Difference between revisions of "Colloquia"
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|Erman | |Erman | ||
|- | |- | ||
− | |Nov | + | |Nov 13 '''Wednesday 4-5pm''' |
|Ananth Shankar (MIT) | |Ananth Shankar (MIT) | ||
|Exceptional splitting of abelian surfaces | |Exceptional splitting of abelian surfaces | ||
Line 84: | Line 84: | ||
|Nov 22 | |Nov 22 | ||
| Jeffrey Danciger (UT Austin) | | Jeffrey Danciger (UT Austin) | ||
− | | [[#Jeffrey Danciger (UT Austin) | " | + | | [[#Jeffrey Danciger (UT Austin) | "Affine geometry and the Auslander Conjecture"]] |
| Kent | | Kent | ||
+ | |- | ||
+ | |Nov 25 '''Monday 4-5 pm Room 911''' | ||
+ | |Tatyana Shcherbina (Princeton) | ||
+ | | [[# Tatyana Shcherbina (Princeton)| "Random matrix theory and supersymmetry techniques"]] | ||
+ | |Roch | ||
|- | |- | ||
|Nov 29 | |Nov 29 | ||
|Thanksgiving | |Thanksgiving | ||
| | | | ||
+ | |- | ||
+ | |Dec 2 '''Monday 4-5pm''' | ||
+ | |Tingran Gao (University of Chicago) | ||
+ | | [[#Tingran Gao (University of Chicago)| "Manifold Learning on Fibre Bundles"]] | ||
+ | |Smith | ||
+ | |- | ||
+ | |Dec 4 '''Wednesday 4-5 pm Room 911''' | ||
+ | |Andrew Zimmer (LSU) | ||
+ | |[[#Andrew Zimmer (LSU)| "Intrinsic and extrinsic geometries in several complex variables"]] | ||
+ | |Gong | ||
|- | |- | ||
|Dec 6 | |Dec 6 | ||
− | | | + | |Charlotte Chan (MIT) |
− | | | + | |[[#Charlotte Chan (MIT)|"Flag varieties and representations of p-adic groups"]] |
+ | |Erman | ||
+ | |- | ||
+ | |Dec 9 '''Monday 4-5 pm''' | ||
+ | |Hui Yu (Columbia) | ||
+ | |[[#Hui Yu (Columbia)|Singular sets in obstacle problems]] | ||
+ | |Tran | ||
|- | |- | ||
|Dec 11 '''Wednesday''' | |Dec 11 '''Wednesday''' | ||
|Nick Higham (Manchester) | |Nick Higham (Manchester) | ||
− | |LAA lecture | + | |[[#Nick Higham (Manchester)|LAA lecture: Challenges in Multivalued Matrix Functions]] |
|Brualdi | |Brualdi | ||
| | | | ||
Line 117: | Line 138: | ||
|Jan 24 | |Jan 24 | ||
|Reserved for job talk | |Reserved for job talk | ||
+ | | | ||
+ | |- | ||
+ | |Jan 29 '''Wednesday 4-5 pm''' | ||
+ | |[https://ajzucker.wordpress.com/ Andy Zucker] (Lyon) | ||
+ | | | ||
+ | |Soskova/Lempp | ||
| | | | ||
|- | |- | ||
Line 232: | Line 259: | ||
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. | 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)=== | ===Eugenia Cheng (School of the Art Institute of Chicago)=== | ||
Line 238: | Line 266: | ||
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. | 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)=== | ===Omer Mermelstein (Madison)=== | ||
Line 265: | Line 294: | ||
===Jose Rodriguez (UW-Madison)=== | ===Jose Rodriguez (UW-Madison)=== | ||
− | Determining the closest point to a model (subset of Euclidean space) is an important problem in many applications in science, | + | |
+ | 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 | 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. | descent approach. However, when there are multiple local minima, there is no guarantee of convergence to the true global minimizer. | ||
Line 276: | Line 306: | ||
===Ananth Shankar (MIT)=== | ===Ananth Shankar (MIT)=== | ||
− | 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 surfaces. 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. | + | |
+ | 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 surfaces. 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)=== | ===Franca Hoffman (Caltech)=== | ||
Line 285: | Line 317: | ||
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 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. | 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) === | === Jeffrey Danciger (UT Austin) === | ||
− | Title: | + | 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. | ||
+ | |||
+ | |||
+ | === 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. | ||
== Future Colloquia == | == Future Colloquia == |
Latest revision as of 17:06, 2 December 2019
Contents
- 1 Mathematics Colloquium
- 1.1 Fall 2019
- 1.2 Spring 2020
- 1.3 Abstracts
- 1.3.1 Will Sawin (Columbia)
- 1.3.2 Yan Soibelman (Kansas State)
- 1.3.3 Alicia Dickenstein (Buenos Aires)
- 1.3.4 Jianfeng Lu (Duke)
- 1.3.5 Eugenia Cheng (School of the Art Institute of Chicago)
- 1.3.6 Omer Mermelstein (Madison)
- 1.3.7 Shamgar Gurevich (Madison)
- 1.3.8 Jose Rodriguez (UW-Madison)
- 1.3.9 Ananth Shankar (MIT)
- 1.3.10 Franca Hoffman (Caltech)
- 1.3.11 Jeffrey Danciger (UT Austin)
- 1.3.12 Tatyana Shcherbina (Princeton)
- 1.3.13 Tingran Gao (University of Chicago)
- 1.3.14 Andrew Zimmer (LSU)
- 1.3.15 Charlotte Chan (MIT)
- 1.3.16 Hui Yu (Columbia)
- 1.3.17 Nick Higham (Manchester)
- 1.4 Future Colloquia
- 1.5 Past Colloquia
Mathematics Colloquium
All colloquia are on Fridays at 4:00 pm in Van Vleck B239, unless otherwise indicated.
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 | Nick Higham (Manchester) | LAA lecture: Challenges in Multivalued Matrix Functions | Brualdi | |
Dec 13 | Reserved for job talk |
Spring 2020
date | speaker | title | host(s) | |
---|---|---|---|---|
Jan 24 | Reserved for job talk | |||
Jan 29 Wednesday 4-5 pm | Andy Zucker (Lyon) | Soskova/Lempp | ||
Jan 31 | Reserved for job talk | |||
Feb 7 | Reserved for job talk | |||
Feb 14 | Reserved for job talk | |||
Feb 21 | Shai Evra (IAS) | Gurevich | ||
Feb 28 | Brett Wick (Washington University, St. Louis) | Seeger | ||
March 6 | Jessica Fintzen (Michigan) | Marshall | ||
March 13 | ||||
March 20 | Spring break | |||
March 27 | (Moduli Spaces Conference) | Boggess, Sankar | ||
April 3 | Caroline Turnage-Butterbaugh (Carleton College) | Marshall | ||
April 10 | Sarah Koch (Michigan) | Bruce (WIMAW) | ||
April 17 | Song Sun (Berkeley) | Huang | ||
April 24 | Natasa Sesum (Rutgers University) | Angenent | ||
May 1 | Robert Lazarsfeld (Stony Brook) | Distinguished lecture | Erman |
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 surfaces. 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.
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.