Waves on complex structures

Waves provide a useful tool to understand the fine properties of various structures such as manifolds, fractals or graphs. In our recent work we have been studying waves in complex environments and relate the geometric structure to the properties of the wave. A particular interest is to study the properties of waves when they are *chaotic*, that is, like waterwaves during stormy weather near a coast.

Our group has been working on how chaos influences the behaviour of waves arising from the dynamics. In our recent work we have investigated how suitable *nonlinearity* within the dynamics makes the waves very chaotic themselves. This problem is particularly important in Quantum Chaos (Bourgain-Dyatlov, 2017).

We have also been involved in a project led by Etienne Le Masson (Bristol), which relates waves on large graphs to waves on large surfaces. Waves on large graphs are particularly important in applications such as social networks. For example, in community detection they can provide more effective tool than random walks (Sahai-Speranzon-Banaszuk, 2012). With Le Masson we used new ideas of waves on large graphs to provide insight in Quantum Chaos on large surfaces (Le Masson-Sahlsten, 2017).

**Output**

- Thomas Jordan and Tuomas Sahlsten:
*Fourier transforms of Gibbs measures for the Gauss map*, Math. Ann. (2016) Vol 364 (3): 983-1023 - Tuomas Sahlsten and Connor Stevens:
*Fourier decay in nonlinear dynamics*, in preparation - Etienne Le Masson and Tuomas Sahlsten:
*Quantum ergodicity and Benjamini-Schramm convergence of hyperbolic surfaces*, Duke Math. J. 166 (2017), no. 18, 3425-3460. - Etienne Le Masson and Tuomas Sahlsten:
*Equidistribution of Maass cusp forms in the level aspect*, in preparation

Dynamics of affordance learning in cognition

Affordance is a notion in cognitive sciences introduced by Gibson in 1979. The central idea is to describe the possibilities a cognitive agent (human or machine) can do in a given situation. For example, the handles of coffee cup give affordance for holding the coffee cup on your hand. It has been suggested by Gibson that affordances provide foundations for perception in cognition.

Together with Vadim Kulikov (Helsinki) we have been working on mathematical models for learning affordances by realising the process as dynamics in network models with many layers (similar to deep learning). The models build on the recent advancements done on the dynamics of recurrent neural network models based on Hopfield networks and Boltzmann machines.

Output:

- Vadim Kulikov and Tuomas Sahlsten:
*Dynamics of Causality*, in preparation

Thermodynamical formalism in nonsmooth dynamics

When computing entropy or other statistical quantities of chaotic dynamical systems one often has to assume smoothness assumptions, such as \(C^1\). This can be quite restrictive for e.g. uniformly quasiregular dynamics, which provide a natural higher dimensional analogue of complex dynamics. Here the dynamical systems may not even be differentiable and only weak Sobolev regularity exists.

The community working around quasiregular maps (e.g. Rickman 1993) have developed wide variety of methods and tools, which we can use to approach problems in dynamics and recent works of Kangasniemi, Okuyama and Pankka (ref 1, 2, 3) have taken major steps in particular towards ergodic theory of invariant measures for uniformly quasiregular dynamics using Sobolev cohomology theory. In our project with Ilmari Kangasniemi and Pekka Pankka (Helsinki), and Yûsuke Okuyama (Kyoto) we are exploiting these methods to develop thermodynamical formalism in uniformly quasiregular dynamics.

Output:

- Ilmari Kangasniemi, Yusuke Okuyama, Pekka Pankka and Tuomas Sahlsten:
*Entropy of uniformly quasiregular mappings*, in preparation