Published on Department of Mathematics Calendar (https://calendar.math.yale.edu)

Week of December 1, 2023

December 1, 2023
Friday Morning Seminar [3]	Influence in Mixing Measures	Elchanan Mossel - MIT	10:00am - KT 801
Geometric Analysis and Application [4]	Scattering for geometrically constrained Schrödinger operators	Adam Black - Yale University	2:00pm - KT 906

December 6, 2023
Applied Mathematics [5]	Neural encoding manifolds	Luciano Dyballa - Yale University	3:00pm - LOM 214
Colloquium [6]	Using logic to study homeomorphism groups	Thomas Koberda - University of Virginia	4:00pm - KT 219

December 7, 2023
	Learning seminar on D-modules	Do Kien Hoang - Yale	3:50pm - KT 801
Group Actions, Geometry and Dynamics [7]	Effective versions of Ratner’s equidistribution theorem	Lei Yang - IAS	4:00pm - KT205

December 8, 2023
Friday Morning Seminar [3]	Multiplicative and additive determinantal inequalities for totally nonnegative matrices	Daniel Soskin - Lehigh University	10:00am - KT 801
Geometric Analysis and Application [4]	Harmonic branched coverings and uniformization of CAT(k) spheres	Christine Breiner - Brown University	2:00pm - KT 906

December 13, 2023
Applied Mathematics [5]	Inverse obstacle scattering with trapping and dissipative media	Travis Askham - NJIT	3:00pm - LOM 214

Abstracts

Week of December 1, 2023

December 1, 2023
Friday Morning Seminar [3]	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 [4]	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.

December 6, 2023
Applied Mathematics [5]	Neural encoding manifolds	3:00pm - LOM 214	Integrating the data from large numbers of neurons responding to an ensemble of stimuli in behaving animals is one of the key challenges facing computational neuroscience. We introduce neural encoding manifolds, a construct in which each point is a neuron and nearby neurons respond similarly in time to similar stimuli. The advantages of this unsupervised machine learning approach will be demonstrated in two very different neural systems. First, in the mouse, naturalistic stimuli drove both the retina and visual cortex. Encoding manifolds were developed for each, and trajectories across the manifold reveal how stimulus selectivity is organized differently in these two populations. Surprisingly, convolutional neural networks are even more topologically extreme. Second, when applied to the nematode C. elegans, our encoding manifold organizes neurons into neighborhoods that relate to specific functional roles. These inform whether the available anatomical connectomes are sufficient to explain behavior, and suggest direct combinations of neurons that could comprise behavioral modules.
Colloquium [6]	Using logic to study homeomorphism groups	4:00pm - KT 219	I will describe some recent results on the first order rigidity of homeomorphism groups of compact manifolds, and their applications to dynamics of group actions on manifolds. I will also describe how to find “syntactic” invariants of manifolds, and how these can be used to give a conjectural model-theoretic characterization of the genus of a surface.

December 7, 2023
	Learning seminar on D-modules	3:50pm - KT 801	This is the last lecture in the series.
Group Actions, Geometry and Dynamics [7]	Effective versions of Ratner’s equidistribution theorem	4:00pm - KT205	I will talk about recent progress in the study of quantitative equidistribution of unipotent orbits in homogeneous spaces, namely, effective versions of Ratner’s equidistribution theorem. In particular, I will explain the main idea of my proof for unipotent orbits in SL(3,R)/SL(3,Z). The proof combines new ideas from harmonic analysis and incidence geometry. In particular, the quantitative behavior of unipotent orbits is closely related to a Kakeya model.

December 8, 2023
Friday Morning Seminar [3]	Multiplicative and additive determinantal inequalities for totally nonnegative matrices	10:00am - KT 801	Totally positive matrices are matrices in which each minor is positive. Lusztig extended the notion to reductive Lie groups. He also proved that specialization of elements of the dual canonical basis in representation theory of quantum groups at q=1 are totally non-negative polynomials. Thus, it is important to investigate classes of functions on matrices that are positive on totally positive matrices. I will discuss several sources of such functions. One has to do with multiplicative determinantal inequalities (joint work with M.Gekhtman). Another deals with certain partial sums of Plucker relations (joint work with P.K.Vishwakarma). The third source deals with majorizing monotonicity of symmetrized Fischer’s products which are a natural generalization of Hadamard-Fischer inequalities. Majorizing monotonicity of symmetrized Fischer’s products was already known for hermitian positive semidefinite case which brings additional motivation to verify if they hold for totally positive matrices as well (joint work with M.Skandera). The main tools we employed are network parametrization, Temperley-Lieb and monomial trace immanants.
Geometric Analysis and Application [4]	Harmonic branched coverings and uniformization of CAT(k) spheres	2:00pm - KT 906	Abstract: Consider a metric space (S,d) with an upper curvature bound in the sense of Alexandrov (i.e. via triangle comparison). We show that if (S,d) is homeomorphically equivalent to the 2-sphere, then it is conformally equivalent to the 2-sphere. The method of proof is through harmonic maps, and we show that the conformal equivalence is achieved by an almost conformal harmonic map. The proof relies on the analysis of the local behavior of harmonic maps between surfaces, and the key step is to show that an almost conformal harmonic map from a compact Riemann surface onto a surface with an upper curvature bound is a branched covering. This work is joint with Chikako Mese.

December 13, 2023
Applied Mathematics [5]	Inverse obstacle scattering with trapping and dissipative media	3:00pm - LOM 214	Inverse obstacle scattering is the recovery of the boundary of a homogeneous object given scattering measurements far from the object. This can be contrasted with inverse medium scattering where some continuously varying (and compact) material parameter is to be recovered. The obstacle recovery problem has the benefit that the governing PDEs are generally homogeneous boundary value problems; standard boundary integral representations reduce the dimension of the PDE discretization by one. However, the obstacle setting introduces some difficulties. The space of solutions is non-convex and the natural regularizations of the problem are non-linear. In this talk, we’ll review some numerical methods for the obstacle problem that utilize ideas originally applied to the medium problem by Yu Chen. Then we’ll present some numerical methods and results of two recent papers that concern the obstacle scattering problem for trapping and dissipative media, respectively, and discuss some remaining challenges. This work is in collaboration with Carlos Borges, Jeremy Hoskins, and Manas Rachh.

Links
[1] https://calendar.math.yale.edu/list/calendar/grid/week/2023-W47 [2] https://calendar.math.yale.edu/list/calendar/grid/week/2023-W49 [3] https://calendar.math.yale.edu/seminars/friday-morning-seminar [4] https://calendar.math.yale.edu/seminars/geometric-analysis-and-application [5] https://calendar.math.yale.edu/seminars/applied-mathematics [6] https://calendar.math.yale.edu/seminars/colloquium [7] https://calendar.math.yale.edu/seminars/group-actions-geometry-and-dynamics [8] https://calendar.math.yale.edu/list/calendar/grid/week/abstract/2023-W47 [9] https://calendar.math.yale.edu/list/calendar/grid/week/abstract/2023-W49