Colloquia: Difference between revisions
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Many tools have been developed in combinatorics to study global structure in finite graphs. One such tool is called Szemer\'{e}di’s regularity lemma, which gives a structural decomposition for any large finite graph. Beginning with work of Alon-Fischer-Newman, Lov\'{a}sz-Szegedy, and Malliaris-Shelah, it has been shown over the last 15 years that regularity lemmas can be used to detect structural dichotomies in graphs, and that these dichotomies have deep connections to model theory. In this talk, I present extensions of this type of result to arithmetic regularity lemmas, which are analogues of graph regularity lemmas, tailored to the study of combinatorial problems in finite groups. This work uncovered tight connections between tools from additive combinatorics, and ideas from the model theoretic study of infinite groups. | Many tools have been developed in combinatorics to study global structure in finite graphs. One such tool is called Szemer\'{e}di’s regularity lemma, which gives a structural decomposition for any large finite graph. Beginning with work of Alon-Fischer-Newman, Lov\'{a}sz-Szegedy, and Malliaris-Shelah, it has been shown over the last 15 years that regularity lemmas can be used to detect structural dichotomies in graphs, and that these dichotomies have deep connections to model theory. In this talk, I present extensions of this type of result to arithmetic regularity lemmas, which are analogues of graph regularity lemmas, tailored to the study of combinatorial problems in finite groups. This work uncovered tight connections between tools from additive combinatorics, and ideas from the model theoretic study of infinite groups. | ||
<div id="Maldague">'''Friday, December 1. Dominique Maldague''' | <div id="Maldague">'''Friday, December 1. Dominique Maldague''' | ||
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This talk will provide an overview of recent developments in Fourier restriction theory, which is the study of exponential sums over restricted frequency sets with geometric structure, typically arising in pde or number theory. Decoupling inequalities measure the square root cancellation behavior of these exponential sums. I will highlight recent work which uses the latest tools developed in decoupling theory to prove much more delicate sharp square function estimates for frequencies lying in the cone in R^3 (Guth-Wang-Zhang) and moment curves (t,t^2,...,t^n) in all dimensions (Guth-Maldague). | This talk will provide an overview of recent developments in Fourier restriction theory, which is the study of exponential sums over restricted frequency sets with geometric structure, typically arising in pde or number theory. Decoupling inequalities measure the square root cancellation behavior of these exponential sums. I will highlight recent work which uses the latest tools developed in decoupling theory to prove much more delicate sharp square function estimates for frequencies lying in the cone in R^3 (Guth-Wang-Zhang) and moment curves (t,t^2,...,t^n) in all dimensions (Guth-Maldague). | ||
<div id="Webber">'''Wednesday, December 6. Robert Webber''' | |||
'''Randomized matrix decompositions for faster scientific computing''' | |||
Traditional numerical methods based on expensive matrix factorizations struggle with the scale of modern scientific applications. For example, kernel-based algorithms take a data set of size N, form a kernel matrix of size N x N, and then perform an eigendecomposition or inversion at a cost of O(N^3) operations. For data sets of size N >= 10^5, kernel learning is too expensive, straining the limits of personal workstations and even dedicated computing clusters. Randomized iterative methods have emerged as a faster alternative to the classical approaches. These methods combine randomized exploration with information about which matrix structures are important, leading to significant speed gains. | |||
In this talk, I will review recent developments concerning two randomized algorithms. The first is "randomized block Krylov iteration", which uses an array of random Gaussian test vectors to probe a large data matrix in order to provide a randomized principal component analysis. Remarkably, this approach works well even when the matrix of interest is not low-rank. The second algorithm is "randomly pivoted Cholesky decomposition", which iteratively samples columns from a positive semidefinite matrix using a novelty metric and reconstructs the matrix from the randomly selected columns. Ultimately, both algorithms furnish a randomized approximation of an N x N matrix with a reduced rank k << N, which enables fast inversion or singular value decomposition at a cost of O(N k^2) operations. The speed-up factor from N^3 to N k^2 operations can be 3 million. The newest algorithms achieve this speed-up factor while guaranteeing performance across a broad range of input matrices. | |||
<div id="Fraczyk">'''Monday, December 11. Mikolaj Fraczyk''' | <div id="Fraczyk">'''Monday, December 11. Mikolaj Fraczyk''' | ||
Revision as of 17:40, 29 November 2023
UW Madison mathematics Colloquium is on Fridays at 4:00 pm in Van Vleck B239 unless otherwise noted.
Fall 2023
| date | speaker | title | host(s) | |
|---|---|---|---|---|
| Sept 8 | Tushar Das (University of Wisconsin-La Crosse) | Playing games on fractals: Dynamical & Diophantine | Stovall | |
| Sept 15 | John Schotland (Yale) | Nonlocal PDEs and Quantum Optics | Li | |
| Sept 22 | David Dumas(University of Illinois Chicago) | Geometry of surface group homomorphisms | Zimmer | |
| Sept 29 | no colloquium (see Monday) | |||
| Monday Oct 2 at 4 pm | Edriss Titi (Texas A&M University) | Distinguished lectures: On the Solvability of the Navier-Stokes and Euler Equations, where do we stand? | Smith, Stechmann | |
| Oct 13 | Autumn Kent | The 0π Theorem | ||
| Oct 20 | Sara Maloni (UVA) | Some new results in Higher Teichmüller Theory | Dymarz, Uyanik, GmMaW | |
| Wednesday Oct 25 at 4 pm | Gigliola Staffilani (MIT) | The Schrödinger equations as inspiration of beautiful mathematics | Ifrim, Smith | |
| Oct 27 | Rodrigo Bañuelos (Purdue) | Probabilistic tools in discrete harmonic analysis | Stovall | |
| Tuesday Oct 31 at 4 pm | Irit Dinur (The Weizmann Institute of Science) | Gurevich | ||
| Wednesday Nov 1 at 4 pm | Irit Dinur (The Weizmann Institute of Science) | Gurevich | ||
| Tuesday Nov 14 at 4 pm (Stirling 1310) | Ziyang Gao (Leibniz University Hannover) | Sparsity of rational and algebraic points | Arinkin, Yang | |
| Monday Nov 20 | Ruobing Zhang (Princeton) | Metric geometric aspects of Einstein manifolds | Paul | |
| Monday Nov 27 | Yizhe Zhu (UC Irvine) | Asymmetry Helps: Non-Backtracking Spectral Methods for Sparse Matrices and Tensors | Shen | |
| Wednesday Nov 29 | Caroline Terry (OSU) | Measuring combinatorial complexity via regularity lemmas | Andrews | |
| Friday Dec 1 | Dominique Maldague (MIT) | Sharp square function estimates in Fourier restriction theory | Stovall | |
| Wednesday Dec 6 | Robert Webber (Caltech) | Randomized matrix decompositions for faster scientific computing | Smith | |
| Monday Dec 11 | Mikolaj Fraczyk (Jagiellonian University, Krakow, Poland) | Large subgroups in higher rank | Stovall, Zimmer |
Abstracts
Friday, September 8. Tushar Das
Playing games on fractals: Dynamical & Diophantine We will present sketches of a program, developed in collaboration with Lior Fishman, David Simmons, and Mariusz Urbanski, which extends the parametric geometry of numbers (initiated by Wolfgang Schmidt and Leonhard Summerer) to Diophantine approximation for systems of m linear forms in n variables. Our variational principle (arXiv:1901.06602) provides a unified framework to compute Hausdorff and packing dimensions of a variety of sets of number-theoretic interest, as well as their dynamical counterparts via the Dani correspondence. Highlights include the introduction of certain combinatorial objects that we call templates, which arise from a dynamical study of Minkowski’s successive minima in the geometry of numbers; as well as a new variant of Schmidt’s game designed to compute the Hausdorff and packing dimensions of any set in a doubling metric space. The talk will be accessible to students and faculty whose interests contain a convex combination of homogeneous dynamics, Diophantine approximation and fractal geometry.
Friday, September 15. John Schotland
Nonlocal PDEs and Quantum Optics Quantum optics is the quantum theory of the interaction of light and matter. In this talk, I will describe a real-space formulation of quantum electrodynamics with applications to many body problems. The goal is to understand the transport of nonclassical states of light in random media. In this setting, there is a close relation to kinetic equations for nonlocal PDEs with random coefficients.
Friday, September 22. David Dumas
The space of homomorphisms from the fundamental group of a compact surface to a Lie group is a remarkably rich and versatile object, playing a key role in mathematical developments spanning disciplines of algebra, analysis, geometry, and mathematical physics. In this talk I will discuss and weave together two threads of research within this larger story: 1) the study of manifolds that are obtained by taking quotients of symmetric spaces (the "inside view") and 2) those obtained as quotients of domains in flag varieties (the "boundary view"). This discussion will start with classical objects--hyperbolic structures on surfaces---and continue into topics of ongoing research.
Friday, October 13. Autumn Kent
A celebrated theorem of Thurston tells us that among the many ways of filling in cusps of hyperbolic $3$--manfiolds, all but finitely many of them produce hyperbolic manifolds once again. This finiteness may be refined in a number of ways depending on the ``shape’’ of the cusp, and I’ll give a light and breezy discussion of joint work with K. Bromberg and Y. Minsky that allows shapes not covered by any of the previous theorems. This has applications such as answering questions asked in my 2010 job talk here at UW.
Friday, October 20. Sara Maloni
The Teichmüller space of a surface S is the space of marked hyperbolic structure on S, up to equivalence. By considering the holonomy representation of such structures, the Teichmüller space can also be seen as a connected component of (conjugacy classes of) representations from the fundamental group of S into PSL(2,R), consisting entirely of discrete and faithful representations. Generalizing this point of view, Higher Teichmüller Theory studies connected components of (conjugacy classes of) representations from the fundamental group of S into more general semisimple Lie groups which consist entirely of discrete and faithful representations.
We will give a survey of some aspects of Higher Teichmüller Theory and will make links with the recent powerful notion of Anosov representation. We will conclude by focusing on two separate questions: Do these representations correspond to deformation of geometric structures? Can we generalize the important notion of pleated surfaces to higher rank Lie groups like PSL(d, C)? The answer to question 1 is joint work with Alessandrini, Tholozan and Wienhard, while the answer to question 2 is joint work with Martone, Mazzoli and Zhang.
Wednesday, October 25. Gigliola Staffilani
In the last two decades great progress has been made in the study of dispersive and wave equations. Over the years the toolbox used in order to attack highly nontrivial problems related to these equations has developed to include a collection of techniques: Fourier and harmonic analysis, analytic number theory, math physics, dynamical systems, probability and symplectic geometry. In this talk I will introduce a variety of results using as model problem mainly the periodic 2D cubic nonlinear Schrödinger equation. I will start by giving a physical derivation of the equation from a quantum many-particles system, I will introduce periodic Strichartz estimates along with some remarkable connections to analytic number theory, I will move on the concept of energy transfer and its connection to dynamical systems, and I will end with some results following from viewing the periodic nonlinear Schrödinger equation as an infinite dimensional Hamiltonian system.
Friday, October 27. Rodrigo Bañuelos
Probabilistic tools in discrete harmonic analysis
The discrete Hilbert transform was introduced by David Hilbert at the beginning of the 20th century as an example of a singular quadratic form. Its boundedness on the space of square summable sequences appeared in H. Weyl’s doctoral dissertation (under Hilbert) in 1908. In 1925, M. Riesz proved that the continuous version of this operator is bounded on L^p(R), 1 < p < \infty, and that the same holds for the discrete version on the integers. Shortly thereafter (1926), E. C. Titchmarsh gave a different proof and from it concluded that the operators have the same p-norm. Unfortunately, Titchmarsh’s argument for equality was incorrect. The question of equality of the norms had been a “simple tantalizing" problem ever since.
In this general colloquium talk the speaker will discuss a probabilistic construction, based on Doob’s “h-Brownian motion," that leads to sharp inequalities for a collection of discrete operators on the d-dimensional lattice Z^d, d ≥ 1. The case d = 1 verifies equality of the norms for the discrete and continuous Hilbert transforms. The case d > 1 leads to similar questions and conjectures for other Calderón-Zygmund singular integrals in higher dimensions.
Sparsity of rational and algebraic points
It is a fundamental question in mathematics to find rational solutions to a given system of polynomials, and in modern language this question translates into finding rational points in algebraic varieties. This question is already very deep for algebraic curves defined over Q. An intrinsic natural number associated with the curve, called its genus, plays an important role in studying the rational points on the curve. In 1983, Faltings proved the famous Mordell Conjecture (proposed in 1922), which asserts that any curve of genus at least 2 has only finitely many rational points. Thus the problem for curves of genus at least 2 can be divided into several grades: finiteness, bound, uniform bound, effectiveness. An answer to each grade requires a better understanding of the distribution of the rational points. In my talk, I will explain the historical and recent developments of this problem according to the different grades. Another important topic on studying points on curves is the torsion packets. This topic goes beyond rational points. I will also discuss briefly about it in my talk.
Metric geometric aspects of Einstein manifolds
Abstract: This lecture concerns the metric Riemannian geometry of Einstein manifolds, which is a central theme in modern differential geometry and is deeply connected to a large variety of fundamental problems in algebraic geometry, geometric topology, analysis of nonlinear PDEs, and mathematical physics. We will exhibit the rich geometric/topological structures of Einstein manifolds and specifically focus on the structure theory of moduli spaces of Einstein metrics.
My recent works center around the intriguing problems regarding the compactification of the moduli space of Einstein metrics, which tells us how Einstein manifolds can degenerate. Such problems constitute the most challenging part in the metric geometry of Einstein manifolds. We will introduce recent major progress in the field. If time permits, I will propose several important open questions.
Asymmetry Helps: Non-Backtracking Spectral Methods for Sparse Matrices and Tensors
The non-backtracking operator, an asymmetric matrix constructed from an undirected graph, connects to various aspects of graph theory, including random walks, graph zeta functions, and expander graphs. It has emerged as a powerful tool for analyzing sparse random graphs, leading to significant advancements with established results for sparse random matrices using this operator. Additionally, algorithms employing the non-backtracking operator have achieved optimal sample complexity in many low-rank estimation problems. In my talk, I will present my recent work utilizing the non-backtracking operator, demonstrating how theoretical elegance drives the design of innovative algorithms through the introduction of asymmetry into data matrices. The discussion will include estimates of the extreme singular values of sparse random matrices and explore data science applications such as hypergraph community detection and tensor completion.
Measuring combinatorial complexity via regularity lemmas
Many tools have been developed in combinatorics to study global structure in finite graphs. One such tool is called Szemer\'{e}di’s regularity lemma, which gives a structural decomposition for any large finite graph. Beginning with work of Alon-Fischer-Newman, Lov\'{a}sz-Szegedy, and Malliaris-Shelah, it has been shown over the last 15 years that regularity lemmas can be used to detect structural dichotomies in graphs, and that these dichotomies have deep connections to model theory. In this talk, I present extensions of this type of result to arithmetic regularity lemmas, which are analogues of graph regularity lemmas, tailored to the study of combinatorial problems in finite groups. This work uncovered tight connections between tools from additive combinatorics, and ideas from the model theoretic study of infinite groups.
Sharp square function estimates in Fourier restriction theory
This talk will provide an overview of recent developments in Fourier restriction theory, which is the study of exponential sums over restricted frequency sets with geometric structure, typically arising in pde or number theory. Decoupling inequalities measure the square root cancellation behavior of these exponential sums. I will highlight recent work which uses the latest tools developed in decoupling theory to prove much more delicate sharp square function estimates for frequencies lying in the cone in R^3 (Guth-Wang-Zhang) and moment curves (t,t^2,...,t^n) in all dimensions (Guth-Maldague).
Randomized matrix decompositions for faster scientific computing
Traditional numerical methods based on expensive matrix factorizations struggle with the scale of modern scientific applications. For example, kernel-based algorithms take a data set of size N, form a kernel matrix of size N x N, and then perform an eigendecomposition or inversion at a cost of O(N^3) operations. For data sets of size N >= 10^5, kernel learning is too expensive, straining the limits of personal workstations and even dedicated computing clusters. Randomized iterative methods have emerged as a faster alternative to the classical approaches. These methods combine randomized exploration with information about which matrix structures are important, leading to significant speed gains.
In this talk, I will review recent developments concerning two randomized algorithms. The first is "randomized block Krylov iteration", which uses an array of random Gaussian test vectors to probe a large data matrix in order to provide a randomized principal component analysis. Remarkably, this approach works well even when the matrix of interest is not low-rank. The second algorithm is "randomly pivoted Cholesky decomposition", which iteratively samples columns from a positive semidefinite matrix using a novelty metric and reconstructs the matrix from the randomly selected columns. Ultimately, both algorithms furnish a randomized approximation of an N x N matrix with a reduced rank k << N, which enables fast inversion or singular value decomposition at a cost of O(N k^2) operations. The speed-up factor from N^3 to N k^2 operations can be 3 million. The newest algorithms achieve this speed-up factor while guaranteeing performance across a broad range of input matrices.
Large subgroups in higher rank
Let G be a higher-rank semisimple Lie group (for example, SL_n(R), n > 2). Lattices of G are well understood, thanks to the celebrated Margulis’ arithmeticity theorem. The infinite covolume discrete subgroups of G remain much more mysterious. There has been a lot of progress towards understanding some special classes of subgroups, like the Anosov subgroups, but it is still hard to find "large" discrete subgroups other than the lattices themselves. It is natural to ask if this apparent lack of examples could be explained by new rigidity phenomena. In my talk, I'll make this question more precise and present several instances where the answer is yes, for example, the confined discrete subgroups (j.w. Tsachik Gelander) and the discrete subgroups with finite Bowen-Margulis-Sullivan measure (j.w. Minju Lee).