## Research Topics

#Mathematical physics | #Mathematical biology | #Complex analysis | #Geometric measure theory | #Random systems | #Applications

### Mathematical physics

In mathematical physics Kupiainen and his collaborators have for the past 15 years been involved in various aspects of dynamics: finite dimensional, extended and stochastic. Their approach has been to bring powerful ideas from statistical physics and quantum field theory to the mathematical study of problems in dynamics.

Together with K. Gawedzki, Kupiainen made a breakthrough in the field of **turbulence**. They used ideas from quantum field theory to establish the so called intermittency property in a nontrivial turbulent system, the randomly advected passive scalar. This is a stochastic PDE which shares many properties with the Navier Stokes equations and therefore their result was considered a major development in the field.

In collaboration with J. Bricmont he introduced the **renormalization group**, a method originally devised for the physical problems of divergences in quantum field theory and critical phenomena in statistical physics, as a method for analysis of long time asymptotics of **nonlinear parabolic partial differential equations** that had considerable effect in this field. In a work with Bricmont and Gawedzki he used the renormalization group as a new approach to the classical Kolmogorov-Arnold-Moser theory of **quasiperiodic motion** in mechanical systems (such as the solar system).

Together with various collaborators Kupiainen used expansion methods from statistical mechanics to understand **space time chaos** in systems of coupled maps. Finally, with Bricmont and R. Lefevere he used statistical mechanics ideas to prove ergodicity and exponential mixing property for two-dimensional turbulence, i.e. for the **stochastic Navier-Stokes equation**.

### Mathematical biology

The **mathematical biology** group studies attractors in competitive dynamical systems and in structured populations as well as in the theory of adaptive dynamics.

**Competitive dynamical systems** are of great importance for understanding real ecosystems and they provide a host of mathematically interesting and intrinsically hard problems. Recently, Yi Wang of the Biomathematics group proved (together with J. Jiang) in the affirmative a long standing famous conjecture by Hal Smith (1986) concerning the uniqueness of a global attractor of time-discrete competitive dynamical systems in n dimensions. Presently, the investigation of the topological and geometrical structure of the attractor is a central theme in the theory of competitive dynamical systems.

Modelling of **structured populations** leads to infinite-dimensional dynamical systems. Traditionally, the models have been formulated as hyperbolic PDEs with nonlocal boundary conditions describing the birth process. Ecological interactions between individuals are reflected in very complicated nonlinearities in the equations and, in particular, in the boundary conditions. During the past 20 years, Gyllenberg has developed with his collaborators Diekmann and Thieme a perturbation theory for dual semigroups that is well suited to analyse stability and bifurcations in structured population models.

**Adaptive dynamics** is a mathematical theory that explicitly links population dynamics to long-term evolution driven by mutation and natural selection. It provides methods of both model formulation and model analysis as well as mathematical theorems that relate phenomena on an evolutionary time scale to processes and structures defined in ecological and population dynamical terms. Adaptive dynamics is a new and rapidly developing theory that poses interesting and mathematically challenging problems, and that is being applied by a growing number of researchers to a wide variety of concrete ecological-evolutionary problems. Geritz, Gyllenberg and Kisdi play a pioneering role in the development and application of adaptive dynamics. Geritz is one of the co-founders of the theory and together with J.A.J. Metz he gave a complete classification of the singular points in one-dimensional phenotype spaces. Gyllenberg extended the theory to physiologically structured populations. Kisdi has applied the theory to many concrete ecological-evolutionary problems.

### Complex analysis

In geometric **complex analysis**, extreme phenomena often arise from related dynamical models. Conversely, among chaotic dynamical systems holomorphic dynamics, combined with tools from complex analysis, provide one of the few cases where a deep and quantitative understanding and classification is possible. Therefore the last 20 years have seen a strong interaction between analysis and complex dynamics. For instance, S. Sullivan and A. Douady developed in the early 1980's a quasiconformal deformation theory. These methods, termed presently as quasiconformal surgery, still provide the main new method to study the geometry of Julia sets and related topics in complex dynamics.

On the other hand, ideas from complex dynamics are very crucial e.g. in Astala's solution to the well known Gehring-Reich conjecture, which provides basic optimal distortion properties for planar quasiconformal mappings. A fundamental explanation for the role of quasiconformal methods in complex dynamics was provided by the holomorphic motions. Today the motions of curves offer an exiting and fruitful research direction, with a potential towards understanding central questions in complex analysis, such as the multifractal properties of harmonic measure and integral means of conformal mappings. More generally, geometric nonlinear analysis and the mappings of finite distortion provide strong methods for conformally invariant geometric phenomena, from deterministic to the random setting.

### Geometric measure theory

In **geometric measure theory** a point of focus is the characteristics of measures arising naturally from the theory of dynamical systems. Often the attractor of a dynamical system has fractal features, for example, it contains details at arbitrarily small scales. The group members have been intensively involved in the study of **geometric properties of fractal sets and measures**. The methods introduced by P. Mattila have played a central role in many important developments.

Dynamical systems are being widely investigated using different concepts of **dimensions**. Dimensions are natural parameters to measure geometric size. They are often related by precise formulas to other dynamical parameters, such as pressure, entropy and Lyapunov exponents. One crucial aspect is to understand the behaviour of dimensions under transformations such as projections and under intersections. From a practical point of view, this is extremely useful since in experiments all the information concerning dynamical systems is obtained via measurements and measurements may be interpreted as lower dimensional projections or sections of the original high dimensional data. The techniques in this field combine measure theoretic methods with Fourier analytic ones. They have been introduced to the study of invariant measures on Riemannian manifolds and that of SRB-measures of coupled map lattices by E. and M. Järvenpää.

### Random systems

One of the common themes of the group is **randomness**. Most natural systems have noise as an essential component thereby forcing one to consider dynamical models with randomness present in their parameters. In the mathematical physics of **disordered systems**, models we are interested in include Schrödinger equation with random potential and particle motion in the presence of random scatterers. **Stochastic (partial) differential equations** enter the study of turbulent advection and randomly forced fluid dynamics. They model also nonequilibrium phenomena such as heat conduction.

The **adaptive dynamics** theory of biological evolution contains random phenomena on two different timescales. On the ecological timescale demographic stochasticity plays a role during the initial phase of an invasion event when the number of individuals of a new mutant phenotype is still low. On the evolutionary timescale the process of mutation, i.e., the generation of new mutant phenotypes is a random process. Both sources of randomness affect the direction of evolution in the phenotype space.

A particularly promising direction is the study of random geometrical structures such as the **random fractals** of stochastic Loewner equation (SLE) or random quasiconformal maps of the plane pursued by the mathematical physics and geometric analysis teams.

While the group doesn't include pure probabilists it has **strong expertise in random systems**.

Kupiainen is one of the leading researchers in rigorous statistical mechanics and quantum field theory, fields which have provided important applications and new ideas in probability theory. The work by Bricmont and Kupiainen on random walks in asymmetric random environments solved a major problem in probability as did their work on ergodicity of the stochastic Navier-Stokes equation in two dimensions.

Gyllenberg has a long experience in stochastic processes (renewal processes, semi- Markov proccesses, regenerative processes). He is a member of the steering group of the European Science Foundation scientific program Stochastic Dynamics" (STOCHDYN) and Kupiainen likewise is in the steering group of the ESF program RDSES on random systems.

Likewise, Saksman has expertise in stochastics through his research on simulated annealing and Markov chain Monte Carlo (MCMC) methods and Muratore- Ginanneschi through his work in mathematical finance and stochastic control theory as well as through his work on statistical hydrodynamics.

### Applications

Finally, the group has more **practical** interests. It is pursuing Markov Chain Monte Carlo methods that have become indispensable in **computational statistics**, especially in connection with the Bayesian approach. These methods are increasingly applied in various practical situations, from logistics and industrial problems to different natural sciences. In many cases the recent adaptive MCMC methods have been able to solve the practical problem caused by the strong dependence of the effectiveness of MCMC algorithms on the relation between the unknown target distribution and the proposal distribution. Together with his collaborators Saksman originated the non-Markovian adaptive Monte Carlo methods and gave theoretical basis for their validitation. Today the research and applications of adaptive MCMC is in very active phase.

The group has started a TEKES (state industrial research funding organization) funded project on multiphase flow simulation aiming at concrete **industrial applications** and it has launched a collaboration with researchers at Nokia on the topics of ad hoc **wireless networks** and **data compression** of audio signals.

The TEKES funded project is in collaboration with engineering groups and industrial partners (Wärtsilä, Kemira, Metso). The research project on ad-hoc wireless networks and data compression of audio signals are supported by a grant from Teknologiateollisuuden 100-vuotissäätiö (a private foundation funding research that helps high tech industry to improve its competitiveness).