• Speaker: Alexander Vainshtein, Haifa University
• Title: Cluster structures on SL_n and the Belavin-Drinfeld classification
• Date: 09/19/2022
• Time: 12:30 PM - 1:30 PM
• Place: C304 Wells Hall
• Contact: Michael Shapiro (mshapiro@msu.edu)

Cluster structures were discovered by S. Fomin and A. Zelevinsky about twenty years ago and quickly found applications in various fields of mathematics and mathematical physics. In the latter, several advances were made in a study of classical and quantum integrable systems arising in the context of cluster structures. These systems "live" on Poisson-Lie groups and their Poisson homogeneous spaces, hence it is important to understand an interplay between cluster and Poisson structures on such objects. In this talk I will explain a construction of a family of (generalized) cluster structures in the algebra of regular functions on SL_n related to the Belavin-Drinfeld classification of Poisson-Lie structures on SL_n. Based on a joint work with M.~Gekhtman (Notre Dame) and M.~Shapiro (MSU).

• Speaker: Craig Gross, MSU
• Title: All aboard! A mathematical study of transit equity in Baltimore
• Date: 09/19/2022
• Time: 1:00 PM - 2:00 PM
• Place: C117 Wells Hall (Virtual Meeting Link)
• Contact: Craig Gross (grosscra@msu.edu)

In 2015, the governor of Maryland canceled a light rail project through the city of Baltimore that had been planned and funded for over a decade. Instead, the money was diverted to funding highways near the richer, whiter suburbs of the city. As Baltimore is home to some of the most extreme class-disparity and segregation in the country, this decision significantly hurt the potential for a more equitable transit system. But by how much? $\\$ This talk will be a tour through a mathematical investigation of how the canceled light rail might have increased access to jobs across the city. In particular, we use some dimension-reduction and clustering algorithms on CDC Social Vulnerability Indices to explore which parts of the city may be socioeconomically disadvantaged. We then compute job accessibility metrics to determine how the light rail would have affected these regions. We also give some considerations for converting a collection of many relevant indicators into more interpretable, manageable metrics for future transit studies. $\\$ This is joint work with Adam Lee, Kethaki Varadan, and Yangxinyu Xie. $\\$ This will be a hybrid seminar and take place in C117 Wells Hall and via Zoom at https://msu.zoom.us/j/99426648081?pwd=ZEljM3BPUXg2MjVUMVM5TnlzK2NQZz09 .

• Speaker: Brent Nelson, MSU
• Title: Worth Their Weight
• Date: 09/19/2022
• Time: 4:00 PM - 5:30 PM
• Place: C517 Wells Hall
• Contact: Brent Nelson (banelson@msu.edu)

I will continue my introduction to weights. I will briefly mention equivalent conditions of normality for weights before moving onto a discussion of semi-cyclic representations and Tomita-Takesaki theory. I will conclude with a detailed examination of Plancherel weights on locally compact groups.

• Speaker: Alberto Takase, MSU and UC Irvine
• Title: Spectral estimates of dynamically-defined and amenable operator families (In collaboration with Siegfried Beckus)
• Date: 09/20/2022
• Time: 11:00 AM - 11:50 AM
• Place: C304 Wells Hall
• Contact: Ilya Kachkovskiy (ikachkov@msu.edu)

For dynamically-defined operator families, the Hausdorff distance of the spectra is estimated by the distance of the underlying dynamical systems while the group is amenable. We prove that if the group has strict polynomial growth and both the group action and the coefficients are Lipschitz continuous, then the spectral estimate has a square root behavior or, equivalently, the spectrum map is $\frac{1}{2}$-Holder continuous. We prove the behavior can be improved resulting in the spectrum map being Lipschitz continuous if the coefficients are locally-constant. In 1990, the square root behavior was established for the Almost Mathieu Operator or, more generally, the quasiperiodic operators with Lipschitz continuous potentials. Our results extend the square root behavior to a bigger class of operators such as (magnetic) discrete Schrodinger operators with finite range and with Lipschitz continuous coefficients.

• Speaker: Bin Sun, Oxford
• Title: $L^2$-Betti numbers of fiber bundles
• Date: 09/20/2022
• Time: 3:00 PM - 4:00 PM
• Place: C304 Wells Hall (Virtual Meeting Link)
• Contact: Vijay B Higgins (higgi231@msu.edu)

We study the $L^2$-Betti numbers of fiber bundles $F \rightarrow E \rightarrow B$ of manifolds. Under certain conditions (e.g., when $F$ is simply connected), $b_*^{(2)}(E)$ can be computed using the twisted $L^2$-Betti numbers of $B.$ We relate the twisted and untwisted $L^2$-Betti numbers of $B$ when $\pi_1(B)$ is locally indicable. As an application, we compute $b_*^{(2)}(E)$ when $B$ is either a surface or a non-positively curved $3-$manifold. This is a joint work with Dawid Kielak.

• Speaker: 
• Title: Organizational Meeting
• Date: 09/20/2022
• Time: 4:00 PM - 5:00 PM
• Place: C517 Wells Hall
• Contact: Ivan So (soivan@msu.edu)

No abstract available.

• Speaker: Alex Wilson, Dartmouth
• Title: A Diagram-Like Basis for the Multiset Partition Algebra
• Date: 09/21/2022
• Time: 3:00 PM - 3:50 PM
• Place: Online (virtual meeting) (Virtual Meeting Link)
• Contact: Bruce E Sagan (bsagan@msu.edu)

There's a classical connection between the representation theory of the symmetric group and the general linear group called Schur-Weyl Duality. Variations on this principle yield analogous connections between the symmetric group and other objects such as the partition algebra and more recently the multiset partition algebra. The partition algebra has a well-known basis indexed by graph-theoretic diagrams which allows the multiplication in the algebra to be understood visually as combinations of these diagrams. I will present an analogous basis for the multiset partition algebra and show how this basis can be used to describe generators and construct representations for the algebra.

• Speaker: Michael McNulty, MSU
• Title: Conditionally Stable Self-Similar Blowup for the Supercritical Quadratic Wave Equation Beyond the Light Cone
• Date: 09/21/2022
• Time: 4:10 PM - 5:00 PM
• Place: C304 Wells Hall
• Contact: Willie Wai-Yeung Wong (wongwil2@msu.edu)

Nonlinear wave equations of power-type serve as excellent toy models for geometric PDEs such as the Yang-Mills and wave maps equations. Of great interest in the energy supercritical setting is that of threshold phenomena. In this setting, unstable self-similar blowup solutions are believed to play an essential role in describing the threshold of singularity formation. We will discuss the stability of an explicitly known, unstable self-similar blowup solution of the energy supercritical quadratic wave equation in a region of spacetime which extends beyond the time of blowup. To overcome this instability, we introduce a new canonical method to investigate unstable self-similar solutions. This work represents the first steps toward an understanding of threshold phenomena in the energy supercritical setting.

• Speaker: Rayan Saab, University of California, San Diego (UCSD)
• Title: Quantizing neural networks
• Date: 09/22/2022
• Time: 2:30 PM - 3:30 PM
• Place: Online (virtual meeting) (Virtual Meeting Link)
• Contact: Mark A Iwen (iwenmark@msu.edu)

Neural networks are highly non-linear functions often parametrized by a staggering number of weights. Miniaturizing these networks and implementing them in hardware is a direction of research that is fueled by a practical need, and at the same time connects to interesting mathematical problems. For example, by quantizing, or replacing the weights of a neural network with quantized (e.g., binary) counterparts, massive savings in cost, computation time, memory, and power consumption can be attained. Of course, one wishes to attain these savings while preserving the action of the function on domains of interest. We present data-driven and computationally efficient methods for quantizing the weights of already trained neural networks and we prove that our methods have favorable error guarantees under a variety of assumptions. We also discuss extensions and provide the results of numerical experiments, on large multi-layer networks, to illustrate the performance of our methods. Time permitting, we will also discuss open problems and related areas of research.

• Speaker: Peikai Qi, MSU
• Title: Modular form and Hecke operator1
• Date: 09/22/2022
• Time: 3:00 PM - 4:00 PM
• Place: C204A Wells Hall
• Contact: Peikai Qi (qipeikai@msu.edu)

In the talk, we will review the definition of modular form and hecke algebra.

• Speaker: Chenyang Xu, Princeton University
• Title: K-stability and birational geometry
• Date: 09/22/2022
• Time: 4:10 PM - 5:00 PM
• Place: Online (virtual meeting) (Virtual Meeting Link)
• Contact: Joseph Waldron (waldro51@msu.edu)

The notion of K-stability for a Fano varieties was introduced by differential geometers in late 90s, to capture the existence of a Kähler-Einstein metric. In the last decade, it has gradually become clear to algebraic geometers that K-stability provides a rich algebraic theory in higher dimensional geometry. In particular, it can be used to solve the longstanding question of constructing moduli spaces for Fano varieties. I will survey the background of K-stability and how algebraic geometers’ understanding of it has evolved. In particular, I will explain algebraic geometry plays a key role of establishing the equivalence between K-stability and the existence of a Kähler-Einstein metric, i.e. the Yau-Tian-Donaldson Conjecture, for all Fano varieties. If time permits, I want to also discuss the construction of K-moduli spaces parametrizing Fano varieties, and how the recipe given by K-stability can be used to resolve the issues that mystify people for a long time.