Waves on complex structures
In modern physics waves commonly represent a propagation of some form of energy through physical space, but they can occur in many other models like large social networks (facebook, collaboration networks), where waves could be used to model propagation of information. If the physical space or network is very complex (like a fractal or large clustered network), propagating waves of various frequencies through them can help to pick up some information, which would be normally hard to see. For example in large networks waves can be particularly useful in community detection by providing faster algorithms than random walks (Sahai-Speranzon-Banaszuk, 2012).
In the case when the physical space is geometrically large (say, in volume), the properties of wave propagation through the physical space seem to be similar to analogous wave/information propagation models on large graphs. We verified this more precisely in 2017 (ref) with Etienne Le Masson (Cergy). We are continuing in this project led by Le Masson and hope to provide more evidence towards this connection, which we hope to give a new insight to Quantum Chaos.
In signal processing waves can be used to provide a representation of a complicated signal. The Uncertainty Principle roughly states that the frequencies of the waves cannot be bounded at the same time as the actual length (time) of the signal. In our recent work we have been involved in a project led by Semyon Dyatlov (Berkeley), where he introduced the idea of Fractal Uncertainty Principle, which asks about the same question as the Uncertainty principle, but restricts the time-domain of the signal onto a fractal set. This problem is particularly important in understanding wave propagation in unbounded space, where the wave could propagate without bounds (Bourgain-Dyatlov, 2017).
- 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.
- 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.
- Ilmari Kangasniemi, Yusuke Okuyama, Pekka Pankka and Tuomas Sahlsten: Entropy of uniformly quasiregular mappings, in preparation