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Mathematics Colloquium

The colloquium runs Thursday afternoons at 3:30PM in Phillips Hall 332 with a tea preceding at 3:00 PM in Phillips Hall 330. The talks are aimed at general audiences with different backgrounds.
In the year 2022-2023, most talks will be held in-person, and some talks may be on zoom with a virtual tea time preceding at 3:15pm.

Colloquium committee (2022 fall): Jiuzu Hong, David Rose, Andrey Smirnov
Colloquium committee (2023 spring): Jiuzu Hong, Casey Rodriguez, Andrey Smirnov
Zoom meeting ID: 985 5728 2488, Password: The first six digits of pi

2023 Spring

Date Speaker Affiliation Mode Title
May 3 Jeff Cheeger New York University In-person Quantitative differentiation
April 27 Colleen Robles Duke In-person Hodge theoretic generalizations of Satake-Baily-Borel
April 20 Marcelo Disconzi Vanderbilt In-person General-relativistic viscous fluids
March 30 Rebecca Goldin George Mason In-person On positivity for flag manifolds and Hessenberg spaces
Feb 9 Weiqiang Wang UVA In-person Quantum Schur dualities ABC
Feb 2 Jingfang Huang UNC In-person Mathematical and Statistical Analysis of Compressible Data on Compressive Network
Jan 26 Pavel Etingof MIT In-person Lie theory in tensor categories with applications to modular representation theory

Jeff Cheeger
Abstract: Quantitative differentiation deals with the behavior of function or geometric object at all locations and
on all scales. We will give a relatively short elementary proof of the main theorem in the simplest case,
f : [0, 1] → R, with say \int_{0}^{1} | f ′|2 ≤ 1. By a location and scale, we will mean a dyadic subinterval. Since for 0
each n = 1, 2, . . . there are 2n such subintervals, each of length 2^{−n}, it follows that the sum of the lengths of all dyadic intervals is infinite. Quantitative differentiation states that for all ε > 0, the sum of the lengths of those intervals on which f fails to be ε-linear is ≤ 2ε^{−2}. Here, f is ε-linear on an interval of length r if it differs from the best linear approximation by at most εr. Instances of the basic quantitative differentiation idea have appeared in several relatively advanced contexts going back to work of Dorronsoro in 1985. However, the above simplest case seems not to be widely known. We will also describe a 2-dimensional example from riemannian geometry which typifies applications to geometric analysis over the last 10 years. These concern partial regularity theory for various nonlinear elliptic and parabolic geometric pde, as well as a bi-Lipschitz nonembedding theorem for the Heisenberg group in the target L1. They are joint with (subsets of) Toby Colding, Bruce Kleiner, Assaf Naor, Aaron Naber, Daniele Valtorta, Bob Haslhofer, and Wenshuai Jiang.

Collen Robbles
Abstract: here are many ways to compactify a locally hermitian symmetric space. Distinguished amongst this is a minimal compactification due to Satake. I will give a general audience introduction to a program to construct Hodge theoretically meaningful generalizations of this minimal construction. This is joint work with Mark Green, Phillip Griffiths and Radu Laza.

Marcelo Disconzi
Abstract: In this talk, we will review some recent developments at the intersection of mathematics and physics regarding theories of relativistic fluids with viscosity. The discovery of the quark-gluon plasma that forms in heavy-ion collision experiments provides a unique opportunity to study properties of matter under extreme conditions, as the quark-gluon plasma is the hottest, smallest, and densest fluid known to humanity. Studying the quark-gluon plasma also provides a window into the earliest moments of the universe, since microseconds after the Big Bang the universe was filled with matter in the form of the quark-gluon plasma. For more than two decades, the community has intensely studied the quark-gluon plasma with the help of a rich interaction between experiments, theory, phenomenology, and numerical simulations. From these investigations, a coherent picture has emerged, indicating that the quark-gluon plasma behaves essentially like a relativistic liquid with viscosity. More recently, state-of-the-art numerical simulations strongly suggested that viscous and dissipative effects can also have non-negligible effects on gravitational waves produced by binary neutron star mergers. But despite the importance of viscous effects for the study of such systems, a robust and mathematically sound theory of relativistic fluids with viscosity is still lacking. This is due, in part, to difficulties in preserving causality upon the inclusion of viscous and dissipative effects into theories of relativistic fluids. In this talk, we will survey the history of the problem and report on a new approach to relativistic viscous fluids that addresses these issues.

Weiqiang Wang
Abstract: The classical Schur duality admits a q-deformation due to Jimbo, which is a duality between a quantum group and Hecke algebra of type A. A new quantum Schur duality between an i-quantum group (arising from quantum symmetric pairs) and Hecke algebra of type B was formulated by Huanchen Bao and myself. In this talk, I will explain these dualities (with geometric incarnation) and applications to super Kazhdan-Lusztig theories of type ABC.

Rebecca Goldin
Abstract: “Positivity” is a phenomenon involving the intersection of subvarieties in the presence of enough structure to ensure that intersection points are positively oriented, among other generalizations. It has a direct combinatorial interpretation, which has engaged mathematicians for at least 150 years, in one form or another. Such phenomena occur with frequency in the context of complex homogenous spaces such as flag varieties. In this talk, I will introduce and define positivity in a limited context, including some generalizations to the (torus) equivariant cohomology ring* of G/B, where G is a complex reductive group and B is a Borel subgroup. I will then turn to Hessenberg varieties, a special class of subvarietes of G/B. In some cases, these subvarieties also exhibit positivity properties, and have combinatorial formulas describing it. In other cases, positivity is unknown but lots of hints exist, leading to conjectural behavior. I will close with several open problems. The work I present consists of outcomes from various joint projects with L. Mihalcea, R. Singh, B. Gorbutt, M. Precup, and J. Tymoczko.

* (Almost) everything you need to know about equivariant cohomology will be presented during the talk.

Jingfang Huang, Shahar Kovalsky, and Jeremy Marzuola
Abstract: We present an overview of the FRG project on the “Mathematical and Statistical Analysis of Compressible Data on Compressive Networks”. Compressible features of data include the low-rank, low-dimension, sparsity, and features from the classification/categorization/clustering process. Discovering such compressible features is a major challenge in data analysis, which we will address using hierarchical decompositions derived from spectral, statistical, and algebraic geometric analysis of data. We also study how to construct optimally defined compressive networks, specifically tailored to the discovered compressible features, to enable an accurate and efficient extraction and manipulation of sparse representations in complex and high dimensional systems in an inherently interpretable manner. Sample ongoing projects include the accurate and efficient representation of layer potential using algebraic variety, spectral flow and fast computation of the eigensystems, frequency domain based statistical analysis, fast algorithms for high dimensional truncated multivariate Gaussian expectations, and recursive tree algorithms for orthogonal matrix generation and matrix-vector multiplications in rigid body Brownian dynamics simulations.

Pavel Etingof

2022 Fall

Date Speaker Affiliation Mode Title
Nov 17 Andras Szenes University of Geneva In-person Polytopes, toric varieties, and the intersection homology of moduli spaces of semistable bundles
Nov 10 Daniel Tataru Berkeley In-person Long time solutions for one dimensional dispersive flows
Oct 27 Yannick Sire Johns Hopkins In-person Geometric variational problems: regularity vs singularity formation
Sep 29 Sharon Lubkin NC State In-person Cell packing in the notochord: morphometry, pattern, and forces
Sep 22 Andras Juhasz Oxford Zoom Knot theory and machine learning
Sep 15 Kirsten Wickelgren Duke In-person The Weil Conjectures and A1-homotopy theory

Andras Szenes
Abstract: Toric varieties are intimately related to the combinatorics of polytopes, and I will describe an important example of such a relation linked to certain natural Lie-theoretic data. I will show then how, calculating intersection homologies of these varieties brings us to a beautiful new formula for the intersection Betti numbers of moduli spaces of semistable bundles on Riemann surfaces. This problem has a long history, which began with the works of Frances Kirwan in the 1980’s, and saw a recent breakthrough by Mozgovoy and Reineke. I will be reporting on joint work with Camilla Felisetti and Olga Trapeznikova.

Daniel Tataru
Abstract: The question of long time or global existence of solutions is one of the fundamental ones in the study of
partial differential equations. For this talk I will try to present an overview of the ideas that have been used
in the study of such problems, from classical to the most recent ones.

Yannick Sire
Abstract: I will describe in a very informal way some techniques to deal with the existence ( and more qualitatively regularity vs singularity formation) in different geometric problems and their heat flows motivated by (variations of) the harmonic map problem, the construction of Yang-Mills connections or nematic liquid crystals. I will emphasize in particular on recent results on the construction of very fine asymptotics of blow-up solutions via a new gluing method designed for parabolic flows. I’ll describe several open problems and many possible generalizations, since the techniques are rather flexible.

Sharon Lubkin
Abstract: The notochord, the defining feature of chordates, is a pressurized tube which actuates elongation of the chordate embryo. The zebrafish notochord consists of large vacuolated cells surrounded by a thin sheath. We characterized the patterns of the cells’ packing, and their relationship to the known regular patterns from the study of foams, and irregular patterns in a gel bead system. Disruption of the wild type packing pattern leads to developmental defects. By constructing a suite of models of the physics of cell packing in regular patterns the notochord, we have determined key parameter ratios governing the packing pattern, cell and tissue morphometry, and derived surprisingly simple expressions for key morphometric quantities in terms of tension ratios.

Andras Juhasz
Abstract: The signature of a knot K in the 3-sphere is a classical invariant that gives a lower bound on the genera of compact, oriented surfaces in the 4-ball with boundary K. We say that K is hyperbolic if its complement admits a complete, finite volume hyperbolic metric. I will explain how we have used methods from machine learning to find an unexpected relationship between the signature and the cusp shape of a hyperbolic knot. This is joint work with Alex Davies, Marc Lackenby, and Nenad Tomasev.

Kirsten Wickelgren
Abstract: In a celebrated paper from 1948, André Weil proposed a beautiful connection between algebraic topology and the number of solutions to equations over finite fields: the zeta function of a variety over a finite field is simultaneously a generating function for the number of solutions to its defining equations and a product of characteristic polynomials of endomorphisms of cohomology groups. The ranks of these cohomology groups are the number of holes of each dimension of the associated complex manifold.
This talk will describe the Weil conjectures and then enrich the zeta function to have coefficients in a group of bilinear forms. The enrichment provides a connection between the solutions over finite fields and the associated real and complex manifolds. It is formed using A1-homotopy theory. No knowledge of A1-homotopy theory is necessary. The new work in this talk is joint with Margaret Bilu, Wei Ho, Padma Srinivasan, and Isabel Vogt.