Abstracts

Week of December 3, 2023

December 6, 2023
Applied Mathematics 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 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 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 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 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.