Abstracts

Week of February 5, 2023

February 6, 2023
Group Actions, Geometry and Dynamics Short closed geodesics in higher rank arithmetic locally symmetric spaces 4:00pm -

A well-known conjecture of Margulis predicts the existence of a uniform lower bound on the systole of any irreducible arithmetic locally symmetric space. In joint work with F. Thilmany, we proved that this conjecture is equivalent to a weak version of the Lehmer conjecture, a well-known problem from Diophantine geometry.
In joint work with M. Fraczyk, we recently established a uniform lower bound for simple Lie groups of higher rank conditional on a uniform lower bound on Salem numbers, a much weaker – but still open – problem. I will discuss these results and some tools used in the proofs and present additional results which highlight the structure of the bottom of the length spectrum.
Geometry, Symmetry and Physics Mirror symmetry for Q-Fano 3-folds 4:30pm -
LOM 214

This is a report on work of my graduate student Cristian Rodriguez. A Q-Fano 3-fold is a complex projective variety with mild singularities such that its 1st Chern class is positive. Q-Fano 3-folds with b_2=1 arise as end products of Mori's minimal model program. Thousands of families are expected, whereas there are only 17 in the smooth case. We will describe mirror symmetry for Q-Fano 3-folds in terms of the Strominger-Yau-Zaslow conjecture and Kontsevich's homological mirror symmetry conjecture, building on work of Auroux. The mirror of a Q-Fano 3-fold is a K3 fibration over the affine line such that the total space is log Calabi--Yau and some power of the monodromy at infinity is maximally unipotent. In 95 cases the Q-Fano is realized as a hypersurface in weighted projective space and we describe the mirror K3 fibration explicitly.
February 7, 2023
Geometry & Topology The symplectic structure of the SL_n(R)-Hitchin component 4:15pm -
LOM 206

The SL_n(R)-Hitchin component of a closed surface is a special component in the character variety consisting of homomorphisms from the fundamental group of the surface to the Lie group SL_n(R). It carries a symplectic structure, defined by the Atiyah-Bott-Goldman form. I will provide an explicit computation of this symplectic form in terms of the generalized Fock-Goncharov coordinates of the Hitchin component (associated to a geodesic lamination on the surface).
February 8, 2023
Applied Mathematics A new perturbation theory for statistical problems 1:00pm -
AKW 200

Abstract: Many statistical models for real-world data have the following structure. Let A be a low rank matrix, and E a matrix of the same dimension. The objective is to approximate a parameter f(A) from the noisy data A' = A + E. A represents the ground truth, A' the observable data, and E the noise. In statistical settings, E is taken to be random. While this set-up is simple, it represents an extremely rich environment in which to study problems in data science.

In this talk, I will discuss how spectral perturbation theory is employed to solve problems in statistics and data science. However, classical perturbation bounds, like the Davis-Kahan theorem, are wasteful in the random E setting. This motivates us to look at the problem through the lens of random matrix theory. I will discuss our improved spectral perturbation bounds and applications.
Colloquium The critical exponent: old and new. 4:15pm -
LOM 214
The critical exponent is an important numerical invariant of discrete groups acting on negatively curved Hadamard manifolds, Gromov hyperbolic spaces, and higher-rank symmetric spaces. In this talk, I will focus on discrete groups acting on hyperbolic spaces (i.e., Kleinian groups), which is a family of important examples of these three types of spaces. In particular, I will review the classical result relating the critical exponent to the Hausdorff dimension using the Patterson-Sullivan theory and introduce new results about Kleinian groups with small or large critical exponents. 
February 10, 2023
Friday Morning Seminar Friday Morning Seminar 9:30am -
LOM 215

A relaxed-pace seminar on impromptu subjects related to the interests of the audience.

Everyone is welcome.

The subjects are geometry, probability, combinatorics, dynamics, and more!