Isospectral connections, frame flow ergodicity, and polynomial maps between spheres

Speaker: 

Thibault Lefeuvre

Institution: 

Sorbonne Université

Time: 

Thursday, February 29, 2024 - 4:00pm to 4:50pm

Location: 

RH 306

Classifying real polynomial maps between spheres is a challenging problem in real algebraic geometry. Remarkably, this question has found recent applications in two seemingly unrelated fields:

- in spectral theory, it allowed to solve Kac's celebrated isospectral problem (Can one hear the shape of a drum?) for the connection Laplacian.

- in dynamical systems, it allowed to prove ergodicity for a certain class of partially hyperbolic flows (extensions of the geodesic flow on negatively-curved manifolds).

I will explain these problems and how they all connect together. No prerequisite required -- the talk is intended for a broad audience.

Joint work with Mihajlo Cekić.

Dynamics on homogeneous spaces: a quantitative viewpoint

Speaker: 

Amir Mohammadi

Institution: 

UC San Diego

Time: 

Thursday, April 18, 2024 - 4:00pm

Host: 

Location: 

RH 306

Rigidity phenomena in homogeneous spaces have been extensively studied over the past few decades with several striking results and applications. We will give an overview of activities pertaining to the quantitative aspect of the analysis in this context with an emphasis on recent developments.

Fractalization, Quantization, and Revivals in Dispersive Systems

Speaker: 

Peter Olver

Institution: 

University of Minnesota

Time: 

Thursday, February 8, 2024 - 4:00pm to 4:50pm

Location: 

RH 306

Dispersive quantization, also known as the Talbot effect describes the remarkable evolution, through spatially periodic linear dispersion, of rough initial data, producing fractal, non-differentiable profiles at irrational times and, for asymptotically polynomial dispersion relations, quantized structures and revivals at rational times.  Such phenomena have been observed in dispersive waves, optics, and quantum mechanics, and have intriguing connections with number theoretic exponential sums.   I will present recent results on the analysis and numerics for linear and nonlinear dispersive wave models, both integrable and non-integrable, as well as integro-differential equations modeling interface dynamics and Fermi-Pasta-Ulam-Tsingou systems of coupled nonlinear oscillators.

TBA

Speaker: 

Gunther Uhlmann

Institution: 

University of Washington

Time: 

Thursday, March 14, 2024 - 4:00pm to 4:50pm

Location: 

RH 306

Anti-classification Results in Smooth Ergodic Theory

Speaker: 

Marlies Gerber

Institution: 

Indiana University

Time: 

Thursday, January 25, 2024 - 4:00pm to 4:50pm

Host: 

Location: 

306 RH

There are two well-known equivalence relations on the family of measure-preserving ergodic automorphisms (of a given Lebesgue space): isomorphism and Kakutani equivalence. Two such automorphisms are said to be isomorphic if there is a measure-preserving relabeling of the points in the space that changes one automorphism into the other. Kakutani equivalence is a weaker equivalence relation in which we are also allowed to replace the automorphisms by their first return maps to measurable subsets of the space before relabeling the points.

We consider the complexity of the problem of classifying ergodic automorphisms up to isomorphism or up to Kakutani equivalence. For example, are these equivalence relations, considered as subsets of the Cartesian product of the space of ergodic automorphisms with itself, Borel sets? The first breakthroughs were the anti-classification results of Foreman, Rudolph, and Weiss for isomorphism of ergodic automorphisms, and the subsequent anti-classification results of Foreman and Weiss for isomorphism of ergodic automorphisms that are also smooth diffeomorphisms preserving a given smooth measure on a manifold. 

I will describe further anti-classification results for the Kakutani equivalence relation, and for both isomorphism and Kakutani equivalence, when we restrict to smooth diffeomorphisms with additional properties, such as the mixing property or the K property.

Quasi-Critical Points of Toroidal Belyi Maps

Speaker: 

Edray Goins

Institution: 

Pomona College

Time: 

Thursday, November 30, 2023 - 4:00pm to 5:00pm

Host: 

Location: 

RH 306
A Belyi map \( \beta: \mathbb{P}^1(\mathbb{C}) \to \mathbb{P}^1(\mathbb{C}) \) is a rational function with at most three critical values; we may assume these values are \( \{ 0, \, 1, \, \infty \} \).  Replacing \( \mathbb{P}^1 \) with an elliptic curve \( E: \ y^2 = x^3 + A \, x + B \), there is a similar definition of a Belyi map \( \beta: E(\mathbb{C}) \to \mathbb{P}^1(\mathbb{C})\).  Since \( E(\mathbb{C}) \simeq \mathbb T^2(\mathbb {R}) \) is a torus, we call \( (E, \beta) \) a Toroidal Belyi pair. There are many examples of Belyi maps \( \beta: E(\mathbb{C}) \to \mathbb P^1(\mathbb{C}) \) associated to elliptic curves; several can be found online at LMFDB. Given such a Toroidal Belyi map of degree \( N \), the inverse image \( G = \beta^{-1} \bigl( \{ 0, \, 1, \, \infty \} \bigr) \) is a set of \( N \) elements which contains the critical points of the Belyi map. In this project, we investigate when \( G \) is contained in \( E(\mathbb{C})_{\text{tors}} \).
 
This is joint work with Tesfa Asmara (Pomona College), Erik Imathiu-Jones (California Institute of Technology), Maria Maalouf (California State University at Long Beach), Isaac Robinson (Harvard University), and Sharon Sneha Spaulding (University of Connecticut).  This was work done as part of the Pomona Research in Mathematics Experience (NSA H98230-21-1-0015).

Curvature and harmonic analysis on compact manifolds

Speaker: 

Chris Sogge

Institution: 

Johns Hopkins University

Time: 

Thursday, December 7, 2023 - 4:00pm to 4:50pm

Host: 

Location: 

RH 306

We shall explore the role that curvature plays in harmonic analysis on compact manifolds.
We shall focus on estimates that measure the concentration of eigenfunctions.  Using them we are able to affirm the classical Bohr correspondence principle and obtain a new classification of compact space forms in terms of the growth rates of various norms of (approximate) eigenfunctions.

Mean estimation in high dimension

Speaker: 

Shahar Mendelson

Institution: 

Australian National University

Time: 

Thursday, October 5, 2023 - 4:00pm to 5:00pm

Host: 

Location: 

RH 306

Consider an unknown random vector $X$ that takes values in $R^d$. Is it possible to "guess" its mean accurately if the only information one is given consists of $N$ independent copies of $X$? More accurately, given an arbitrary norm on $R^d$, the goal is to find a mean estimation procedure upon receiving a wanted confidence parameter $\delta$ and $N$ independent copies $X_1,\cdots,X_N$ of an unknown random vector $X$ (that has a finite mean $\mu$ and finite covariance) the procedure returns $\hat{\,\,\mu}$ for which the norm of the error $\hat{\,\mu} - \mu$ is as small as possible, with high probability $1-\delta$.

This mean estimation problem has been studied extensively over the years. I will present some of the ideas that have led to its solution. An obvious choice is to set $\hat{\,\mu}$ to be the empirical mean, i.e. the arithmetic mean of the sample vectors $X_i$. Surprisingly, the empirical mean is a terrible option for small confidence parameters $\delta$ -- most notably, when X is "heavy-tailed". What is even more surprising is that one can find an optimal estimation procedure that performs as if the (arbitrary) random vector X were Gaussian. (A joint work with G. Lugosi)

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