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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 |
| Carsten Wiuff (University of Copenhagen) | TBAAbstract: TBA |
Tuesday, 14 May 2024 |
14:00 |
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 |
| Juan Patino Echeverria (University of Auckland) | Transitions to wild chaos in a four-dimensional Lorenz-like systemAbstract: Wild chaos is a form of higher-dimensional chaotic dynamics that can only arise in vector fields of dimension at least four. This talk explores wild chaos in a four-dimensional system of differential equations, which is an extension of the classic Lorenz equations. Recently, Gonchenko, Kazakov and Turaev (2021) showed, via the computation of Lyapunov exponents, that this system has a wild chaotic attractor at a particular point in parameter space.
To explain how this wild chaotic attractor arises geometrically, we perform a bifurcation analysis of the system in a two-parameter setting. As a starting point, we continue the one-parameter bifurcation structure of the classic Lorenz equations when the relevant new parameter is “switched on”. We find that the well-known homoclinic explosion point of the Lorenz system unfolds and gives rise to infinite cascades of curves of Shilnikov-type global connections in the four-dimensional system. These connections are formed by the unstable manifold of the origin, which plays an essential role in the emergence of complicated dynamics in the system. We also compute the kneading diagram that encodes how this one-dimensional manifold repeatedly moves around a pair of equilibria. In combination with the direct computation of curves of global bifurcations, the kneading diagram provides insight that helps identify regions where wild chaos may occur. |
Tuesday, 4 June 2024 |
13:00 |
Huxley 140 |
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