Helsinki): Complex stretching and quasicircles
Abstract:
We can classify the local behaviour of quasiconformal mappings near a point by considering two different quantities. The stretching behaviour is bounded by the Hölder continuity, and rotation can be at most like some logarithmic spiral. These two can be combined into what we call a complex stretching exponent. After recalling some previously known results, we will consider the problem of finding the bounds for the complex stretching on a line where the situation is more constrained than on a general 1-dimensional set.
-------
Thursday 5.11.2020 12.15-13.15 o'clock
Aleksis Koski (Helsinki): The Sobolev Jordan-Schönflies problem
Abstract:
We will (once again) discuss the problem of extending a given boundary homeomorphism between two Jordan curves as a Sobolev homeomorphism of the plane, dubbed the Sobolev Jordan-Schönflies problem. Besides recalling the basic known results, we will explore some new constructions such as a counterexample for the full range of Sobolev-exponents p and providing positive results for quasidisk targets and those with piecewise smooth boundary.
-------
Thursday 29.10.2020 12.15-13.15 o'clock
Eino Rossi (Helsinki): Visible part of self-affine sets
Abstract:
Consider a compact set in the plane. Given a direction, we can ask what is the dimension of the part of the set that we see, when looking from that given direction. The visibility conjecture states that if the original set has dimension greater than one, then the dimension of the visible part equals one for almost all directions. The conjecture is open in full generality, but it has been confirmed for some special fractal sets. In this talk, I will discuss a new approach to the visibility problem and the application of this method to visible parts of dominated self-affine sets that satisfy a projection condition.
-------
Thursday 22.10.2020 17.15-18.15 o'clock (Please note the exceptional time)
Duncan Dauvergne (Princeton University): The Archimedean limit of random sorting networks
Abstract:
Consider a list of n particles labelled in increasing order. A sorting network is a way of sorting this list into decreasing order by swapping adjacent particles, using as few swaps as possible. Simulations of large-n uniform random sorting networks reveal a surprising and beautiful global structure involving sinusoidal particle trajectories, a semicircle law, and a theorem of Archimedes.
Based on these simulations, Angel, Holroyd, Romik, and Virag made a series of conjectures about the limiting behaviour of sorting networks. In this talk, I will discuss how to use the local structure of random sorting networks to prove these conjectures.
-------
Thursday 15.10.2020 12.15-13.15 o'clock
Raffael Hagger (University of Reading): Boundedness and Compactness of Toeplitz + Hankel Operators
Abstract:
-------
Thursday 10.9.2020 12.15-13.15 o'clock
Akseli Haarala (Helsinki): On the electrostatic Born-Infeld equations and the Lorentz mean curvature operator
Abstract:
In 1930's Born and Infeld proposed a new model of nonlinear electrodynamics. In the electrostatic case the Born-Infeld equations lead to the study of a certain quasilinear, non-uniformly elliptic operator that comes with a natural gradient constraint. The same operator appears also as the mean curvature operator of spacelike surfaces in the Lorentz-Minkowski space, the setting of special relativity. We will explain both of these contexts to motivate the mathematical study of said operator.
Our main focus will be on the regularity of the solutions of the electrostatic Born-Infeld equations. We will talk about some now classical results as well as some recent developments. We hope to give some ideas on the problems and methods involved.
-------
Thursday 3.9.2020 12.15-13.15 o'clock
Pavel Kurasov (Stockholm University): Crystalline measures: quantum graphs, stable polynomials and explicit examples
Abstract:
TALKS SPRING 2020
------
Thursday 12.3.2020 C124 12-14 o'clock
Pekka Pankka (Helsinki): Quasiregular curves
Abstract:
The analytic definition of quasiconformal, and more generally quasiregular, mappings between Riemannian manifolds requires that the domain and range of the map have the same dimension. This equidimensionality presents itself also in the local topological properties on quasiregular mappings such as discreteness and openness. In this talk, I will discuss an extension of quasiregular mappings, called quasiregular curves, for which the range may have higher dimension than the domain.
------
Thursday 5.3.2020 C124 12-14 o'clock (joint with the Mathematical Physics Seminar)
Paul Dario (Tel Aviv University): Homogenization of Helffer-Sjöstrand equations and application to the Villain model.
Abstract:
In this talk, we will study the Villain rotator model in dimension larger than three and prove that, at low temperature, the truncated two-point function of the model decays asymptotically like |x|^{2-d}, with an algebraic rate of convergence. The argument starts from the observation that the asymptotic properties of the Villain model are related to the large-scale behavior of a vector-valued random surface with uniformly elliptic and infinite range potential, following the arguments of Fröhlich, Spencer and Bauerschmidt. We will then see that this behavior can then be studied quantitatively by combining two sets of tools: the Helffer-Sjöstrand PDE, initially introduced by Naddaf and Spencer to identify the scaling limit of the discrete Ginzburg-Landau model, and the techniques of the quantitative theory of stochastic homogenization developed by Armstrong, Kuusi and Mourrat.
-------
Wednesday 4.3.2020 C124 14-16 o'clock (joint Mathematical Physics seminar and Geometric and Functional Analysis seminar)
Istvan Prause (University of Eastern Finland): Integrability of limit shapes
Abstract:
Limit shape formation is a ubiquitous feature of highly correlated statistical mechanical systems. It says that in the macroscopic limit the random system settles into a fixed deterministic limit. These geometric limit shapes often (known or conjectured to) exhibit arctic boundaries, sharp transitions from ordered (frozen) to disordered (liquid) phases. The guiding theme of the talk is to ask how integrability of the model is reflected in the integrability of the limit shape PDE. I'll show that for the dimer model and the isoradial 5-vertex model limit shapes have strikingly simple parametrizations in terms of the underlying conformal coordinate. The talk is based on joint work with Rick Kenyon.
-------
Thursday 27.2.2020 C124 12-14 o'clock
Jonas Tölle (Helsinki): Variability of paths and differential systems with general BV-coefficients
Abstract:
We study existence and regularity of generalized Lebesgue-Stieltjes integrals $$\int_0^t \varphi(X_s)\,dY_s,\quad t\ge 0,$$ where $X$ is a multidimensional Hölder continuous path, $Y$ is a Hölder continuous driving path and $\varphi$ is a function of (locally) bounded variation. We shall give a meaningful definition for the compositions $\varphi(X)$ and prove with the help of harmonic analysis, fractional calculus and certain fine properties of BV-functions that they are sufficiently regular for the above integral to make sense.
The key idea to manage this is a relative and quantitative condition between the coefficient $\varphi$ on the one hand and the path $X$ on the other hand. This condition ensures that the path $X$ spends very little time in regions where the coefficient is particularly irregular, and is made precise and discussed systematically in terms of mutual Riesz energy of the occupation measure of the path $X$ and the gradient measure of the coefficient function $\varphi$, where we shall provide sufficient conditions and examples in terms of upper regularity estimates for Borel measures.
Furthermore, we shall prove a change of variable formula and, given slightly higher regularity, provide a quantitive approximation scheme by Riemann-Stieltjes sums. Under further conditions, we also establish existence, regularity and uniqueness results for Hölder continuous solutions to systems of differential equations determined by integrals of the above type. The talk is based on a joint work with Michael Hinz (Bielefeld University) and Lauri Viitasaari (Aalto University).
-------
Thursday 20.2.2020 C124 12-14 o'clock
Peter Lindqvist (NTNU Trondheim): Old and "new" about the p-Laplace Equation: an overview.
--------
Thursday 13.2.2020 C124 12-14 o'clock
Olli Hirviniemi (Helsinki): On localized regularity properties of planar mappings of finite distortion
Abstract:
Mappings of finite distortion generalize the notion of quasiregular mappings by only requiring that the distortion function is finite almost everywhere. In this talk, I consider planar mappings with exponentially integrable distortion, i.e. mappings f such that exp(K(z,f)) is in L^p for some p. Earlier theorem by Astala, Gill, Rohde and Saksman showed that such f lies in the local Sobolev space W^{1,2}_{loc} if p > 1 but not necessarily if p ≤ 1. We show a weighted version which holds in the borderline case p = 1. With holomorphic interpolation we obtain L^2-integrability for the derivative with the weight being any power of K(z,f)^-1. We also discuss a technique for improving the weight to a logarithmic one. This talk is based on joint work with István Prause and Eero Saksman.
--------
Thursday 6.2.2020 C124 12-14 o'clock
Ilmari Kangasniemi (Helsinki): On the entropy of uniformly quasiregular maps
Abstract:
In this talk, I discuss a joint work with Yusuke Okuyama, Pekka Pankka and Tuomas Sahlsten, where we study the entropy of uniformly quasiregular (UQR) maps. Results from holomorphic dynamics suggest a question of whether the topological entropy h(f) of every UQR map on a closed, connected, oriented Riemannian n-manifold equals log(deg f). Our results give a positive answer when the ambient manifold is not a rational cohomology sphere. The lower bound h(f) ≥ log(deg f) uses measure theoretic entropy, along with recent results on the invariant measure of a UQR map. The upper bound h(f) ≤ log(deg f) is due to a result of Gromov, behind which lies a quantitative Ahlfors-regularity bound for the image of a mapping from n-space to kn-space with quasiregular coordinate functions.
-------
Thursday 30.1.2020 C124 12-14 o'clock
Gohar Aleksanyan (Helsinki): Regularity theory of free boundary problems
Abstract:
In the first part of the talk I shall discuss some well known methods that have been used to obtain regularity both of the solution and of the free boundary for the so called obstacle-type problems. The second part of the talk will be devoted to other types of free boundary problems, where new methods are needed. Following a linearisation technique due to John Andersson, I shall describe an iterative argument which implies the regularity of the free boundary for the biharmonic obstacle problem.
--------
Tuesday 21.1.2020 C124 14-16 o'clock
Ekaterina Mukoseeva (SISSA, Trieste): Minimality of the ball for a model of charged liquid droplets
Abstract: see
TALKS AUTUMN 2019
Thursday 12.12.2019 C124 12-14 o'clock
Aleksis Koski (Jyväskylä University): Further homeomorphic Sobolev extensions
Abstract:
I will discuss the problem of extending a given boundary map between two Jordan domains as a Sobolev homeomorphism between their interiors - a question which is of fundamental nature in Nonlinear Elasticity. This ties in to my previous talk in the GAFA-seminar, but I will start from the basics again with new audience members in mind. The main result to be presented is an extension theorem for the case where the target domain is a quasidisk or John domain.
--------
Thursday 5.12.2019 C124 12-14 o'clock
Jari Taskinen (Helsinki):
Schauder bases and the decay rate of the heat equation
Abstract:
We consider the classical Cauchy problem for the linear heat equation and integrable initial data in the Euclidean space $\mathbb{R}^N$. We show that given a weighted $L^p$-space $L_w^p(\mathbb{R}^N)$ with $1 \leq p < \infty$ and a fast growing weight $w$, there are Schauder bases $(e_n)_{n=1}^\infty$ in $L_w^p(\mathbb{R}^N)$ with the following property: given a positive integer $m $ there exists $n_m > 0$ such that, if the initial data $f$ belongs to the closed linear span of $e_n$ with $n \geq n_m$, then the decay rate of the solution of the heat equation is at least $t^{-m}$. Actually such a basis can be found as a small perturbation of any given Schauder basis of the space. The proof is based on a construction of a basis of $L_w^p(\mathbb{R}^N)$, which annihilates an infinite sequence of bounded functionals.
This is joint work with Jos\’e Bonet (Valencia) and Wolfgang Lusky (Paderborn) published in J. Evol. Equ. 19 (2019).
---------
Thursday 21.11.2019 C124 12-14 o'clock
Kari Astala (Aalto University): Random tilings and non-linear Beltrami equations
Abstract:
In this talk, based on joint work with E. Duse, I. Prause and X. Zhong, we study scaling limits of random tilings and other dimer models. It turns out that the geometry of the limit regions, i.e. the boundaries between the ordered and disordered (or frozen and liquid) domains can be described by a non-linear and degenerate Beltrami equation with curious properties.
---------
Thursday 14.11.2019 C124 12-14 o'clock
Antonio J. Fernandez (University of Bath): On a class of elliptic problems with critical growth in the gradient
Abstract: see
--------
Thursday 7.11.2019 C124 12-14 o'clock
Jean-Baptiste Casteras (Helsinki): Radial solutions to the Keller-Segel equation
Abstract :
In this talk, we will be interested in the Keller-Segel equation. This equation arises when looking for steady states to the Keller-Segel system which describes chemiotaxis phenomena. We will make a radial bifurcation analysis of this equation and describe the asymptotic behavior of the solutions. Joint works with Denis Bonheure, Juraj Földes, Benedetta Noris and Carlos Roman.
--------
Thursday 3.10.2019 C124 12-14 o'clock
Marti Prats (Aalto University): Measuring Triebel-Lizorkin fractional smoothness on domains in terms of first-order differences
Abstract:
---------
Thursday 26.9.2019 C124 12-14 o'clock
Ilkka Holopainen (Helsinki): Asymptotic Plateau problem for prescribed mean curvature hypersurfaces
Abstract:
I will talk on a recent joint paper with Jean-Baptiste Casteras and Jaime Ripoll.
We consider an $n$-dimensional Cartan-Hadamard manifold $N$ that satisfies the so-called strict convexity condition and has strictly negative upper bound for sectional curvatures, $K\le-\alpha^2<0$. Given a suitable subset $L\subset\partial_\infty N$ of the asymptotic boundary of $N$ and a continuous function $H\colon N\to [-H_0,H_0],\ H_0<(n-1)\alpha$, we prove the existence of an open subset $Q\subset N$ of locally finite perimeter whose boundary $M$ has generalized mean curvature $H$ towards $N\setminus Q$ and $\partial_\infty M=L$. By regularity theory, $M$ is a $C^2$-smooth $(n-1)$-dimensional submanifold up to a closed singular set of Hausdorff dimension at most $n-8$. In particular, $M$ is $C^2$-smooth if $n\le 7$. Moreover, if $H\in [-H_0,H_0]$ is constant and $n\le 7$, there are at least two disjoint hypersurfaces $M_1, M_2$ with constant mean curvature $H$ and $\partial_\infty M_i=L,\ i=1,2$. Our results generalize those of Alencar and Rosenberg, Tonegawa, and others.
---------
Thursday 12.9.2019 No seminar (because of the 2nd Helsinki - Saint Petersburg Math Colloquium 11.9-13.9.2019 in Exactum, Kumpula)
------
Tuesday 10.9.2019 C124 14-16 o'clock (joint with the Mathematical Physics seminar)
Benny Avelin (Uppsala Univ.): Neural ODEs as the deep limit of ResNets
Abstract:
In the deep limit, the stochastic gradient descent on a ResNet type deep neural network, where each layer share the same weight matrix, converges to the stochastic gradient descent for a Neural ODE and that the corresponding value/loss functions converge. Our result gives, in the context of minimization by stochastic gradient descent, a theoretical foundation for considering Neural ODEs as the deep limit of ResNets. Our proof is based on certain decay estimates for associated Fokker-Planck equations.
--------
Thursday 5.9.2019 C124 12-14 o'clock
Daniel Faraco (Universidad Autonoma de Madrid): MHD equations
Abstract:
I will discuss weak solutions in magneto hydrodynamics, with more focused in the three dimensional situation. Special emphasis will be made on the 2 form formalism of the problem. This is a joint work with Sauli Lindberg and László Székelyhidi Jr.
---------
Tuesday 27.8.2019 C124 14-16 o'clock
Yulia Meshkova (Chebyshev Laboratory, St. Petersburg State University): On quantitative homogenization of periodic hyperbolic systems
Abstract:
The talk is devoted to homogenization of periodic differential operators. We study the quantitative homogenization for the solutions of the hyperbolic system with rapidly oscillating coefficients. In operator terms, we are interested in approximations of the cosine and sine operators in suitable operator norms. Approximations for the resolvent of the generator of the cosine family have been already obtained by T. A. Suslina. So, we rewrite hyperbolic equation as parabolic system and consider corresponding unitary group. For this group, we adopt the proof of the Trotter-Kato theorem by introduction of some correction term and derive hyperbolic results from elliptic ones.
-------
TALKS SPRING 2019
Thursday 13.6.2019 C124 12-14 o'clock
Sylvester Eriksson-Bique (UCLA): Constructing and Uniformizing Loewner carpets
Abstract:
Carpets are metric spaces that are homeomorphic to the standard Sierpinski carpet. By a theorem of Whyburn, they have a natural topological characterization. Consequently, they arise in many contexts involving dynamics, self similarity or geometric group theory. In these contexts, the Loewner property would have many structural and geometric implications for the space. However, unfortunately, we do not know if the Loewner property would be satisfied, or even could be satisfied, in many of the applications of interest. I will discuss explicit constructions of infinitely many Loewner carpets with arbitrary conformal dimension, and whose snowflake embeddings in the plane are explicit. I will further discuss rigidity results that ensure that even for the same conformal dimension we have infinitely many quasisymmetrically distinct carpets. The first part will present the general setting, construction and results. The latter part will give proofs of some of the parts and introduce the main technical tools used.
-------
Monday 27.5.2019 C124 14-16 o'clock (please note: exceptional time)
Odi Soler (UA Barcelona): Distortion of Sets under Inner Functions
Abstract:
An inner function f is a holomorphic function on the unit disk for which the radial limit exists and has modulus one almost everywhere. This means that it maps the disk onto the disk and the circle onto the circle. It is a classical result known as the Denjoy-Wolff Theorem that such functions have a unique fixed point in the closed unit disk with derivative at most one (in the angular sense when it lies on the boundary).
Another classical result, known as Löwner's Lemma, states that if an inner function fixes the origin, it keeps the Lebesgue measure on the boundary invariant. Moreover, Fernández and Pestana obtained estimates on the distortion of the Hausdorff content of sets on the boundary in this same case. In this talk, we will consider the case in which an inner function fixes no points on the disk, and hence its Denjoy-Wolff fixed point is at the boundary. In particular, we will present a measure on the circle that is almost invariant under the action of such an inner function and an analogue of the Hausdorff content to study sets of dimension less than one. This talk is based on joint work with Matteo Levi and Artur Nicolau.
----
Friday 24.5.2019 C124 14-16 o'clock (joint with the Harmonic Analysis seminar)
Enrico Le Donne (University of Jyväskylä and University of Pisa): Mathematical appearances of sub-Riemannian geometries
Abstract:
Sub-Riemannian geometries are a generalization of Riemannian geometries. Roughly speaking, in order to measure distances in a sub-Riemannian manifold, one is allowed to only measure distances along curves that are tangent to some subspace of the tangent space.
These geometries arise in many areas of pure and applied mathematics (such as algebra, geometry, analysis, mechanics, control theory, mathematical physics, theoretical computer science), as well as in applications (e.g., robotics, vision). This talk introduces sub-Riemannian geometry from the metric viewpoint and focus on a few classical situations in pure mathematics where sub-Riemannian geometries appear. For example, we shall discuss boundaries of rank-one symmetric spaces and asymptotic cones of nilpotent groups. The goal is to present several metric characterizations of sub-Riemannian geometries so to give an explanation of their natural manifestation. We first give a characterization of Carnot groups, which are very special sub-Riemannian geometries. We extend the result to self-similar metric Lie groups (in collaboration with Cowling, Kivioja, Nicolussi Golo, and Ottazzi). We then present some recent results characterizing boundaries of rank-one symmetric spaces (in collaboration with Freeman).
----
Thursday 23.5.2019 C124 12-13 o'clock
Luca Capogna (Worcester Polytechnic Institute): Existence and uniqueness of Green functions for the Q-Laplacian in PI metric measure spaces.
Abstract:
In an ongoing joint project with Mario Bonk and Xiaodan Zhou, we prove existence and uniqueness of Green functions for the Q-Laplacian in Q-Ahlfors regular PI spaces. Our strategy is largely based on early work of Ilkka Holopainen who established the results in the Riemannian case. In our work we deal with weak solutions of the relevant PDE in the context of Cheeger’s framework for differentiability.
---------
Tuesday 21.5.2019 C124 14-16 o'clock
Ville Tengvall (Helsinki): Compactness of the branch set of branched covers and a question of Vuorinen
Abstract:
We study the compactness of the branch set (i.e. the set of points where a mapping is not a local homeomorphism) of branched covers (i.e. continuous, discrete and open mappings) in Euclidean spaces. Also quasiregular mappings and mappings of finite distortion are considered in the talk. The talk is based on the following joint works with Aapo Kauranen and Rami Luisto:
1) Mappings of finite distortion: compactness of the branch set (to appear in J. Anal. Math.). [arxiv:1709.08724]
2) On proper branched coverings and a question of Vuorinen (preprint). [arxiv:1904.12645]
---------
Monday 20.5.2019 C124 12-14 o'clock (note: exceptional time)
Niels Martin Møller (University of Copenhagen): Mean curvature flow and Liouville-type theorems
Abstract:
In the first part we review the basics of mean curvature flow and its important solitons, which are model singularities for the flow, with a view towards minimal surface theory and elliptic PDEs. These solitons have been studied since the first examples were found by Mullins in 1956, and one may consider the more general class of ancient flows, which arise as singularity models by blow-up. Insight from gluing constructions indicate that classifying them as such is not viable, except e.g. under various curvature assumptions.
In the talk's second part, however, without restrictions on curvature, we will show that if one applies certain "forgetful" operations - discard the time coordinate and take the convex hull - then there are only four types of behavior. To show this, we prove a natural new "wedge theorem" for proper ancient flows, which adds to a long story: It is reminiscent of a Liouville theorem, and generalizes our own wedge theorem for self-translaters from 2018 (a main motivating example throughout the talk) that implies the minimal surface case by Hoffman-Meeks (1990) which in turn contains the classical theorems by Omori (1967) and Nitsche (1965).
This is joint work with Francesco Chini (U Copenhagen).
---------
Friday 17.5.2019 C124 14-16 o'clock (joint with the Harmonic Analysis seminar)
Luis Alias (Universidad de Murcia): Trapped submanifolds in de Sitter spacetime
Abstract: See attached
---------
Thursday 16.5.2019 C124 12-13 o'clock
Alexey Karapetyants (Southern Federal University, Rostov-on-Don): A class of Hausdorff - Berezin operators on the unit disc.
Abstract:
We introduce and study the class of Hausdorff-Berezin operators on the unit disc in the Lebesgue p-spaces with Haar measure. We discuss certain algebraic properties of such operators, and also give sufficient, and, in some cases necessary boundedness conditions for such operators. Joint work with Profs. K. Zhu and S. Samko.
-------
Wednesday 15.5.2019 C124 14.15-15.15 o'clock (joint with Mathematical Physics seminar)
David Fisher (Indiana University): Arithmeticity, Superrigidity and Totally Geodesics Submanifolds
Abstract: All compact negatively curved manifolds admit infinitely many closed geodesics. I will discuss a recent result showing that hyperbolic manifolds admitting infinitely many closed totally geodesic submanifolds of codimension one are very special and in fact arithmetic. In fact a slightly more technical version holds for closed totally geodesic submanifolds of any dimension greater than 1. If time permits, I will explain how the proof involves a new superrigidity theorem and results from homogeneous dynamics.
-------
Tuesday 14.5.2019 C124 14-16 o'clock
Alexei Poltoratski (Texas A & M University): Beurling-Malliavin theory II
Abstract: This is a continuation of the previous seminar talk on 2.5.
-------
Thursday 9.5.2019 C124 12-14 o'clock
Jose Andre Rodriguez Migueles (Helsinki): Hyperbolicity of links complements in circle bundles over hyperbolic 2-orbifolds.
Abstract:
Let $L$ be a link in $M$ a circle bundle over a hyperbolic 2-orbifolds, that projects injectively to a filling multicurve of closed geodesics. We prove that the complement of $L$ in $M$ admits a hyperbolic structure of finite volume and give combinatorial bounds of its volume.
------
Tuesday 7.5.2019 C124 14-16 o'clock (joint with the Mathematical Physics seminar)
Marianna Russkikh (University of Geneva): Dimers and embeddings.
Abstract:
One of the main questions in the context of the universality and conformal invariance of a critical 2D lattice model is to find an embedding which geometrically encodes the weights of the model and that admits “nice” discretizations of Laplace and Cauchy-Riemann operators. We establish a correspondence between dimer models on a bipartite graph and circle patterns with the combinatorics of that graph. We describe how to construct a circle pattern embedding of a dimer planar graph using its Kasteleyn weights. This embedding is the generalization of the isoradial embedding and it is closely related to the T-graph embedding.
Based on: “Dimers and Circles” joint with R. Kenyon, W. Lam, S. Ramassamy; and “Holomorphic functions on t-embeddings of planar graphs” joint with D. Chelkak, B. Laslier.
-------
Thursday 2.5.2019 C124 12-14 o'clock
Alexei Poltoratski (Texas A & M University): Beurling-Malliavin theory I
Abstract: I will discuss the history and the proof and some application of the celebrated theorem of Beurling and Malliavin on completeness of exponential functions in $L^2(a,b)$.
-------
Tuesday 30.4.2019 C124 14-15 o'clock (60 min.)
Burak Hatinoglu (Texas A & M University): Mixed Data in Inverse Spectral Problems for the Schroedinger Operators
Abstract: We consider the Schroedinger operator, Lu = -u''+qu on (0,pi) with a potential q in L^1(0,\pi). Borg's theorem says that q can be uniquely recovered from two spectra. By Marchenko, q can be uniquely recovered from spectral measure. After recalling some results from inverse spectral theory of one dimensional Schroedinger operators, we will discuss the following problem: Can q be recovered from support of spectral measure, which is a spectrum, and partial data on another spectrum and the set of point masses of the spectral measure?
--------
Thursday 18.4.2019 C124 12-14 o'clock
Kari Vilonen (Helsinki & University of Melbourne): Geometric Satake equivalence, part 2
Abstract: see below
--------
Tuesday 16.4.2019 C124 14-16 o'clock
Kari Vilonen (Helsinki & University of Melbourne): Geometric Satake equivalence, part 1
Abstract:
The goal of these lectures is to explain the geometric Satake equivalence. It is important in several areas of mathematics. For example, it provides the foundation for the work of Vincent Lafforgue for which he won the Breakthrough Prize.
The geometric Satake equivalence gives a canonical construction of the dual group making use of the geometry of the affine Grassmannian. I will explain the ingredients which go into this construction.
--------
Thursday 11.4.2019 C124 12-14 o'clock
Esko Heinonen (Universidad de Granada): Jenkins-Serrin problem for translating graphs
Abstract:
A translating soliton is a smooth oriented hypersurface S in MxR whose mean curvature satisfies H = <X,N>, where X is a given vector and N denotes the unit normal to the surface S. In the case S is a graph of a function u, u satisfies the so-called translating soliton equation and S will be called a translating graph. These solitons play an important role in the study of the singularities of the mean curvature flow, but recently they have gained also a lot of interest on their own.
On the other hand, the Jenkins-Serrin problem asks for solutions (of certain PDE) to a Dirichlet problem on a domain D such that the boundary data can be also infinite on some parts of the boundary of D. This problem has been considered earlier e.g. for the minimal surfaces but in this talk I will discuss about existence results for the translating soliton equation in Riemannian products MxR. The talk is based on recent joint works with E.S. Gama, J. de Lira and F. Martin.
-------
Thursday 4.4.2019 C124 12-14 o'clock
Otte Heinävaara (Helsinki): Matrix monotone functions on general sets
Abstract:
‘Matrix monotonicity’ is the notion one gets by combining functional calculus and Loewner order on Hermitian matrices. We discuss new ways to interpret and approach Loewner's classic results on matrix monotone functions, and examine ways to generalize these results to general subsets of the real line.
-------
Thursday 28.3.2019 C124 12-14 o'clock
Odi Soler (UA Barcelona): Approximation in the Zygmund Class
Abstract:
Let L* denote the Zygmund Class on a compact support, the unit circle for instance. It is known that the space I(BMO) of functions with BMO derivative (in the distributional sense) is a subspace of L*. In this talk, based on a joint work with A. Nicolau, we give an estimate for the distance of a given function f in L* to the subspace I(BMO). We will do so by means of a discretisation similar to another used previously by J. Garnett and P. Jones to study the space BMO.
--------
Tuesday 26.3.2019 C124 14-16 o'clock (joint with the Mathematical Physics seminar)
Erik Aurell (KTH Stockholm): Quantum heat and path integrals
Abstract:
Quantum fluctuation relations in the style of Kurchan rely on measuring the energy of system before and after a process. Analogously, quantum heat can be defined as the change of energy of a bath, or baths, during a process. I will discuss how the distribution function of this quantity can be computed in a path integral formulation originally developed by Feynman and Vernon for the open system quantum state. It is hence a functional of the system only, the bath or baths having been integrated out.
If time allows I will consider the special case of thermal power of the heat flow through a two-state system (a qubit), interacting with two baths as in the spin-boson problem. I will then discuss the qualitative similarities and differences between when the qubit interacts weakly or strongly with the baths. Most of the material in the talk can be found in the two papers
E Aurell "Characteristic functions of quantum heat with baths at different temperatures", Physical Review E vol 97 p 062117 (2018)
E Aurell and F Montana "Thermal power of heat flow through a qubit" arXiv:1901.05896 (2019)
----------
Thursday 21.3.2019 C124 12-14 o'clock
Henrik Wirzenius (Helsinki): The quotient algebra K(X)/A(X) for Banach spaces X failing the approximation property
Abstract:
Let X be a Banach space. The quotient algebra K(X)/A(X) of compact-by-approximable operators is a non-unital radical Banach algebra, which can only be non-trivial within the class of Banach spaces failing the approximation property.
We will discuss some of the structural properties of the quotient algebra K(X)/A(X) and give various examples of Banach spaces for which the quotient algebra is infinite-dimensional. This is a joint work with Hans-Olav Tylli.
--------
Tuesday 19.3.2019 C124 14-16 o'clock
Wilhelm Schlag (Yale): On the Bourgain-Dyatlov fractal uncertainty principle
Abstract:
I will discuss uncertainty principles, including the Bourgain-Dyatlov breakthrough result from late 2016. Using harmonic analysis of the type which arises in the Beurling-Malliavin theorem, they showed (in a quantitative way) that a function on the line cannot be supported on fractal sets both in the physical and Fourier variables. The higher-dimensional version of this theorem remains open. I will describe some partial progress by Rui Han and myself.
--------
Thursday 14.3.2019 C124 12-14 o'clock
David Bate (Helsinki): A short proof of Cheeger's differentiation theorem.
Abstract:
We give a new proof of Cheeger's generalisation of Rademacher's theorem to doubling metric measure spaces that satisfy a Poincaré inequality. Our approach uses Guth's short proof of the multilinear Kakeya inequality to show that any measure with n independent Alberti representations has Hausdorff dimension at least n.
All relevant definitions will be given during the talk. This is based on joint work with Ilmari Kangasniemi and Tuomas Orponen.
--------
Thursday 28.2.2019 C124 12-14 o'clock
Rami Luisto (Jyväskylä University): Characterization of branched covers with simplicial branch sets
Abstract:
By a branched cover we refer to a continuous, open and discrete mapping, and the set of points where it fails to be locally injective its branch set. By the classical Stoilow Theorem, a branched cover between planar domains is locally equivalent to the winding map and the equivalence is even quasiconformal when the original mapping is quasiregular. In higher dimensions the claim is not true, except for some special cases. Indeed, by the classical theorems of Church-Hemmingsen amd Martio-Rickman-Väisälä, a branched cover between euclidean n-domains is locally equivalent to a winding map when the image of the branch set is an (n-2)-dimensional hyperplane.
In this talk we discuss a recent result, joint with Eden Prywes, showing that in all dimensions the a branched cover is equivalent to a PL-mapping when the image of the branch set is an (n-2)-dimensional simplicial complex. This extends a three-dimensional result of Martio and Srebro.
---------
Friday 15.2.2019 C124 14-15 o'clock (joint with the Harmonic Analysis seminar)
Giovanna Citti (University of Bologna, Italy): Schauder estimates at the boundary in Carnot groups
Abstract: Internal Schauder estimates have been deeply studied in subriemannian setting, while estimates at the boundary are known only in the Heisenberg groups. The proof of these estimates in the Heisenberg setting, due to Jerison, is based on the Fourier transform technique and can not be repeated in general Lie groups. Here we built a Poisson kernel starting from the fundamental solution, from which we deduce the Schauder estimates at non characteristic boundary points.This is a joint work with Baldi and Cupini.
---------
Thursday 14.2.2019 C124 12-14 o'clock
Christoph Bandt (Universität Greifswald): Computer-assisted search and analysis of self-similar sets and tiles
---------
Thursday 7.2.2019 C124 12-14 o'clock
Tuomo Kuusi (Helsinki): Homogenization, linearization and large-scale regularity for nonlinear elliptic equations
Abstract:
I will consider nonlinear, uniformly elliptic equations with variational structure and random, highly oscillating coefficients satisfying a finite range of dependence, and discuss the corresponding homogenization theory. I will recall basic ideas how to get quantitative rates of homogenization for nonlinear uniformly convex problems. After this I will discuss our recent work proving that homogenization and linearization commute in the sense that the linearized equation (linearized around an arbitrary solution) homogenizes to the linearization of the homogenized equation (linearized around the corresponding solution of the homogenized equation). These results lead to a better understanding of differences of solutions to the nonlinear equation. As a consequence, we obtain a large-scale C^{0,1}-type estimate for differences of solutions and improve the regularity of the homogenized Lagrangian by showing that it has the same regularity as the original heterogeneous Lagrangian, up to C^{2,1}.
----------
Tuesday 5.2.2019 C124 14-16 o'clock
Julian Weigt (Aalto University): Almost-Orthogonality of Restricted Haar Functions
Abstract:
We consider the Haar functions h_I on dyadic intervals. We show that if p > 2/3 and E ⊂ [0,1] then the set of all functions h_I*1_E with |I ∩ E| >= p|I| is a Riesz sequence. For p <= 2/3 we provide a counterexample.
-----------
Thursday 31.1.2019 C124 12-14 o'clock
Jani Virtanen (University of Reading): An operator-theoretic approach to Szegö's limit theorems and generalizations
Abstract:
Szegö’s limit theorems and their generalizations have played an important role in many parts of mathematics and mathematical physics since the early 1900s. We focus on the strong Szegö limit theorem and present one of its six main proofs based on operator-theoretic methods à la Widom. While this approach may be the most elegant of them all, it is the Riemann-Hilbert method that provides the most powerful apparatus to deal with generalizations to functions with singularities. Some of these recent generalizations will be discussed in some detail.
--------
Tuesday 29.1.2019 C124 14-16 o'clock
Santeri Miihkinen (Åbo Akademi University): On the Hilbert matrix operator on analytic function spaces
Abstract:
The Hilbert matrix is a classical (one-sided) infinite matrix introduced by Hilbert in 1900's. Historically, its properties have been studied in the sequence spaces $\ell^p$ by Hardy and Riesz. It can also be defined on spaces of analytic functions by its action on their Taylor coefficients and it is one of the central linear operators investigated in operator theory. In recent years, there has been active research on determination of the exact value of its operator norm on different analytic function spaces. We will discuss these results on Hardy and Bergman spaces and our contribution regarding the value of its norm in the Bergman spaces. The talk is partly based on a joint work with Mikael Lindström and Niklas Wikman (Åbo Akademi University).
--------
Tuesday 22.1.2019 C124 14-16 o'clock
Ilmari Kangasniemi (Helsinki): Restrictions for automorphic quasiregular maps and Lattès maps.
Abstract:
In 1975, Martio proved that a k-periodic quasiregular (QR) map f: R^n -> S^n can have finite multiplicity in a period strip only if k=n or k=n-1. We present a generalization of this result to the setting of QR maps automorphic with respect to a discrete group of Euclidean isometries. Additionally, we discuss the application of results of this type to the theory of Lattès-type uniformly quasiregular (UQR) maps on closed manifolds.
--------
Thursday 17.1.2019 C124 12-14 o'clock
Håkan Hedenmalm (KTH, Stockholm): Off-spectral analysis of Bergman kernels
Please note: Håkan Hedenmalm will also give the talk "Planar orthogonal polynomials and boundary universality of random normal matrices" in the Mathematical Physics seminar on Wednesday 16.1.2019 C124 14-16 o'clock.
-----------
TALKS AUTUMN 2018
Thursday 13.12.2018 C124 12-14 o'clock
Olavi Nevanlinna (Aalto University): Solvability complexity index - can the spectrum of a bounded operator be computed?
Abstract:
In the talk a new hierarchy on computing is outlined and applied to computation of the spectrum and essential spectrum of bounded operators in separable Hilbert spaces. The interest is in general to be able to classify the “difficulty” of solving problems which in (Turing sense) are non-computable.
Key words: algorithm is in the usual meaning, while a “tower of algorithms” is used in a similar way as in Doyle, McMullen (Acta Math -89). If the tower has k levels, then the problem which is solved with such a tower is in Delta_{k+1}. For example, the spectra of bounded operators in l_2 are in Delta_4, the spectra of self-adjoint bounded operators in Delta_3 and the spectra of compact operators in Delta_2. These are sharp.
References on this approach can be found in arXiv: 1508.03280 Jonathan Ben-Artzi, Anders C. Hansen, Olavi Nevanlinna, Markus Seidel
---------
Tuesday 11.12.2018 C124 14-16 o'clock
György Pal Geher (University of Reading): Wigner's theorem on quantum mechanical symmetry transformations
Abstract:
Wigner’s theorem is a cornerstone in the mathematical foundations of Quantum Mechanics. It states that if a bijective map on the set of all pure states preserves the transition probability, then there exists either a unitary or an antiunitary operator which induces this map in a natural way. In my talk I will present an elementary proof of this famous result and explain its connection to Quantum Mechanics. Then I will present some of its most recent generalisations.
---------
Friday 7.12.2018 C124 14-16 o'clock (NOTE: exceptional time)
Raffael Hagger (Leibniz Universität Hannover): A Hitchhiker's Guide to Limit Operators (and their applications)
Abstract: see enclosed
---------
Thursday 22.11.2018 C124 12-14 o'clock
Pekka Pankka (Helsinki): Quasiregular extension of cubical Alexander maps
Abstract:
This is a continuation to the talk last week. I will discuss quasiregular extension theorems for cubical Alexander maps and its application to higher dimensional versions of constructions of Rickman (1985) and Heinonen-Rickman (1998) related to wild branching. This is joint work with Jang-Mei Wu.
-------
Tuesday 20.11.2018 C124 14-16 o'clock
Aleksis Koski (Jyväskylä): Sobolev homeomorphic extensions
Abstract:
In the mathematical theory of nonlinear elasticity one typically represents elastic bodies as domains in Euclidean space, and the main object of study are deformations (mappings) between two such bodies. The class of acceptable deformations one considers usually consists of Sobolev homeomorphisms between the respective domains, for example, with some given boundary values. It is hence a fundamental question in this theory to ask whether a given boundary map admits a homeomorphic extension in the Sobolev class or not. We share some recent developements on this subject, including sharp existence results and counterexamples.
------
Thursday 15.11.2018 C124 12-14 o'clock
Pekka Pankka (Helsinki): Deformation of cubical Alexander maps
Abstract:
This talk is a continuation to a series of talks in this seminar in spring 2013 on Rickman's Picard construction. This time I will discuss a higher dimensional version of Rickman's deformation theory for two-dimensional Alexander maps. In this talk, I will focus on simple covers, cubical Alexander maps, and results on normal forms of cubical Alexander maps on shellable cubical complexes. Quasiregular applications of this method will be discussed next Thursday. The talk is independent of the previous talks on the Picard construction and requires no familiarity with the topic. This is joint work with Jang-Mei Wu
------
Tuesday 13.11.2018 C124 14-16 o'clock (joint with Stochastics models seminar)
Eero Saksman (Helsinki): Elementary introduction to probabilistic number theory (part I)
Abstract:
This is a first part of series of talks which aim to give an introduction to basics of probabilistic number theory. The talk is aimed especially for students and no previous knowledge on number theory is assumed (and only a little amount of probability is needed).
------
Tuesday 6.11.2018 C124 14-16 o'clock
Karl Brustad (Aalto University): The dominative p-Laplacian and sublinear elliptic operators
--------
Thursday 1.11.2018 C124 12-14 o'clock
Vesa Julin (Jyväskylä): The Gaussian Isoperimetric Problem for Symmetric Sets
Abstract:
The Gaussian isoperimetric inequality states that among all sets with given Gaussian measure the half-space has the smallest Gaussian surface area. Since the half-space is not symmetric with respect to the origin, a natural question is to restrict the problem among symmetric sets. This problem turns out to be surprisingly difficult. In my talk I will discuss how it is related to probability and to the study of singularities of mean curvature flow, and present our recent result which partially solves the problem. This is a joint work with Marco Barchiesi.
-----
Thursday 25.10.2018 C124 12-14 o'clock
Eino Rossi (Helsinki): On measures that improve $L^q$ dimension under convolution
Abstract:
The $L^q$ dimension of a probability measure $\mu$, denoted by $L(\mu,q)$, is one way of measuring the smoothness of $\mu$. Heuristically, convolution is a smoothing operation, so $L^q$ dimension should increase in convolutions. We give two different general criteria which guarantee that the $L^q$ dimension strictly increases in convolution. Some classes that satisfy one of the criteria are for example Ahlfors regular measures, measures supported on porous sets, and Moran construction measures. Our results hold for any finite $q>1$ and thus we also have corollaries about the improvement of the $L^\infty$ dimension, which is the limit of $L(\mu,q)$ as $q\to \infty$, or equivalently the supremum of the Frostman exponents of $\mu$.
The dimension results follow from discrete results about improvement of $L^q$ norms in a given level, and those results in turn are obtained using Shmerkin's inverse theorem for $L^q$ norms. The talk is based on a collaboration with Pablo Shmerkin.
-----
Thursday 18.10.2018 C124 12-14 o'clock
Ole Brevig (NTNU Trondheim): Sharp norm estimates for composition operators on the Hardy space of Dirichlet series
-----
Thursday 11.10.2018 C124 12-14 o'clock
Eino Rossi (Helsinki): On measures that improve $L^q$ dimension under convolution
Please note: this talk has been shifted to Thursday 25.10.2018 C124 12-14 o'clock (see above)
----
Tuesday 9.10.2018 C124 14-16 o'clock
Matthew Romney (Jyväskylä University): Singular quasisymmetric mappings
Abstract:
Quasisymmetric mappings are homeomorphisms between metric spaces which preserve relative distance between points. In the Euclidean case (dimension at least two), quasisymmetric mappings preserve sets of Lebesgue measure zero. In this talk, we construct examples to show that this result fails for mappings from Euclidean space onto metric spaces without further geometric assumptions. Portions of this work are joint with D. Ntalampekos.
----
Thursday 4.10.2018 C124 12-14 o'clock
Jose Andres Rodriguez Migueles (Helsinki): Geodesics on hyperbolic surfaces and knot complements
Abstract: see
-----
Thursday 27.9.2018 C124 12-14 o'clock
Armando Gutierrez (Aalto University): The metric compactification of L_p spaces
Abstract:
I will present a method that permits to compactify metric spaces in a weak sense. The geometry of a metric space can be better understood by knowing the objects that belong to its metric compactification. In this talk I will show explicit formulas for the elements of the metric compactification of the classical Banach spaces L_p in finite and infinite dimensions.
-----
Tuesday 25.9.2018 C124 14-16 o'clock
Wen Wu (South China University of Technology): Hankel determinant of certain automatic sequences
-----
Thursday 20.9.2018 C124 12-14 o'clock
Teri Soultanis (SISSA/Fribourg): Polylipschitz forms, pull-back of metric currents, and homological boundedness of BLD-elliptic spaces
Abstract:
The push-forward of metric currents by a Lipschitz map is a standard tool in the theory of metric currents. We define a local inverse for a subclass of maps, the pull-back of metric currents by a BLD-map. BLD-maps contain bilipschitz maps. Using mass- and flat-norm estimates for pull-backs of normal currents we prove a nonsmooth analogue of the Bonk-Heinonen cohomological boundedness of quasiregularly elliptic manifolds. Our result is in the setting of homology, BLD-maps and oriented cohomology manifolds.
-----
Tuesday 18.9.2018 C124 14-16 o'clock
Shusen Yan (University of New England, Australia): Elliptic problems involving critical growth in domains with small holes
Abstract:
This talk deals with an elliptic problem involving critical Sobolev exponent in domains with small hole. Uniqueness and symmetry of the solutions will be discussed under an energy constraint for the solutions. Existence of solutions with very large energy will also be discussed.
-----
Thursday 13.9.2018 C124 12-14 o'clock
Jan-Steffen Mueller (Universität des Saarlandes): Higher-Order Variational Problems of Linear Growth in Image Analysis: Regularity Aspects
Abstract:
In Image Analysis, variational methods are widely used for the denoising and inpainting of defective image data. Models of linear growth are particularly well suited for these kinds of applications, since their minimizers do not blur sharp contours. However, this may also lead to the unwanted "staircasing effect", which means that the result displays "blocky" structures. To avoid this, various approaches of higher order were studied over the last years. In my talk, which is based on the results of my doctoral thesis, I will discuss the regularity theory of this class of variational problems.
-----
The inaugural Geometric and Functional Analysis Seminar
Thursday 6.9.2018 C124 12-14 o'clock
Mikhail Sodin (Tel Aviv University): Spectral measures of finitely valued stationary sequences and an approximation problem on the circle
Abstract:
We will discuss a somewhat striking spectral property of finitely valued stationary sequences and a related approximation problem on the unit circle. The talk is based on joint works with A. Borichev, A. Nishry, and B. Weiss (arXiv:1409.2736, arXiv:1701.03407) and on an ongoing work with A. Borichev and A. Kononova.