• Speaker: Craig Gross, MSU
• Title: High-dimensional sparse Fourier transforms and spectral methods
• Date: 03/27/2023
• Time: 2:00 PM - 3:00 PM
• Place: C329 Wells Hall (Virtual Meeting Link)
• Contact: Craig Paul Gross (grosscra@msu.edu)

High-dimensional function approximation suffers from the "curse of dimensionality": an exponential dependence on the dimension of the function's domain in an algorithm's runtime. For problems with large dimensions (e.g., >10), approximation based on standard tensor-product-like approaches of one-dimensional theory can be intractable. $\\$ In this talk, we present an alternative algorithm that by passes the computational curse of dimensionality, so long as the function of interest has a sparse Fourier series (or at least can be well approximated by one). Additionally, we show a new way to use this high-dimensional sparse Fourier transform in a classic application of Fourier transforms: spectral methods for solving PDE. Our method allows us to solve extremely high-dimensional (>1000) and multiscale elliptic PDE. $\\$ This is the practice for my dissertation defense! Please come, give feedback, and help make my presentation better! $\\$ This will be a hybrid seminar and take place in C329 Wells Hall and via Zoom at https://msu.zoom.us/j/99426648081?pwd=ZEljM3BPUXg2MjVUMVM5TnlzK2NQZz09 .

• Speaker: Matthew Lorentz, MSU
• Title: Noncommutative Geometry and Uniform Roe Algebras
• Date: 03/27/2023
• Time: 4:00 PM - 5:30 PM
• Place: C517 Wells Hall
• Contact: Brent Nelson (banelson@msu.edu)

"Nice spaces", say smooth Riemannian manifolds, contain a great deal of geometric structure. We can examine their tangent and normal bundles. We can look at their commutative algebra of smooth functions, which are dense in the commutative algebra of continuous functions. In fact, from the continuous functions (which are a commutative C*-algebra) we can recover the topology of the manifold. The primary motivation of noncommutative geometry is to extend this notion of commutative duality to the noncommutative setting. To find an analog of De Rham cohomology. To find suitable definitions of differential forms and multivector fields on a noncommutative space. A natural class of noncommutative spaces to investigate are coarse spaces, which are dual to uniform Roe algebras. In this talk we will define uniform Roe algebras and uniform Roe modules and look at a few examples. We will also define Hochschild (co)homology which will be our analog for differential forms and multivector fields. Lastly, we will discuss current advancements in our understanding of the Hochschild cohomology of uniform Roe algebras.

• Speaker: Selim Sukhtaiev, Auburn University
• Title: The Maslov index in spectral theory: an overview.
• Date: 03/28/2023
• Time: 10:30 AM - 11:30 AM
• Place: C304 Wells Hall
• Contact: Ilya Kachkovskiy (ikachkov@msu.edu)

This talk is centered around a symplectic approach to eigenvalue problems for systems of ordinary differential operators (e.g., Sturm-Liouville operators, canonical systems, and quantum graphs), multidimensional elliptic operators on bounded domains, and abstract self-adjoint extensions of symmetric operators in Hilbert spaces. The symplectic view naturally relates spectral counts for self-adjoint problems to the topological invariant called the Maslov index. In this talk, the notion of the Malsov index will be introduced in analytic terms and an overview of recent results on its role in spectral theory will be given.

• Speaker: Qin Fan, Shanghai Jiao Tong University
• Title: Bracelets are theta functions for surface cluster algebras
• Date: 03/28/2023
• Time: 1:00 PM - 2:00 PM
• Place: C304 Wells Hall
• Contact: Linhui Shen (shenlin1@msu.edu)

The skein algebra of a marked surface admits the basis of bracelet elements constructed by Fock-Goncharov and Musiker-Schiffler-Williams. As a cluster algebra, it also admits the theta basis from the cluster scattering diagram by Gross-Hacking-Keel-Kontsevich. In a joint work with Travis Mandel, we show that the two bases coincide except for the once-punctured torus. Our results extend to quantum cluster algebras with coefficients arising from the surface even in punctured cases. Long-standing conjectures on strong positivity and atomicity follow as corollaries. We also connect our results to Bridgeland's stability scattering diagrams.

• Speaker: Martha Precup, Washington University
• Title: Singularities of regular Hessenberg varieties
• Date: 03/29/2023
• Time: 3:00 PM - 3:50 PM
• Place: Online (virtual meeting) (Virtual Meeting Link)
• Contact: Bruce E Sagan (bsagan@msu.edu)

Regular Hessenberg varieties are subvarieties of the flag variety with important connections to representation theory, algebraic geometry, and combinatorics. Like Schubert varieties, the structure of Hessenberg varieties can be studied using the combinatorics of the symmetric group, but there are also other combinatorial parameters to incorporate. In this talk, we give a criterion for identifying singular points in a particular collection of regular Hessenberg varieties. Our results lead to a classification of all singular Hessenberg varieties in this collection. This talk is based on joint work with Erik Insko and Alex Woo.

• Speaker: Zhonghui Sun, MSU
• Title: Covering spaces
• Date: 03/29/2023
• Time: 3:00 PM - 4:00 PM
• Place: C304 Wells Hall
• Contact: Igor Rapinchuk (rapinchu@msu.edu)

No abstract available.

• Speaker: Sung-Jin Oh, UC Berkeley
• Title: Late time tail of waves on dynamic asymptotically flat spacetimes of odd space dimensions
• Date: 03/29/2023
• Time: 4:10 PM - 5:00 PM
• Place: C304 Wells Hall
• Contact: Willie Wai-Yeung Wong (wongwil2@msu.edu)

I will present an upcoming work with J. Luk (Stanford), where we develop a general method for understanding the late time tail for solutions to wave equations on asymptotically flat spacetimes with odd spatial dimensions, which is applicable to nonlinear problems on dynamical backgrounds. In addition to its inherent interest, such information is crucial for studying problems involving the interaction of waves with a spatially localized object; indeed, our motivation for developing this method comes from the Strong Cosmic Censorship Conjecture. I will explain how our method recovers and refines Price's law for linear problems on stationary backgrounds, and also how it shows that the late time tails are in general different(!) from the linear stationary case in the presence of nonlinearity and/or a dynamical background.

• Speaker: Woongbae Park, University of Pittsburgh
• Title: An integral formula for rational homotopy groups and analytic estimates.
• Date: 03/30/2023
• Time: 12:00 PM - 1:00 PM
• Place: C304 Wells Hall
• Contact: Benjamin I Schmidt (schmi444@msu.edu)

In this talk, I present a rational homotopy group and and its construction using minimal models given by Sullivan. After briefly describing Sullivan's theorem, I will consider the specific example of S^n v S^n and compute few low-dimensional rational homotopy groups. In the second part, I introduce the Novikov integral formula for the rational homotopy group and provide analytic bound obtained from the integral formula. The talk will end with specific examples.

• Speaker: Patel Coupek, MSU
• Title: Rigid analytic geometry
• Date: 03/30/2023
• Time: 2:10 PM - 3:10 PM
• Place: C204A Wells Hall
• Contact: Peikai Qi (qipeikai@msu.edu)

No abstract available.

• Speaker: Tim Hoheisel, McGill University
• Title: The Maximum Entropy on the Mean Method for Linear Inverse Problems (and beyond)
• Date: 03/30/2023
• Time: 4:10 PM - 5:00 PM
• Place: C304 Wells Hall
• Contact: Olga Turanova (turanova@msu.edu)

The principle of ‘maximum entropy’ states that the probability distribution which best represents the current state of knowledge about a system is the one with largest entropy with respect to a given prior (data) distribution. It was first formulated in the context of statistical physics in two seminal papers by E. T. Jaynes (Physical Review, Series II. 1957), and thus constitutes an information theoretic manifestation of Occam’s razor. We bring the idea of maximum entropy to bear in the context of linear inverse problems in that we solve for the probability measure which is close to the (learned or chosen) prior and whose expectation has small residual with respect to the observation. Duality leads to tractable, finite-dimensional (dual) problems. A core tool, which we then show to be useful beyond the linear inverse problem setting, is the ‘MEMM functional’: it is an infimal projection of the Kullback- Leibler divergence and a linear equation, which coincides with Cramér’s function (ubiquitous in the theory of large deviations) in most cases, and is paired in duality with the cumulant generating function of the prior measure. Numerical examples underline the efficacy of the presented framework. The talk encompasses joint work with Rustum Choksi (McGill), Ariel Goodwin (McGill), Carola-Bibiane Schönlieb (Cambridge), and Yakov Vaisbourd (McGill).