# Geometric and Functional Analysis Seminar

##### Organizers: Kari Astala (Aalto University), 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 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 **Tuesday 14-16 o'clock** and **Thursday 12-14 o'clock** (the main time slot).

## TALKS SPRING 2020

PLEASE NOTE: the seminar has been suspended until further notice because of the university emergency measures

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Thursday 19.3.2020 C124 12-14 o'clock

**Olli Martio** (Helsinki): Comments on modulus theory (CANCELLED)

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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.

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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*.*

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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.

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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).

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Thursday 20.2.2020 C124 12-14 o'clock

**Peter Lindqvist **(NTNU Trondheim): Old and "new" about the p-Laplace Equation: an overview.

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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.

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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.

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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.

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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 abstract_mukoseeva.pdf

## 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.

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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).

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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.

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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 Antonio_J_Fernandez_Abstract.pdf

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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.

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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: Abstract-Prats-Aalto.pdf

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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.

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Thursday 12.9.2019 No seminar (because of the **2nd Helsinki - Saint Petersburg Math Colloquium** 11.9-13.9.2019 in Exactum, Kumpula)

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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.

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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.

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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.

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## 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.

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**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.

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**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).

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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.

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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]

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**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).

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**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 LJAlias_Abstract_Helsinki2019.pdf

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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.

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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.

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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.

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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.

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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.

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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 $L^2(a,b)$.

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Tuesday 30.4.2019 C124 14-15 o'clock (60 min.)

**Burak Hatinoglu** (Texas A & M University): *M*ixed 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?

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Thursday 18.4.2019 C124 12-14 o'clock

**Kari Vilonen** (Helsinki & University of Melbourne): Geometric Satake equivalence, part 2

Abstract: see below

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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.

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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.

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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.

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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.

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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)

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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.

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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.

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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.

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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.

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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.

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Thursday 14.2.2019 C124 12-14 o'clock

**Christoph Bandt** (Universität Greifswald): Computer-assisted search and analysis of self-similar sets and tiles

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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}.

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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.

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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.

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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).

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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.

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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.

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## 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__

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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.

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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 Abstract_Hagger.pdf

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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.

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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.

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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

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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).

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Tuesday 6.11.2018 C124 14-16 o'clock

**Karl Brustad** (Aalto University): The dominative p-Laplacian and sublinear elliptic operators

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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.

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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.

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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

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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)

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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.

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Thursday 4.10.2018 C124 12-14 o'clock

**Jose Andres Rodriguez Migueles** (Helsinki): Geodesics on hyperbolic surfaces and knot complements

Abstract: see JARM_41018.pd

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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.

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Tuesday 25.9.2018 C124 14-16 o'clock

**Wen Wu** (South China University of Technology): Hankel determinant of certain automatic sequences

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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.

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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.

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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.

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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.