Abstracts

Week of April 9, 2023

April 10, 2023
Group Actions, Geometry and Dynamics Applications of quantitative disjointness for the trivial action 4:00pm -
LOM206

We will discuss applications of a discrepancy trick, which sometimes can be used to prove quantitative disjointness results. As an illustration, consider the following example. Fix an imaginary quadratic number field and consider a "diagonal" measure on a product of the unit tangent bundles of two modular curves given by the push-forward under two multivariate monomials of the uniform measure on two copies of the class group. If the monomials are independent, assuming GRH for the Dedekind zeta function, this diagonal measure is quantitatively equidistributed in the product in terms of the size of the discriminant of the quadratic field.

This version of the result implies (conditional) bounds on the size of torsion subgroups of the class group, which was proven previously by Ellenberg and Venkatesh. The main dynamical theorem has several applications and, for example, can be used to prove for most dimensions quantitatively the equidistribution of the triples consisting of a rational subspace, the shape of the integer lattice in this subspace, and the shape of the integer lattice in the orthogonal complement.

This is joint work with Menny Aka, Manfred Einsiedler, Philippe Michel, and Andreas Wieser.
Geometry, Symmetry and Physics Perverse sheaves on symmetric products of the plane, Schur algebras and K-theory 4:30pm -
LOM214

In analogy with the (generalized) Springer correspondence relating perverse sheaves on a nilpotent cone to representations of the Weyl group, we consider perverse sheaves on the symmetric product of n copies of the plane C2, constructible with respect to the natural stratification by collision of points. This category is semisimple when the coefficients have characteristic zero, but with positive characteristic coefficients it can be very complicated. We show that this category is equivalent to modules over a convolution algebra given by K-theory of sheaves on the symmetric group, equivariant for the action of Young subgroups on the left and right. Up to Morita equivalence, this algebra has a Schur algebra as a quotient. I will also explain how this algebra arises using the K-theory of Hilbert schemes and a theorem of Bridgeland, King, and Reid. Joint work with Carl Mautner.
April 11, 2023
Geometry & Topology Horocycle flow on the moduli space of translation surfaces 4:15pm -
LOM 206

I will discuss the dynamics of the horocycle flow on a stratum of translation surfaces (which is an invariant subvariety of the bundle Omega M_g of holomorphic one forms over the moduli space of genus g Riemann surfaces). This flow can be defined as the action of upper triangular matrices with eigenvalue 1, acting linearly on flat charts. Work of Ratner on unipotent flows on homogeneous spaces leads to the question of whether the orbit-closures and invariant measures for this action can be meaningfully classified. I will quickly survey both positive and negative results in this direction. The talk will be based on joint work with Bainbridge, Chaika, Smillie, and Ygouf (in various combinations).
April 12, 2023
Applied Mathematics Estimating position-dependent transport properties from molecular dynamics simulations 1:00pm -
AKW 200

The physical properties and structure of materials under nanoscopic confinement can significantly differ from their bulk counterparts. This talk focuses on a class of estimators used to determine the position-dependent diffusivity tensor for a confined fluid. The estimators are suitable for trajectory data obtained from confocal microscopy experiments and molecular dynamics (MD) simulations. We first demonstrate the use of the estimators on trajectories generated from a few toy problems, to highlight some numerical issues associated with them. Specifically, we show that these estimators may lose accuracy near hard boundaries of the simulation domain. To address this issue, we introduce a correction scheme using Diffusion maps. We demonstrate the use of this correction on trajectories obtained from MD simulations of a model system consisting of a pure liquid within a slit pore. The resulting diffusivity profiles yield empirical distributions that are in good agreement with those obtained directly from MD simulations.
Colloquium Self-Similar Blow up Profiles for Fluids via Physics-Informed Neural Networks 4:15pm -
LOM 214

In this talk I will explain a new numerical framework, employing physics-informed neural networks, to find a smooth self-similar solution for different equations in fluid dynamics. The new numerical framework is shown to be both robust and readily adaptable to several situations.
April 13, 2023
Analysis Floquet Hamiltonians - spectrum and dynamics 4:00pm -
LOM 205

We study Schrodinger equations which are periodic in space and time. These models are inspired by recent experimental progress in the study of wave propagation and quantum mechanics under time-periodic forcing. Time-periodic Hamiltonians, however, are not as well understood as their static (autonomous) analogs.

In particular, many discrete models of periodic materials (e.g., of graphene) are known to develop spectral gaps under a time-periodic driving. In PDEs, however, no such gaps are conjectured to form. How do we reconcile these two facts? Using periodic homogenization, we prove that the driven Schrodinger equation has an “effective gap” - a new and physically-relevant relaxation of a spectral gap.

Taking a broader perspective, we ask - how does time-periodic forcing affect a general band structure? We will show that a spectrally-local notion of stability can be formulated and proved, in a way which again corresponds with periodic homogenization theory.
April 14, 2023
Combinatorics Seminar (Joint with the Geometry & Topology Seminar) Skein algebras and Kuperberg webs 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!
Geometric Analysis and Application TBA 2:00pm -
LOM 215

TBA