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| Name |
Title |
Date |
Time |
Room |
| Jonguk Yang (University of Zurich) | Hénon-like RenormalizationAbstract: A 1D smooth map on an interval is unimodal if it maps the interval into itself by folding it once (at the unique critical point). Analogously, a 2D smooth diffeomorphism on a square is Hénon-like if it maps the square into itself by squeezing it along the vertical direction to a thin strip, then bending it into a “C”-shape.
Joint with S. Crovisier, M. Lyubich and E. Pujals, we extended the celebrated renormalization theory of 1D unimodal maps to the 2D setting, so that it can be applied to the study of Hénon-like maps. In this talk, I will give an outline of our main results. This includes renormalization convergence, the uniqueness of the “2D critical point”, and the robustness of the required regularity conditions of the maps (so that they are finite-time checkable). |
Wednesday, 8 May 2024 |
13:00 |
Huxley 340 |
| Tim Austin (University of Warwick) | Notions of entropy in ergodic theory and representation theoryAbstract: Entropy has its origins in thermodynamics and statistical mechanics. It gained mathematical rigour in Shannon's work on the foundations of information theory, and quickly found striking applications to ergodic theory in work of Kolmogorov and Sinai. Many variants and other applications have appeared in pure mathematics since, connecting probability, combinatorics, dynamics and other areas.
I will survey a few recent developments in this story, with an emphasis on some of the basic ideas that they have in common. I will focus largely on (i) Lewis Bowen's "sofic entropy", which helps us to study the dynamics of "large" groups such as free groups, and (ii) a cousin of sofic entropy in the world of unitary representations, which leads to new connections with random matrices.
This talk will be a fairly general survey. I will assume standard background in groups, measure theory and the language of probability, but only a basic awareness of ergodic theory. |
Tuesday, 14 May 2024 |
13:00 |
Huxley 140 |
| Peter Giesl (University of Sussex) | Solving differential inequalities with applications to complete Lyapunov functionsAbstract: Meshfree collocation can be used to solve partial differential equations in a Reproducing Kernel Hilbert space by discretising the problem, which leads to a system of linear equations. In this talk, we seek to solve partial differential inequalities which leads to a quadratic programming problem. We discuss the discretised problem and prove the convergence of solutions of the discretised problem to the original one.
Furthermore, we apply the theory to the computation of complete Lyapunov functions for ODEs. These are functions which characterise the dynamics of the ODE; in particular, they serve to find the connected components of the chain-recurrent set, consisting of attractors and repellers, and to determine their stability. This is joint work with Holger Wendland (Bayreuth), Stefan Suhr (Bochum), Sigurdur Hafstein and Carlos Argaez (Iceland). |
Tuesday, 21 May 2024 |
13:00 |
Huxley 140 |
| Emmanuel Fleurentin (George Mason University and UNC Chapel Hill) | Investigating Most Probable Escape Paths over Periodic Boundaries: a Dynamical Systems ApproachAbstract: Noise-induced tipping (N-tipping) emerges when random fluctuations prompt transitions from one (meta)stable state to another, potentially as a rare event. In this talk, we delineate new techniques for determining Most Probable Escapes Paths (MPEPs) in stochastic differential equations over periodic boundaries. We utilize a dynamical system approach to unravel MPEPs for the intermediate noise regime. We discuss the framework for computing the MPEPs by first looking at intersections of stable and unstable manifolds of invariant sets of a Hamiltonian system derived from the Euler-Lagrange equations of the Freidlin-Wentzell (FW) functional. The Maslov index helps identify which critical points of the FW functional are local minimizers and assists in explaining the effects of the interaction of noise and the deterministic flow. The Onsager-Machlup functional, which is treated as a perturbation of the FW functional, will provide a selection mechanism to pick out the MPEP. We will illustrate our approach and compare our theoretical prediction with Monte Carlo simulations in the Inverted Van der Pol system and a carbon cycle model. |
Tuesday, 18 June 2024 |
13:00 |
Huxley 140 |
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