Abstracts

Week of February 13, 2022

February 14, 2022
Group Actions and Dynamics Counting and boundary limit theorems for representations of Gromov-hyperbolic groups 4:00pm -
Zoom

Let $\Gamma$ be a Gromov-hyperbolic group and $S$ a finite symmetric generating set. The choice of $S$ determines a metric on $\Gamma$ (namely the graph metric on the associated Cayley graph). Given a representation $\rho: \Gamma \to GL_d(\mathbb R)$, we are interested in obtaining results analogous to random matrix products theory (RMPT) but for the deterministic sequence of spherical averages (with respect to $S$-metric). We will discuss a general law of large numbers and more refined limit theorems such as central limit theorem and large deviations. If time allows, we will also see boundary limit theorems and convergence of interpolated matrix norms along geodesic rays to the standard Brownian motion. The connections with (and results in) the classical RMPT, a result of Lubotzky–Mozes–Raghunathan and a question of Kaimanovich–Kapovich–Schupp will be discussed. Joint work with S. Cantrell.
Geometry, Symmetry and Physics The dual Lagrangian fibration of compact hyper-Kähler manifolds 4:30pm -
Zoom

Abstract: A compact hyper-Kähler manifold is a higher dimensional generalization of a K3 surface. An elliptic fibration of a K3 surface correspondingly generalizes to the so-called Lagrangian fibration of a compact hyper-Kähler manifold. It is known that an elliptic fibration of a K3 surface is always "self-dual" in a certain sense. This turns out to be not the case for higher-dimensional Lagrangian fibrations. In this talk, I will propose a construction for the dual Lagrangian fibration of all currently known examples of compact hyper-Kähler manifolds, and try to justify this construction.
February 15, 2022
Algebra and Number Theory Seminar Eichler-Shimura relations 4:30pm -

The well-known classical Eichler-Shimura relation for modular curves asserts that the Hecke operator $T_p$ is equal, as an algebraic correspondence over the special fiber, to the sum of Frobenius and Verschiebung. Blasius and Rogawski proposed a generalization of this result for Shimura varieties with good reduction at $p$, and conjectured that the Frobenius satisfies a certain Hecke polynomial. I will talk about a recent proof of this conjecture for a large class of Shimura varieties of abelian type, and how this proves semisimplicity of cohomology for some Shimura varieties.
February 16, 2022
Applied Mathematics Multi-Reference Alignment In High Dimensions 12:00pm -
https://yale.zoom.us/j/97458245891

Abstract:  In multi-reference alignment (MRA), one wishes to recover a signal (L-dimensional vector) from circularly shifted and noisy measurements of itself.  This problem has received considerable attention in recent years, being a “toy model” for problems in imaging (e.g., cryo-em). It is known that the sample complexity, namely the number of measurements needed to achieve some prescribed estimation error, displays a qualitatively different behavior with respect to the SNR between a “high-SNR” and a “low-SNR” regime. A seminal result of [Perry el al. 2017] has shown that for “generic” signals, as SNR << 1, the sample complexity scales like n ~ 1/SNR^3 ; whereas when SNR >> 1 it is known to scale like n ~ 1/SNR (Note: 1/SNR is the sample complexity scaling for estimating a signal in only additive noise, *without* circular shifts). These bounds hide a (polynomial) dimensional dependence, which hasn’t been characterized tightly.  Motivated by the high dimension of contemporary imaging datasets, we study MRA in a high-dimensional framework (L -> inf). Our main result unveils a phase-transition behavior with respect to the statistical difficulty of MRA. Let alpha=SNR/log(L) ; then as L -> inf, we identify two SNR regimes:
(i) “High-SNR” : alpha > 2, where the sample complexity of MRA is essentially the same as estimating a signal in only additive noise.
(ii) “Low-SNR” : alpha <= 2, where the sample complexity is much worse – in this regime, MRA is a substantially more difficult problem, in an information-theoretic sense.

This is joint work with Tamir Bendory (TAU) and Or Ordentlich (HUJI).
February 17, 2022
Algebra and Geometry lecture series An informal introduction to categorical actions of groups, Lecture 3 4:00pm -
https://yale.zoom.us/j/92613729337
Analysis Geodesic beams and Weyl remainders 4:15pm -

Abstract:

In this talk we discuss quantitative improvements for Weyl remainders under dynamical assumptions on the geodesic flow. We consider a variety of Weyl type remainders including asymptotics for the eigenvalue counting function as well as for the on and off diagonal spectral projector. These improvements are obtained by combining the geodesic beam approach to understanding eigenfunction concentration together with an appropriate decomposition of the spectral projector into quasimodes for the Laplacian. One striking consequence of these estimates is a quantitatively improved Weyl remainder on all product manifolds. This is joint work with Y. Canzani
February 18, 2022
Friday Mornings Friday Morning Seminar 9:00am -

We discuss topics of common interest in the areas of geometry, probability, and combinatorics.
Graduate Student Seminar 3 implies Chaos 12:00pm -

It is well known that chaos arises from 3-body problems (both in the planetary sense as in Reuben's talk and in the romantic sense). In this talk, we will discuss the (not) more general statement that "3 implies chaos". Topics to be surveyed are the classical example of the logistic function, connections to the Mandelbrot set and the classical results of Li--Yorke and Sharkovsky, giving a streamlined proof of the latter due to Burns-Hasselblatt.
Geometric Analysis and Application Minimal hypersurfaces in a generic 8-dimensional closed manifold 2:00pm -

Abstract:

In the recent decade, the Almgren-Pitts min-max theory has advanced the existence theory of minimal surfaces in a closed Riemannian manifold $(M^{n+1}, g)$. When $2 \leq n+1 \leq 7$, many properties of these minimal hypersurfaces (geodesics), such as areas, Morse indices, multiplicities, and spatial distributions, have been well studied. However, in higher dimensions, singularities may occur in the constructed minimal hypersurfaces. This phenomenon invalidates many techniques helpful in the low dimensions to investigate these geometric objects. In this talk, I will discuss how to overcome the difficulty in a generic 8-dimensional closed manifold, utilizing various deformation arguments. En route to obtaining generic results, we prove the generic regularity of minimal hypersurfaces in dimension 8. This talk is partially based on joint works with Zhihan Wang.