Abstracts

Week of November 26, 2023

November 29, 2023
Applied Mathematics A High-Order Close Evaluation Scheme of Laplace and Helmholtz Layer Potentials in 3D 3:00pm -
LOM 214

We present an efficient high-order discretization scheme for the evaluation of Laplace and Helmholtz layer potentials on smooth surfaces in three dimensions. The scheme is panel based and applies an analytical surface to line integral conversion on each panel to evaluate single layer, double layer, and their normal derivatives accurately. A new basis approximation scheme tailed for these layer potential kernels is proposed. Both nearly singular and singular cases are supported via a unified recursive framework. The Laplace part of the scheme is joint work with Shravan Veerapaneni, and the Helmholtz part is joint work with Shidong Jiang.
Colloquium The Amplituhedron and Cluster Algebras 4:00pm -

In this talk we will discuss how two objects of great interest to both mathematicians and physicists are connected.
On one hand, amplituhedra are the image under a linear map of the positive part of the Grassmannian – where all the Pluckers are nonnegative. Introduced by physicists to encode the probability of certain particle interactions – scattering amplitudes – in Quantum Field Theory, they are semialgebraic sets which generalize polytopes inside the Grassmannian.
On the other hand, cluster algebras are a remarkable class of commutative rings with very nice combinatorics introduced by Fomin and Zelevinsky motivated by the study of total positivity. Many nice algebraic varieties are known to have a cluster algebra structure, including the Grassmannian. They also emerged in physics in the context of scattering amplitudes, where they contributed to both conceptual and computational advances.
We will show how Amplituhedra possesses surprisingly rich cluster structures and how they relate to their geometry and combinatorics.
November 30, 2023
Group Actions, Geometry and Dynamics Explicit spectral gap for Schottky subgroups of SL(2,Z). 4:00pm -
KT205

Let F be a family of finite coverings of a hyperbolic surface S. A spectral gap of F is an interval I = [0, epsilon] such that the eigenvalues in I  (counted with multiplicity) of the Laplacian \Delta_S of S and \Delta_X, any X ∈ F, are the same. I will present a joint work with M. Magee where we give a spectral gap for congruence coverings when S is the surface associated to a Schottky subgroup of SL(2, Z) with thick enough limit set. The proof exploits the link between eigenvalues of the Laplacian and zeros of dynamical zeta functions attached to S via the thermodynamic formalism.
Learning seminar on D-modules 4:00pm -
KT 801

This is 11th lecture in the series.
Analysis New improvement to Falconer’s distance set conjecture in higher dimensions 9:00pm -
KT 219

Falconer’s distance set conjecture says that a compact set in $\mathbb{R}^d$ whose Hausdorff dimension larger than $d/2$ must have a distance set of positive measure. The conjecture is still open in all dimensions. In this talk, I’ll discuss some recent progress towards it in dimension three and higher, which involves new techniques from the theory of radial projections and decoupling. This is based on joint works with Xiumin Du, Kevin Ren, and Ruixiang Zhang.
December 1, 2023
Friday Morning Seminar Influence in Mixing Measures 10:00am -
KT 801

Abstract:

The theory of influences in product measures has profound applications in
theoretical computer science, combinatorics, and discrete probability.
This deep theory is intimately connected to functional inequalities and to the Fourier analysis of discrete groups.
Originally, influences of functions were motivated by the study of social choice theory, wherein a Boolean function represents a voting scheme, its inputs represent the votes,
and its output represents the outcome of the elections. Thus, product measures represent a scenario in which the votes of the parties are randomly and independently
distributed, which is often far from the truth in real-life scenarios.

We begin to develop the theory of influences for more general measures under mixing or spectral independence conditions.
More specifically, we prove analogues of the KKL and Talagrand influence theorems for Markov Random Fields on bounded degree graphs
when the Glauber dynamics mix rapidly.
We thus resolve a long standing challenge, stated for example by Kalai and Safra (2005).
We show how some of the original applications of the theory of in terms of voting and coalitions extend to these general dependent measures.
Our results thus shed light both on voting with correlated voters and on the behavior of general functions of Markov Random Fields (also called “spin-systems”) where the Glauber dynamics mixes rapidly.

Based on joint work with Frederic Koehler, Noam Lifshitz and Dor Minzer 
Geometric Analysis and Application Scattering for geometrically constrained Schrödinger operators 2:00pm -
KT 906

Abstract: Classical and quantum scattering theory studies how the nature of the potential energy of a mechanical system affects its longtime dynamics. I will survey some aspects of this vast theory, emphasizing the method of Enss, which has a “geometric” (as opposed to functional-analytic) flavor.  Then, I will describe work with Tal Malinovitch on quantum scattering in Euclidean space off of potentials satisfying certain geometric conditions on their support. The potentials we consider are anisotropic in that they decay at infinity, but only within a collection of cones. For such potentials, we obtain microlocal descriptions of the scattering states, which behave like free waves as time goes to infinity, as well their complement, which consists of states that interact with the potential at long time scales.