Weekly Bulletin
The FIM provides a Newsletter called FIM Weekly Bulletin, which is a selection of the mathematics seminars and lectures taking place at ETH Zurich and at the University of Zurich. It is sent by e-mail every Tuesday during the semester, or can be accessed here on this website at any time.
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| Monday, 11 May | |||
|---|---|---|---|
| Time | Speaker | Title | Location |
| 15:15 - 16:30 |
Beomjun Sohn RWTH Aachen |
Abstract
In this talk, we present a new robustness result for the topological entropy of Hamiltonian diffeomorphisms on closed surfaces. Topological entropy is a fundamental measure of orbital complexity in dynamical systems, capturing chaotic behavior through a single non-negative value. We prove that if a Hamiltonian diffeomorphism has positive entropy, then any perturbation supported in a topological disk of sufficiently small area still has positive entropy. The proof relies on a new braid stability result derived from Floer-theoretic braids of fixed points, which can be related to the area of the support. This talk is based on ongiong joint work with Marcelo Alves and Matthias Meiwes.
Symplectic Geometry SeminarTopological entropy under small area perturbationread_more |
HG G 43 |
| 17:15 - 18:15 |
Stefan Steinerberger UW Seattle |
Abstract
Dimensionality reduction, moving data from very high dimensions to intermediate dimensions, is well-established. It is not too difficult to accurately map n points into ~log(n) dimensions. The problem becomes a lot more difficult if one insists that the high-dimensional data be embedded into 2 dimensions which is what researchers in the biomedical fields need. Moreover, there are algorithms (most prominently tSNE and UMAP) that "semi-successfully" do this: they tend to work (though they also tend to fail on very simple examples). An added difficulty is that it is difficult to assess from the output when they work. I will argue that these algorithms reduce to understanding an attraction-repulsion functional acting on a large number of particles; this viewpoint immediately suggests new ideas and algorithms as well as a few benchmark problems.
ETH-FDS seminar Particle Dynamics and Dimensionality Reductionread_more |
HG F 3 |
| Tuesday, 12 May | |||
|---|---|---|---|
| Time | Speaker | Title | Location |
| 15:15 - 16:15 |
Dr. Massimo Sorella Imperial College London |
Abstract
For the passive vector equation, the fast dynamo conjecture predicts exponential-in-time growth of the L^2 norm of the solution under the Lipschitz flow generated by a vector field, at a rate independent of the resistivity. We prove this conjecture for the pulsed diffusion model with a time-periodic stretch-fold-shear (SFS) vector field. Our approach relies on anisotropic Banach spaces adapted to the underlying flow dynamics. In the zero-diffusivity regime, we establish the existence of a distributional eigenfunction of the time-one solution operator corresponding to a discrete eigenvalue of modulus greater than one, and then treat the resistive term as a perturbation in these spaces.
Analysis SeminarFast dynamo action on the 3-torus for pulsed-diffusionsread_more |
HG G 43 |
| 16:30 - 18:30 |
Dr. Vanessa Piccolo EPFL |
Abstract
<div>Random matrix theory provides a useful framework for studying high-dimensional models arising in machine learning. In this talk, I will discuss conjugate kernel, or random features, matrices of the form YY*, with Y=f(WX), where X represents the data, W represents the weights, and f is a nonlinear activation function applied entrywise. Such matrices arise naturally from the feature representation produced by a two-layer neural network, and their spectrum captures important information about the geometry, correlations, and effective dimension of these features.</div> <div> </div> <div>The aim of the talk is to give an introduction to the spectral behavior of such nonlinear random matrices. I will begin with an overview of some classical examples from random matrix theory, including Wigner matrices and sample covariance matrices, whose limiting spectra are described by the semicircle and Marchenko-Pastur laws. I will then turn to the effect of the nonlinearity. When both W and X have light-tailed entries, the limiting spectral distribution is well understood and coincides with that of a simpler Gaussian-equivalent model. I will also discuss the regime in which the weights are heavy-tailed while the data remain light-tailed. This setting is motivated by empirical observations of heavy-tailed behavior in trained neural networks. In this case, the spectrum no longer follows the light-tailed universality picture: new terms appear in the moment expansion, leading to different limiting behavior and a richer combinatorial structure.</div>
Zurich Graduate ColloquiumWhat are... the spectral properties of conjugate kernel random matrices?read_more |
KO2 F 150 |
| Wednesday, 13 May | |||
|---|---|---|---|
| Time | Speaker | Title | Location |
| 13:30 - 14:30 |
Prof. Dr. Marco Lenci University of Bologna |
Abstract
<p><span style="caret-color: #ffffff; color: #000000; font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none; display: inline !important; float: none;">Exactness and the K-property (a.k.a. K-mixing) are strong ergodic properties whose significance is that a dynamical system loses all information about its initial conditions, as time flows. These properties were the subject of intense investigation in the second half of the last century, before being somewhat superseded by the stronger Bernoulli property, which was at the time proved for a number of popular systems, including many billiards. But the Bernoulli property can only be formulated for dynamical systems preserving a finite measure, while exactness and the K-property work equally well in infinite-measure contexts, which partly explains a revived interest in them.</span><br style="caret-color: #ffffff; color: #ffffff; font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration-line: none; text-decoration-thickness: auto; text-decoration-style: solid;"><br style="caret-color: #ffffff; color: #ffffff; font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration-line: none; text-decoration-thickness: auto; text-decoration-style: solid;"><span style="caret-color: #ffffff; color: #000000; font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none; display: inline !important; float: none;">I will first present two general theorems on the exact and K decomposition of extensions of dynamical systems. One straightforward application is that, in large generality, a recurrent _periodic_ “Lorentz gas” whose finite-measure factor (Sinai billiard) is hyperbolic is K-mixing. Here the expression “Lorentz gas” is intended in a general sense, including virtually all previously studied 2-dimensional periodic Lorentz gases and d-dimensional periodic Lorentz tubes, as well as many d-dimensional periodic Lorentz slabs. (Joint work with Daniele Galli)</span><br style="caret-color: #ffffff; color: #ffffff; font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration-line: none; text-decoration-thickness: auto; text-decoration-style: solid;"><br style="caret-color: #ffffff; color: #ffffff; font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration-line: none; text-decoration-thickness: auto; text-decoration-style: solid;"><span style="caret-color: #ffffff; color: #000000; font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none; display: inline !important; float: none;">Later, if time permits, I will present a K decomposition theorem specifically tailored to 2-dimensional uniformly hyperbolic billiards in infinite measure. Its main application is the K-property for a general class of planar _aperiodic_ Lorentz gases. (Work in progress with Giovanni Canestrari)</span></p>
Ergodic theory and dynamical systems seminarExact and K decomposition theorems, with application to Lorentz gasesread_more |
HG G 19.1 |
| 15:30 - 16:30 |
Daniela Paiva Peñuela Universität Basel |
Abstract
Gizatullin's problem consists of determining which automorphisms of a smooth quartic surface S in P^3 arise from birational transformations of P^3. This problem fits into a more general question: to characterize which automorphisms of a smooth hypersurface in projective space are restrictions of projective transformations.
This question was completely solved by Matsumura and Monsky (1964), and Chang (1978), except for two cases: that of an elliptic curve in P^2 and that of a quartic surface in P^3. For elliptic curves, it is known that although their automorphisms do not come from projective transformations of P^2, they do arise from birational transformations. Therefore, the only open case corresponds to quartic surfaces in P^3, which we refer to as Gizatullin's problem.
In this seminar, I will provide a general background on the theory of K3 surfaces and the birational geometry of P^3, and explain how the interplay between these areas can be exploited to address the problem. In particular, I will present a solution to Gizatullin's problem in certain specific cases.
The results I will present are part of a series of joint works with Carolina Araujo, Ana Quedo, and Sokratis Zikas.
Geometry SeminarAutomorphisms of quartic surfaces and Cremona transformationsread_more |
HG G 43 |
| 16:15 - 18:00 |
Prof. Dr. Yan Guo Brown University |
Abstract
<p>We review recent progress in the PDE study of gravitational collapse in both Newtonian as well as relativistic contexts.</p>
PDE and Mathematical PhysicsGravitational Collapse of Gaseous Starsread_more |
Y27 H 35/36 |
| Thursday, 14 May | |||
|---|---|---|---|
| Time | Speaker | Title | Location |
| 10:15 - 12:00 |
Sylvain Crovisier Université Paris-Saclay |
HG G 43 |
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| Friday, 15 May | |||
|---|---|---|---|
| Time | Speaker | Title | Location |
| 10:15 - 12:00 |
Tom Hutchcroft California Institute of Technology (Caltech) |
HG G 43 |
|