• Speaker: Daniel Soskin, University of Notre Dame
• Title: Determinantal inequalities for totally positive matrices
• Date: 11/07/2022
• Time: 12:30 PM - 1:30 PM
• Place: C304 Wells Hall
• Contact: Linhui Shen (shenlin1@msu.edu)

Totally positive matrices are matrices in which each minor is positive. Lusztig extended the notion to reductive Lie groups. He also proved that specialization of elements of the dual canonical basis in representation theory of quantum groups at q=1 are totally non-negative polynomials. Thus, it is important to investigate classes of functions on matrices that are positive on totally positive matrices. I will discuss two sourses of such functions. One has to do with multiplicative determinantal inequalities (joint work with M.Gekhtman). Another deals with majorizing monotonicity of symmetrized Fischer's products which are known for hermitian positive semidefinite case which brings additional motivation to verify if they hold for totally positive matrices as well (joint work with M.Skandera). The main tools we employed are network parametrization, Temperley-Lieb and monomial trace immanants.

• Speaker: Yuta Hozumi, MSU
• Title: Ensemble Clustering Methods in Transcriptomics Data
• Date: 11/07/2022
• Time: 1:00 PM - 2:00 PM
• Place: C117 Wells Hall (Virtual Meeting Link)
• Contact: Craig Gross (grosscra@msu.edu)

Transcriptomic data, more specifically single cell RNA sequencing (scRNA-seq), is an emerging field in biology that is used to obtain molecular understanding of cells. Analyzing scRNA-seq gives insight to protein and gene regulatory networks, protein expression and diseases. In this talk, I will present ensemble clustering methods used to find clusters in scRNA-seq data, which can then be used for further analysis into differential gene expression, cell trajectory and cell-cell communication. $\\$ 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: Calvin McPhail-Snyder , Duke University
• Title: RTG Seminar: Quantum and hyperbolic invariants of knots (Introductory talk)
• Date: 11/07/2022
• Time: 2:00 PM - 3:00 PM
• Place: C304 Wells Hall (Virtual Meeting Link)
• Contact: Efstratia Kalfagianni (kalfagia@msu.edu)

This talk consists of two related but distinct parts, and should be accessible if you know some algebraic topology and/or differential geometry. The first part is about quantum invariants: I will sketch how to compute the colored Jones polynomials of a knot and discuss their origin in representation theory. The second part is about hyperbolic geometry: I will discuss the basics of hyperbolic knot theory and explain how to compute hyperbolic structures and their volumes using ideal triangulations. The goal is to motivate the volume conjecture discussed in my main talk, which relates the colored Jones polynomials to the hyperbolic volume.

• Speaker: Qingjing Chen, University of California Santa Barbara
• Title: Kuznetsov components of some Fano fourfolds
• Date: 11/07/2022
• Time: 3:00 PM - 4:00 PM
• Place: C304 Wells Hall
• Contact: Laure Flapan (flapanla@msu.edu)

Kuznetsov component A_X of an algebraic variety X is defined to be the right orthogonal of some exceptional collection in the bounded derived category of X. When X is a cubic fourfold or Gushel Mukai fourfold, A_X is a noncommutative K3 surface in the sense that its Serre functor is given by "shifting by 2". Whether or not A_X is equivalent to the bounded derived category of an actual K3 surface is believed to be related to the rationality of the variety X , therefore it has received extensive studies. Yet not many studies seem to answer the question of when the Kuznetsov component of a cubic fourfold is equivalent to that of a Gushel Mukai fourfold, we believe that the answer of this question should be interesting for it will give a part of "Torelli theorem for noncommutative K3 surfaces". In this talk, I will present some partial results which address the previous question.

• Speaker: Brent Nelson, MSU
• Title: Free transport, I
• Date: 11/07/2022
• Time: 4:00 PM - 5:30 PM
• Place: C517 Wells Hall
• Contact: Brent Nelson (banelson@msu.edu)

In operator algebras, specifically free probability, free transport is a technique for producing state-preserving isomorphisms between C* and von Neumann algebras that was developed by Guionnet and Shlyakhtenko in their 2014 Inventiones paper. The inspiration for their work comes from the field of optimal transport, specifically work of Brenier from 1991 who showed that under very mild assumptions one can push forward a probability measure on $\mathbb{R}^n$ to the Gaussian measure. In the non-commutative case, Guionnet and Shlyakhtenko showed that if $x_1,\ldots, x_n$ are self-adjoint operators in a tracial von Neumann algebra $(M,\tau)$ whose distribution satisfies an "integration-by-parts" formula up to a small perturbation, then these operators generate a copy of the free group factor $L(\mathbb{F}_n)$. In this series of talks, I will give an overview of their proof, discuss some applications of their result, and survey the current state of free transport theory.

• Speaker: Calvin McPhail-Snyder , Duke University
• Title: Hyperbolic tensor networks and the volume conjecture
• Date: 11/08/2022
• Time: 3:00 PM - 4:00 PM
• Place: C304 Wells Hall (Virtual Meeting Link)
• Contact: Efstratia Kalfagianni (kalfagia@msu.edu)

Quantum invariants of links like the colored Jones polynomial (which arise from the quantum Chern-Simons theory of Witten-Reshetikhin-Turaev) have a purely algebraic construction in terms of the representation theory of quantum groups. Despite this algebraic nature they appear to be connected to geometry: a class of related volume conjectures assert that their semi-classical asymptotics determine geometric invariants like the hyperbolic volume. To better understand these conjectures a number of authors have studied ways to twist quantum invariants by geometric data. In particular, Blanchet, Geer, Patureau-Mirand, and Reshetikhin recently defined quantum holonomy invariants depending on a link in S^3 and a flat 𝔰𝔩₂ connection on its complement. Their construction uses certain unusual cyclic modules of quantum 𝔰𝔩₂. For technical reasons the invariants are quite difficult to compute. In this talk (based on joint work with Nicolai Reshetikhin) I will explain how to effectively compute them using hyperbolic tensor networks constructed from quantum dilogarithms. Our construction reveals deep connections with hyperbolic geometry and suggests a way to break the Kashaev-Murakami-Murakami volume conjecture into two simpler pieces.

• Speaker: Lubashan Pathirana Karunarathna, MSU
• Title: Limiting Theorems for Compositions of Stationary and Ergodic Random Maps With Applications in Quantum Processes
• Date: 11/09/2022
• Time: 3:00 PM - 3:50 PM
• Place: C405 Wells Hall
• Contact: Dapeng Zhan (zhan@msu.edu)

A discrete parameter quantum process is represented by a sequence of quantum operations, which are completely positive maps that are trace non-increasing. Given a stationary and ergodic sequence of such maps, an ergodic theorem describing convergence to equilibrium for a general class of such processes was recently obtained by Movassagh and Schenker. Under irreducibility conditions, we obtain a law of large numbers that describes the asymptotic behavior of the processes involving the Lyapunov exponent. Furthermore, a central limit-type theorem is obtained under mixing conditions. These results do not require the sequences of quantum operations that describe the quantum process to be trace non-increasing and hence can be applied to a larger class of compositions of random positive maps. In the continuous-time parameter, a quantum process is represented by a double-indexed family of positive map-valued random variables. For a stationary and ergodic family of such maps, we extend the results by Movassagh and Schenker to the continuous case.

• Speaker: Dimitris Vardakis, MSU
• Title: Buffon's needle problem for a random planar disk-like Cantor set
• Date: 11/09/2022
• Time: 4:10 PM - 5:00 PM
• Place: C304 Wells Hall
• Contact: Willie Wai-Yeung Wong (wongwil2@msu.edu)

The Favard length of the planar $1/4$-corner Cantor set is $0$. Estimates exists about the rate with which the Favard length of the previous steps goes to $0$, but the exact rate of decay is unknown. However, if one considers a random construction of the $1/4$-corner Cantor set, things might seem better. In fact, Peres and Solomyak showed that the rate of decay for the average Favard length for the random $1/4$-corner Cantor set is of order exactly $1/n$. We show that the rate of decay for a random disk-like analogue has again order $1/n$. This suggests that any ``reasonable'' random Cantor set of positive and finite length might decay at the same rate.

• Speaker: Keping Huang, MSU
• Title: Classical modular symbols
• Date: 11/10/2022
• Time: 3:00 PM - 4:00 PM
• Place: C204A Wells Hall
• Contact: Keping Huang (huangk23@msu.edu)

No abstract available.