Colloquia/Fall2019

Fall 2019

date	speaker	title	host(s)
Sept 6 Room 911	Will Sawin (Columbia)	On Chowla's Conjecture over F_q[T]	Marshall
Sept 13	Yan Soibelman (Kansas State)	Riemann-Hilbert correspondence and Fukaya categories	Caldararu
Sept 16 Monday Room 911	Alicia Dickenstein (Buenos Aires)	Algebra and geometry in the study of enzymatic cascades	Craciun
Sept 20	Jianfeng Lu (Duke)	How to "localize" the computation?	Qin
Sept 26 Thursday 3-4 pm Room 911	Eugenia Cheng (School of the Art Institute of Chicago)	Character vs gender in mathematics and beyond	Marshall / Friends of UW Madison Libraries
Sept 27
Oct 4
Oct 11	Omer Mermelstein (Madison)	Generic flat pregeometries	Andrews
Oct 18	Shamgar Gurevich (Madison)	Harmonic Analysis on GL(n) over Finite Fields	Marshall
Oct 25
Nov 1	Elchanan Mossel (MIT)	Distinguished Lecture	Roch
Nov 8	Jose Rodriguez (UW-Madison)	Nearest Point Problems and Euclidean Distance Degrees	Erman
Nov 13 Wednesday 4-5pm	Ananth Shankar (MIT)	Exceptional splitting of abelian surfaces
Nov 20 Wednesday 4-5pm	Franca Hoffman (Caltech)	Gradient Flows: From PDE to Data Analysis	Smith
Nov 22	Jeffrey Danciger (UT Austin)	"Affine geometry and the Auslander Conjecture"	Kent
Nov 25 Monday 4-5 pm Room 911	Tatyana Shcherbina (Princeton)	"Random matrix theory and supersymmetry techniques"	Roch
Nov 29	Thanksgiving
Dec 2 Monday 4-5pm	Tingran Gao (University of Chicago)	"Manifold Learning on Fibre Bundles"	Smith
Dec 4 Wednesday 4-5 pm Room 911	Andrew Zimmer (LSU)	"Intrinsic and extrinsic geometries in several complex variables"	Gong
Dec 6	Charlotte Chan (MIT)	"Flag varieties and representations of p-adic groups"	Erman
Dec 9 Monday 4-5 pm	Hui Yu (Columbia)	Singular sets in obstacle problems	Tran
Dec 11 Wednesday 2:30-3:30pm Room 911	Alex Waldron (Michigan)	Gauge theory and geometric flows	Paul
Dec 11 Wednesday 4-5pm	Nick Higham (Manchester)	LAA lecture: Challenges in Multivalued Matrix Functions	Brualdi
Dec 13	Chenxi Wu (Rutgers)	Kazhdan's theorem on metric graphs	Ellenberg
Dec 18 Wednesday 4-5pm	Ruobing Zhang (Stony Brook)	Geometry and analysis of degenerating Calabi-Yau manifolds	Paul

Abstracts

Will Sawin (Columbia)

Title: On Chowla's Conjecture over F_q[T]

Abstract: The Mobius function in number theory is a sequences of 1s, -1s, and 0s, which is simple to define and closely related to the prime numbers. Its behavior seems highly random. Chowla's conjecture is one precise formalization of this randomness, and has seen recent work by Matomaki, Radziwill, Tao, and Teravainen making progress on it. In joint work with Mark Shusterman, we modify this conjecture by replacing the natural numbers parameterizing this sequence with polynomials over a finite field. Under mild conditions on the finite field, we are able to prove a strong form of this conjecture. The proof is based on taking a geometric perspective on the problem, and succeeds because we are able to simplify the geometry using a trick based on the strange properties of polynomial derivatives over finite fields.

Yan Soibelman (Kansas State)

Title: Riemann-Hilbert correspondence and Fukaya categories

Abstract: In this talk I am going to discuss the role of Fukaya categories in the Riemann-Hilbert correspondence for differential, q-difference and elliptic difference equations in dimension one. This approach not only gives a unified answer for several versions of the Riemann-Hilbert correspondence but also leads to a natural formulation of the non-abelian Hodge theory in dimension one. It also explains why periodic monopoles should appear as harmonic objects in this generalized non-abelian Hodge theory. All that is a part of the bigger project ``Holomorphic Floer theory", joint with Maxim Kontsevich.

Alicia Dickenstein (Buenos Aires)

Title: Algebra and geometry in the study of enzymatic cascades

Abstract: In recent years, techniques from computational and real algebraic geometry have been successfully used to address mathematical challenges in systems biology. The algebraic theory of chemical reaction systems aims to understand their dynamic behavior by taking advantage of the inherent algebraic structure in the kinetic equations, and does not need the determination of the parameters a priori, which can be theoretically or practically impossible. I will give a gentle introduction to general results based on the network structure. In particular, I will describe a general framework for biological systems, called MESSI systems, that describe Modifications of type Enzyme-Substrate or Swap with Intermediates, and include many networks that model post-translational modifications of proteins inside the cell. I will also outline recent methods to address the important question of multistationarity, in particular in the study of enzymatic cascades, and will point out some of the mathematical challenges that arise from this application.

Jianfeng Lu (Duke)

Title: How to ``localize" the computation?

It is often desirable to restrict the numerical computation to a local region to achieve best balance between accuracy and affordability in scientific computing. It is important to avoid artifacts and guarantee predictable modelling while artificial boundary conditions have to be introduced to restrict the computation. In this talk, we will discuss some recent understanding on how to achieve such local computation in the context of topological edge states and elliptic random media.

Eugenia Cheng (School of the Art Institute of Chicago)

Title: Character vs gender in mathematics and beyond

Abstract: This presentation will be based on my experience of being a female mathematician, and teaching mathematics at all levels from elementary school to grad school. The question of why women are under-represented in mathematics is complex and there are no simple answers, only many many contributing factors. I will focus on character traits, and argue that if we focus on this rather than gender we can have a more productive and less divisive conversation. To try and focus on characters rather than genders I will introduce gender-neutral character adjectives "ingressive" and "congressive" to replace masculine and feminine. I will share my experience of teaching congressive abstract mathematics to art students, in a congressive way, and the possible effects this could have for everyone in mathematics, not just women.

Omer Mermelstein (Madison)

Title: Generic flat pregeometries

Abstract: In model theory, the tamest of structures are the strongly minimal ones -- those in which every equation in a single variable has either finitely many or cofinitely many solution. Algebraically closed fields and vector spaces are the canonical examples. Zilber’s conjecture, later refuted by Hrushovski, states that the source of geometric complexity in a strongly minimal structure must be algebraic. The property of "flatness" (strict gammoid) of a geometry (matroid) is that which guarantees Hrushovski's construction is devoid of any associative structure. The majority of the talk will explain what flatness is, how it should be thought of, and how closely it relates to hypergraphs and Hrushovski's construction method. Model theory makes an appearance only in the second part, where I will share results pertaining to the specific family of geometries arising from Hrushovski's methods.

Shamgar Gurevich (Madison)

Title: Harmonic Analysis on GL(n) over Finite Fields.

Abstract: There are many formulas that express interesting properties of a finite group G in terms of sums over its characters. For evaluating or estimating these sums, one of the most salient quantities to understand is the character ratio:

trace(ρ(g)) / dim(ρ),

for an irreducible representation ρ of G and an element g of G. For example, Diaconis and Shahshahani stated a formula of the mentioned type for analyzing certain random walks on G.

Recently, we discovered that for classical groups G over finite fields there is a natural invariant of representations that provides strong information on the character ratio. We call this invariant rank.

This talk will discuss the notion of rank for the group GLn over finite fields, demonstrate how it controls the character ratio, and explain how one can apply the results to verify mixing time and rate for certain random walks.

This is joint work with Roger Howe (Yale and Texas AM). The numerics for this work was carried by Steve Goldstein (Madison)

Jose Rodriguez (UW-Madison)

Abstract: Determining the closest point to a model (subset of Euclidean space) is an important problem in many applications in science, engineering, and statistics. One way to solve this problem is by minimizing the squared Euclidean distance function using a gradient descent approach. However, when there are multiple local minima, there is no guarantee of convergence to the true global minimizer. An alternative method is to determine the critical points of an objective function on the model. In algebraic statistics, the models of interest are algebraic sets, i.e., solution sets to a system of multivariate polynomial equations. In this situation, the number of critical points of the squared Euclidean distance function on the model’s Zariski closure is a topological invariant called the Euclidean distance degree (ED degree). In this talk, I will present some models from computer vision and statistics that may be described as algebraic sets. Moreover, I will describe a topological method for determining a Euclidean distance degree and a numerical algebraic geometry approach for determining critical points of the squared Euclidean distance function.

Ananth Shankar (MIT)

Abstract: An abelian surface 'splits' if it admits a non-trivial map to some elliptic curve. It is well known that the set of abelian surfaces that split are sparse in the set of all abelian surfa​ces. Nevertheless, we prove that there are infinitely many split abelian surfaces in arithmetic one-parameter families of generically non-split abelian surfaces. I will describe this work, and if time permits, mention generalizations of this result to the setting of K3 surfaces, as well as applications to the dynamics of hecke orbits. This is joint work with Tang, Maulik-Tang, and Shankar-Tang-Tayou.

Franca Hoffman (Caltech)

Title: Gradient Flows: From PDE to Data Analysis.

Abstract: Certain diffusive PDEs can be viewed as infinite-dimensional gradient flows. This fact has led to the development of new tools in various areas of mathematics ranging from PDE theory to data science. In this talk, we focus on two different directions: model-driven approaches and data-driven approaches. In the first part of the talk we use gradient flows for analyzing non-linear and non-local aggregation-diffusion equations when the corresponding energy functionals are not necessarily convex. Moreover, the gradient flow structure enables us to make connections to well-known functional inequalities, revealing possible links between the optimizers of these inequalities and the equilibria of certain aggregation-diffusion PDEs. In the second part, we use and develop gradient flow theory to design novel tools for data analysis. We draw a connection between gradient flows and Ensemble Kalman methods for parameter estimation. We introduce the Ensemble Kalman Sampler - a derivative-free methodology for model calibration and uncertainty quantification in expensive black-box models. The interacting particle dynamics underlying our algorithm can be approximated by a novel gradient flow structure in a modified Wasserstein metric which reflects particle correlations. The geometry of this modified Wasserstein metric is of independent theoretical interest.

Jeffrey Danciger (UT Austin)

Title: Affine geometry and the Auslander Conjecture

Abstract: The Auslander Conjecture is an analogue of Bieberbach’s theory of Euclidean crystallographic groups in the setting of affine geometry. It predicts that a complete affine manifold (a manifold equipped with a complete torsion-free flat affine connection) which is compact must have virtually solvable fundamental group. The conjecture is known up to dimension six, but is known to fail if the compactness assumption is removed, even in low dimensions. We discuss some history of this conjecture, give some basic examples, and then survey some recent advances in the study of non-compact complete affine manifolds with non-solvable fundamental group. Tools from the deformation theory of pseudo-Riemannian hyperbolic manifolds and also from higher Teichmüller theory will enter the picture.

Tatyana Shcherbina (Princeton)

Title: Random matrix theory and supersymmetry techniques

Abstract: Starting from the works of Erdos, Yau, Schlein with coauthors, the significant progress in understanding the universal behavior of many random graph and random matrix models were achieved. However for the random matrices with a special structure our understanding is still very limited. In this talk I am going to overview applications of another approach to the study of the local eigenvalues statistics in random matrix theory based on so-called supersymmetry techniques (SUSY). SUSY approach is based on the representation of the determinant as an integral over the Grassmann (anticommuting) variables. Combining this representation with the representation of an inverse determinant as an integral over the Gaussian complex field, SUSY allows to obtain an integral representation for the main spectral characteristics of random matrices such as limiting density, correlation functions, the resolvent's elements, etc. This method is widely (and successfully) used in the physics literature and is potentially very powerful but the rigorous control of the integral representations, which can be obtained by this method, is quite difficult, and it requires powerful analytic and statistical mechanics tools. In this talk we will discuss some recent progress in application of SUSY to the analysis of local spectral characteristics of the prominent ensemble of random band matrices, i.e. random matrices whose entries become negligible if their distance from the main diagonal exceeds a certain parameter called the band width.

Tingran Gao (University of Chicago)

Title: Manifold Learning on Fibre Bundles

Abstract: Spectral geometry has played an important role in modern geometric data analysis, where the technique is widely known as Laplacian eigenmaps or diffusion maps. In this talk, we present a geometric framework that studies graph representations of complex datasets, where each edge of the graph is equipped with a non-scalar transformation or correspondence. This new framework models such a dataset as a fibre bundle with a connection, and interprets the collection of pairwise functional relations as defining a horizontal diffusion process on the bundle driven by its projection on the base. The eigenstates of this horizontal diffusion process encode the “consistency” among objects in the dataset, and provide a lens through which the geometry of the dataset can be revealed. We demonstrate an application of this geometric framework on evolutionary anthropology.

Andrew Zimmer (LSU)

Title: Intrinsic and extrinsic geometries in several complex variables

Abstract: A bounded domain in complex Euclidean space, despite being one of the simplest types of manifolds, has a number of interesting geometric structures. When the domain is pseudoconvex, it has a natural intrinsic geometry: the complete Kaehler-Einstein metric constructed by Cheng-Yau and Mok-Yau. When the domain is smoothly bounded, there is also a natural extrinsic structure: the CR-geometry of the boundary. In this talk, I will describe connections between these intrinsic and extrinsic geometries. Then, I will discuss how these connections can lead to new analytic results.

Charlotte Chan (MIT)

Title: Flag varieties and representations of p-adic groups

Abstract: In the 1950s, Borel, Weil, and Bott showed that the irreducible representations of a complex reductive group can be realized in the cohomology of line bundles on flag varieties. In the 1970s, Deligne and Lusztig constructed a family of subvarieties of flag varieties whose cohomology realizes the irreducible representations of reductive groups over finite fields. I will survey these stories, explain recent progress towards finding geometric constructions of representations of p-adic groups, and discuss interactions with the Langlands program.

Hui Yu (Columbia)

Title: Singular sets in obstacle problems

Abstract: One of the most important free boundary problems is the obstacle problem. The regularity of its free boundary has been studied for over half a century. In this talk, we review some classical results as well as exciting new developments. In particular, we discuss the recent resolution of the regularity of the singular set for the fully nonlinear obstacle problem. This talk is based on a joint work with Ovidiu Savin at Columbia University.

Alex Waldron (Michigan)

Title: Gauge theory and geometric flows

Abstract: I will give a brief introduction to two major areas of research in differential geometry: gauge theory and geometric flows. I'll then introduce a geometric flow (Yang-Mills flow) arising from a variational problem with origins in physics, which has been studied by geometric analysts since the early 1980s. I'll conclude by discussing my own work on the behavior of Yang-Mills flow in the critical dimension (n = 4).

Nick Higham (Manchester)

Title: Challenges in Multivalued Matrix Functions

Abstract: In this lecture I will discuss multivalued matrix functions that arise in solving various kinds of matrix equations. The matrix logarithm is the prototypical example, and my first interaction with Hans Schneider was about this function. Another example is the Lambert W function of a matrix, which is much less well known but has been attracting recent interest. A theme of the talk is the importance of choosing appropriate principal values and making sure that the correct choices of signs and branches are used, both in theory and in computation. I will give examples where incorrect results have previously been obtained.

I focus on matrix inverse trigonometric and inverse hyperbolic functions, beginning by investigating existence and characterization. Turning to the principal values, various functional identities are derived, some of which are new even in the scalar case, including a “round trip” formula that relates acos(cos A) to A and similar formulas for the other inverse functions. Key tools used in the derivations are the matrix unwinding function and the matrix sign function.

A new inverse scaling and squaring type algorithm employing a Schur decomposition and variable-degree Pade approximation is derived for computing acos, and it is shown how it can also be used to compute asin, acosh, and asinh.

Chenxi Wu (Rutgers)

Title: Kazhdan's theorem on metric graphs

Abstract: I will give an introduction to the concept of canonical (arakelov) metric on a metric graph, which is related to combinatorial questions like the counting of spanning trees, and generalizes the corresponding concept on Riemann surfaces. I will also present a recent result in collaboration with Farbod Shokrieh on the convergence of canonical metric under normal covers.

Ruobing Zhang (Stony Brook)

Title: Geometry and analysis of degenerating Calabi-Yau manifolds

Abstract: This talk concerns a naturally occurring family of degenerating Calabi-Yau manifolds. A primary tool in analyzing their behavior is to combine the recently developed structure theory for Einstein manifolds and multi-scale singularity analysis for degenerating nonlinear PDEs in the collapsed setting. Based on the algebraic degeneration, we will give precise and more quantitative descriptions of singularity formation from both metric and analytic points of view.