Symplectic and Poisson Geometry Seminar: Beyond Semitoric
A compact four dimensional completely integrable system $f : M \to R^2$ is semitoric if it has only nondegenerate singularities, without hyperbolic blocks, and one of the components of $f$ generates a circle action. Semitoric systems have been extensively studied and have many nice properties: for example, the preimages $f^{1}(x)$ are all connected. Unfortunately, although there are many interesting examples of semitoric systems, the class has some limitation. For example, there are blowups of $S^2 \times S^2$ with Hamiltonian circle actions which cannot be extended to semitoric systems. We expand the class of semitoric systems by allowing certain degenerate singularities, which we call ephemeral singularities. We prove that the preimage $f^{1}(x)$ is still connected for this larger class. We hope that this class will be large enough to include not only all compact four manifolds with Hamiltonian circle actions, but more generally all complexity one spaces. Based on joint work with D. Sepe. 

Number Theory Seminar: Bias in cubic Gauss sums: Patterson's conjecture
We prove, in this joint work with Maksym Radziwill, a 1978 conjecture of S. Patterson (conditional on the Generalised Riemann hypothesis) concerning the bias of cubic Gauss sums. This explains a wellknown numerical bias in the distribution of cubic Gauss sums first observed by Kummer in 1846. There are two important byproducts of our proof. The first is an explicit level aspect Voronoi summation formula for cubic Gauss sums, extending computations of Patterson and Yoshimoto. Secondly, we show that HeathBrown's cubic large sieve is sharp under GRH. This disproves the popular belief that the cubic large sieve can be improved. An important ingredient in our proof is a dispersion estimate for cubic Gauss sums. It can be interpreted as a cubic large sieve with correction by a nontrivial asymptotic main term. This estimate relies on the Generalised Riemann Hypothesis and is one of the fundamental reasons why our result is conditional. 

Graph Theory and Combinatorics Seminar: The Turán number of Berge book hypergraphs
Given a graph G, a Berge copy of G is a hypergraph obtained by enlarging the edges arbitrarily. Gyori in 2006 showed that for r=3 or r=4, an runiform nvertex Berge trianglefree hypergraph has at most floor[n^2/8(r2)] hyperedges if n is large enough, and this bound is sharp. The book graph B_t consists of t triangles sharing an edge. Very recently, Ghosh, Győri, NagyGyörgy, Paulos, Xiao and Zamora showed that a 3uniform nvertex Berge B_tfree hypergraph has at most n^2/8+o(n^2) hyperedges if n is large enough. They conjectured that this bound can be improved to floor[n^2/8]. 

PhD Defense (Felix Clemen): Variations of Turán's Theorem
Abstract: A fundamental result in extremal combinatorics is Turán's theorem: Every K_{r+1}free graph on n vertices has at most as many edges as the complete balanced nvertex rpartite graph. In this talk, we discuss several variations of Turán's theorem. 

Student Cluster Algebra Seminar: Compactifications of cluster varieties and dualities between them
Abstract: Cluster varieties are log CalabiYau varieties which are unions of algebraic tori glued by birational "mutation" maps. In particular, they are blowups of toric varieties. We will show how to generalize the polytope construction of toric varieties to cluster varieties. As an application, we will see that the nonintegral vertex in the NewtonOkounkov body of the Grassmannian comes from the socalled broken line convexity. We will also discuss mirror dualities of cluster varieties from the symplectic perspective. This talk will be based on a series of joint works with BardwellEvans, Bossinger, Hong, Lin, Magee, and NajeraChavez. 

Grad Probability Seminar: Large deviation principles and the wellknown GartnerEllis theorem
In the first part of my talk, I will review large deviation principles and the wellknown GartnerEllis theorem. The main technique is to make a detailed analysis of the log moment generating function and its Fenchel dual. As an application, we recover some large deviation principles we established before. 

Colloquium: Linear Divisibility Sequences
Abstract: The American Journal of Mathematics was America's first mathematics journal. It was founded by J.J. Sylvester in 1878 at Johns Hopkins U. It was planned that each volume would contain 384 pages in 4 parts. In the first volume Sylvester himself wrote 69 pages in three parts on binary forms (and applications to the theory of atomic structure), while Edouard Lucas wrote 88 pages in two parts (in French) on linear recurrence sequences. Lucas' articles placed Fibonacci numbers and other ``wellknown'' linear recurrence sequences into a much broader context. He examined their behaviour locally as well as globally, and he asked several questions that influenced much research in the century and a half to come. In a sequence of papers in the 1930s, Marshall Hall further developed several of Lucas' themes, and in volume 58 (in 1936) studied and tried to classify third order linear divisibility sequences; that is, linear recurrences like the Fibonacci numbers which have the property that $F_m$ divides $F_n$ whenever $m$ divides $n$. Because of many special cases, Hall was unable even to conjecture what a general theorem should look like, and despite developments over the years by various authors, such as Lehmer, Morgan Ward, van der Poorten, Bezivin, Petho, Richard Guy, Hugh Williams,... with higher order linear divisibility sequences, even the formulation of the classification has remained mysterious. In this talk we will present the speaker's ongoing work in classifying all linear divisibility sequences, the key new input coming from an application of the Schmidt/Schlickewei subspace theorem from the theory of diophantine approximation. 

Graduate Analysis Seminar: An introduction to padic analysis
Abstract: This talk is meant to be an introduction padic number and padic analysis. These arise via completing the rational numbers using a metric (relative to a prime number p) which is very different from the ordinary euclidean metric. The first part of the talk will introduce this number system and focus on ways in which it differs from our "Euclidean" intuition. We then will discuss padic integration. We will then define the padic zeta function. The goal will be to describe (without proof) how these are related to the Riemann zeta function. This talk is aimed at 1st year graduate students. 

Graduate Student Homotopy Theory Seminar: Formalism of six operations and derived algebraic stacks
The formalism of six operations was originally introduced by A.Grothendieck and his collaborators in the study of \'etale cohomology. It naturally leads to many wellknown results in cohomology theory like duality and Lefschetz trace formula. This partially justifies the slogan that the formalism of six operations are enhanced cohomology theories. In this talk, I will introduce the formalism of six operations. I will explain the relation between it and some cohomology theories (Topological, coherent, ladic). Moreover, I will talk about the application of it to a nice class of derived algebraic stacks. And show that this leads to some nontrivial results of algebraic (homotopy) Ktheory for stacks. 

Symplectic and Poisson geometry seminar: Nonformal deformation quantization of Poisson manifolds
The Kontsevich formality theorem, established more than 20 years ago, implies that every Poisson manifold has a formal deformation quantization. The existence of nonformal deformations, on the other hand, remains largely an open problem. We study star products defined by semiclassical integral Fourier operators. Our main result states that a Poisson manifold which admits such a star product must be integrable by a symplectic groupoid. This is ongoing joint work with Alejandro Cabrera (UFRJ). 

Number Theory Seminar: The polynomials X^2+(Y^2+1)^2 and X^2 + (Y^3+Z^3)^2 also capture their primes
Abstract: We show that there are infinitely many primes of the form X^2+(Y^2+1)^2 and X^2 + (Y^3+Z^3)^2. Our work builds on the famous FriedlanderIwaniec result on primes of the form X^2+Y^4. More precisely, Friedlander and Iwaniec obtained an asymptotic formula for the number of primes of this form. For the argument we need to estimate Type II sums, which is achieved by an application of the Weil bound, both for pointcounting and for exponential sums over curves. The type II information we get is too narrow for an asymptotic formula, but we can apply Harman's sieve method to establish a lower bound of the correct order of magnitude for the number of primes of the form X^2+(Y^2+1)^2 and X^2 + (Y^3+Z^3)^2. 

Graph Theory and Combinatorics Seminar: Separating the online and offline DPchromatic numbers
The DPcoloring problem, also known as the correspondence coloring problem, is a graph coloring problem invented by Dvořák and Postle that generalizes list coloring. We consider the online and offline versions of the DPcoloring problem, and we show the existence of graphs whose online and offline DPchromatic numbers have arbitrarily large differences. We also show some new results for the online DPchromatic number of bipartite graphs. Joint work with Ladislav Stacho and Bojan Mohar. 

AWM Graduate Colloquium: A model theorist’s tale about the squarefree integers
Abstract: I will speak about joint work with ChieuMinh Tran, a former grad student at UIUC, where we explore the randomness of the squarefree integers and evaluate whether four natural firstorder structures are tame (read domesticable) or not. I will primarily focus on the very accessible numbertheoretic phenomenon we established: the analogue of Dickson’s conjecture for the squarefree integers, and finish by stating the modeltheoretic consequences it entailed. The talk will be accessible to undergraduates, in particular, no prior knowledge of firstorder logic will be assumed. About Neer Bhardwaj: Final year PhD student at Math UIUC, working with the interactions of model theory with algebra and number theory. 

Conversation Series: Eugene Lerman
Join us for the first Conversation Seminar: this will be a series of informal interviews with faculty in the math department, discussing things like one’s career path, challenges, professional inspiration, advice for mathematicians, etc. All department members and visitors are welcome. 

Student Cluster Algebra Seminar: Bethe Ansatz Equation and the Algebraic Bethe Ansatz
This seminar is a continutation of previous topic in quantum integrability. I will give a brief introduction to Bethe ansatz equation and how it is applied to the Heisenberg XXX model. 

Algebra, Geometry & Combinatorics: sCatalan numbers and LittlewoodRichardson polynomials
The sCatalan numbers are a generalization of the wellknown Catalan numbers that were recently introduced after they appeared in certain quantum physics problems. We give a combinatorial description of the sCatalan numbers in terms of LittlewoodRichardson coefficients and explain some of the properties they exhibit in terms of LittlewoodRichardson polynomials. 

Dimension distortion and applications
Abstract: Since Hausdorff dimension was first introduced in 1918, many different notions of dimension have been defined and used throughout many areas of Mathematics. An interesting topic has always been the distortion of said dimensions of a given set under a specific class of mappings. More specifically, Gehring and V\"ais\"al\"a proved in 1973 a theorem concerning the distortion of Hausdorff dimension under quasiconformal maps, while Kaufman in 2000 proved the analogous result for Boxcounting dimension. In this talk, an introduction to the different types of dimensions will be presented, along with the results of of Gehring, V\"ais\"al\"a and Kaufman. We will then proceed to discuss analogous theorems we proved for the Assouad dimension and spectrum, which describe how Kquasiconformal maps change these notions of a given subset of $\mathbb{R}^n$. We will conclude the talk by demonstrating how said theorems can be applied to fully classify polynomial spirals up to quasiconformal equivalence. 

Graduate Student Homotopy Theory Seminar: Introduction to Equivariant Homotopy Theory and RO(G)graded Cohomology
In its simplest form, equivariant homotopy theory is the study of homotopy theory with the addition of actions by a group G. By mixing representation theory into homotopy theory, we create additional structure and complexities to consider. To every representation of our group, we may take the one point compactification and the group will then act on the resulting sphere. In so called “genuine” Gspectra, these representation spheres are invertible, so in particular we may grade cohomology theories on (virtual) Grepresentations. In this introductory talk, I will go over some of the additional properties and structures of equivariant homotopy theory and if time permits illustrate these structures with a cohomology computation. 

Symplectic and Poisson geometry seminar: A new approach to codimension one symplectic foliations
The existence of codimension one foliations on compact manifolds has a nice history, starting with Reeb's foliation on $S^3$ that marked the birth of Foliation Theory, then on general 3manifolds, then Lawson's foliations on $S^5$ and then on all odddimensional sphere, up to the general characterization of Thurston via the Euler characteristic. The analogous question for symplectic foliations is far from being understood it is only rather recently that Lawson's foliation on $S^5$ was turned onto a symplectic one, by Mitsumatsu, but the construction is rather involved. On the other hand, the confoliations of EliashbergThurston revealed very close relationship (via actual deformations) between foliations foliations and contact structure. That theory is well developed only in dimension 3 and that is due, I believe (at least in part), to not noticing that when moving to higher dimensions one should be looking not only at foliations, but at symplectic ones. As a side remark: also Mitsumatsu's construction exploits the geometry of contact forms and adapted open book decompositons. The aim of the talk will be to present a new, we believe much simpler and more conceptual, approach to Lawson's foliation on $S^5$, based on logsymplectic geometry/stable generalized geometry instead of contact geometry. That is joint work with my colleague Gil Cavalcanti. 

Number Theory Seminar: HardyLittlewood problems with almost primes
Abstract: The HardyLittlewood problem asks for the number of representations of an integer as the sum of a prime and two squares. We consider the HardyLittlewood problem where the two squares are restricted to almost primes. This leads to the study of primes in arithmetic progressions to large moduli and automorphic analogue of the Titchmarsh divisor problems. We also consider the number of representations of an integer as the sum of a smooth number and two almost prime squares. This is based on joint work with AssingBlomer and BlomerRydin Myerson. 

Graph Theory and Combinatorics Seminar: Trianglefree graphs: many questions, few answers
We will present problems (and solve some of them) about trianglefree graphs related to Erdős' Sparse Half Conjecture: Every trianglefree graph on n vertices has an induced subgraph on n/2 vertices with at most n^2/50 edges. Among others we prove the following variant of it: For every sufficiently large even integer n the following holds. Every trianglefree graph on n vertices has a partition V(G)=A U B with A=B=n/2 such that e(G[A])+e(G[B]) <= n^2/16. 

Special Colloquium: FokkerPlanck Equations and Machine Learning
Abstract: As the continuous limit of many discretized algorithms, PDEs can provide a qualitative description of algorithm's behavior and give principled theoretical insight into many mysteries in machine learning. In this talk, I will give a theoretical interpretation of several machine learning algorithms using FokkerPlanck (FP) equations. In the first one, we provide a mathematically rigorous explanation of why resampling outperforms reweighting in correcting biased data when stochastic gradienttype algorithms are used in training. In the second one, we propose a new method to alleviate the double sampling problem in modelfree reinforcement learning, where the FP equation is used to do error analysis for the algorithm. In the last one, inspired by an interactive particle system whose meanfield limit is a nonlinear FP equation, we develop an efficient gradientfree method that finds the global minimum exponentially fast. 

Meeting with Dean Patton
Dean Patton will meet with the faculty and staff of Mathematics. 

The Limitations of Diversity/Equity/Inclusion Framework
Most can agree that diversity is good, and in particular, better than its absence. But something is sincerely missing when we devote energy to creating a more diverse community without thinking deeply about what it is our community actually does. For example, I am not in the least bit interested in seeing a woman or a person of color as CEO of Lockheed Martin or Raytheon. In this case, filling a leadership position with a BIPOC/nonmale/nonbinary person not only has no impact on the fundamentally murderous role these companies play in the world, but even worse, it grants cover, distraction, and ligitimacy to the crimes against humanity they routinely commit. For this reason, I believe we need to sincerely ask ourselves: what is it that mathematicians actually do? How does their work and intellectual production impact human beings in the world, and how is their reputation as scientists and experts used by people in power to uphold an unjust status quo? I don't claim to know the answers, but asking these sorts of questions is what animates much of the organizing work of the Just Mathematics Collective. I will talk a bit about this work, our origins as a collective, and our ongoing campaigns. 

GGD/GEAR/Quantum Topology Seminar: Genus bounds from twisted Drinfeld doubles
It is a classical result that the degree of the Alexander polynomial gives a lower bound to the Seifert genus. This theorem does not hold, however, for the Jones polynomial and other quantum knot invariants. In this talk, we will explain how to build quantum knot polynomials that do satisfy a genus bound. Our construction relies on the "twisted" (or equivariant) ReshetikhinTuraev construction specialized at the twisted Drinfeld double of a Zgraded Hopf algebra H, or equivalently, a relative Drinfeld center of a crossed product. When H is an exterior algebra, our invariant generalizes twisted Alexander polynomials, hence our theorem recovers FriedlKim's genus bounds. This is work in progress with Roland van der Veen. 

Student Cluster Algebra Seminar: Trigonometric
This presentation will be the continuation of the last 2 talks where we discuss the idea of integrability. We will discuss another Lax operator arise from the XXZ Heisenberg model and the trigonometric Rmatrix. The set of relations arise from this operator and the YangBaxter equation can be viewed as alternative definition of the qdeformed sl_q(2) algebra. 

Colloquium: Diffusion and diffraction in algebraic topology
Algebraic topologists compute the very basic geometric invariants, such as the genus of a surface or the Euler characteristic of a finite simplicial complex. The most basic invariant of all is the stable homotopy groups of spheres, but these have a wellearned reputation of being a chaotic mess, seemingly without any discernible order. It turns out, however, that with the right diffraction principle and a good deal of hard computation, we can separate out some very intriguing largescale patterns. What these patterns are and how they should fit together is the subject of a number of conjectures, some of which have seen remarkable progress in the past few years. In this talk I’ll display the patterns, lay out the conjectures, and talk about the recent progress, touching on the work of a whole host of people, many of whom live and work here in the Midwest. 

Traveling wave solutions for discrete and nonlocal diffusive LotkaVolterra system
Abstract: We will introduce a biological model called LotkaVolterra competition system. We construct an Nshaped constrained region. By using this region we give an a priori estimate for this system and show that the total population of this 2species system has a nontrivial lower bound. Finally, we know that one of the important issues in ecology is biodiversity. By using this estimate, we find necessary conditions for coexistence of 3species LotkaVolterra system. 

Actuarial Science and Financial Mathematics Seminar: Fairness and Discrimination in Actuarial Pricing
Abstract: This talk will be based on two recent papers. In the first one "The Fairness of Machine Learning in Insurance: New Rags for an Old Man?" (coauthor Laurence Barry), we present an overview of issues actuaries face when dealing with discrimination. Since the beginning of their history, insurers have been known to use data to classify and price risks. As such, they were confronted early on with the problem of fairness and discrimination associated with data. This issue is becoming increasingly important with access to more granular and behavioural data, and is evolving to reflect current technologies and societal concerns. By looking into earlier debates on discrimination, we show that some algorithmic biases are a renewed version of older ones, while others seem to reverse the previous order. Paradoxically, while the insurance practice has not deeply changed nor are most of these biases new, the machine learning era still deeply shakes the conception of insurance fairness. In the second one "A fair pricing model via adversarial learning" (coauthors Vincent Grari, Sylvain Lamprier and Marcin Detyniecki), we suggest a technique to construct a fair pricing model using maximal correlation based techniques and adversarial learning. About: Arthur Charpentier, PhD, is professor at UQAM, Montreal, Canada, and Fellow of the French Institute of Actuaries. He is member of the editorial board of the ASTIN Bulletin, author of several research articles and editor of Computational Actuarial Science with R. He is also the former director of the Data Science for Actuaries program of the French Institute of Actuaries. 

Symplectic and Poisson geometry seminar: Homotopic Stability Chambers in Blowup Ruled Symplectic Surfaces
This talk focuses on homotopic properties of symplectomorphism groups of blow ups of ruled surfaces. The scaffolding that allows to establish homotopic stability chambers inside their symplectic cones expands from existing literature results on their minimal counterparts. We extend inflation and Jholomorphic techniques (and other homotopic tools) to spaces of nonrational, nonminimal symplectic ruled surfaces. We will explain new results regarding $\pi_*$ of the symplectomorphism groups and discuss future results. The results are in collaboration with Jun Li.


Graph Theory and Combinatorics Seminar: Enumerating Min Cutsets and kcutsets in Hypergraphs
Given a hypergraph H=(V,E), we consider two problemsHypergraph Mincut, where the goal is to find a 2partition of V that minimizes the number of hyperedges crossing the 2partition, and Hypergraph kcut, where the goal is to find a kpartition of V that minimizes the number of hyperedges crossing the kpartition. Our goal is to enumerate all optimum solutions in polynomial time. For each problem, the number of optimum solutions could be exponential, but the number of hyperedge sets corresponding to optimum solutions (denoted mincutsets and minkcutsets, respectively) is polynomial (k is a fixed constant for Hypergraph kcut). Our algorithms enumerate mincutsets and minkcutsets (instead of 2partitions and kpartitions, respectively), relying on structural results that relate optimum mincutsets and minkcutsets to min (S,T)cut for small vertex sets S and T. This is joint work with Calvin Beideman and Karthekeyan Chandrasekaran. 

Probability Seminar: Mean field spin glass models under weak external field
Abstract: We study the fluctuation and limiting distribution of free energy in meanfield spin glass models with Ising spins under weak external fields. We prove that at high temperature, there are three subregimes concerning the strength of external field $h \approx \rho N^{\alpha}$ with $\rho,\alpha\in (0,\infty)$. In the supercritical regime $\alpha < 1/4$, the variance of the logpartition function is $\approx N^{14\alpha}$. In the critical regime $\alpha = 1/4$, the fluctuation is of constant order but depends on $\rho$. Whereas, in the subcritical regime $\alpha>1/4$, the variance is $\Theta(1)$ and does not depend on $\rho$. We explicitly express the asymptotic mean and variance in all three regimes and prove Gaussian central limit theorems. Our proofs mainly follow two approaches. One utilizes quadratic coupling and Guerra's interpolation scheme for Gaussian disorder, extending to many other spin glass models. However, this approach can prove the CLT only at very high temperatures. The other one is a clusterbased approach for general symmetric disorders, first used in the seminal work of Aizenman, Lebowitz, and Ruelle~(Comm.~Math.~Phys.~112 (1987), no.~1, 320) for the zero external field case. It was believed that this approach does not work if the external field is present. We show that if the external field is present but not too strong, it still works with a new cluster structure. In particular, we prove the CLT up to the critical temperature in the SherringtonKirkpatrick (SK) model when $\alpha \ge 1/4$. We further address the generality of this clusterbased approach. Specifically, we give similar results for the multispecies SK model and diluted SK model. Based on joint work with Partha S. Dey. 

Conversation Series: Sue Tolman
This is a series of informal interviews with faculty in the math department, discussing things like one's career path, challenges, professional inspiration, advice for mathematicians at various career stages, etc. All department members and visitors are welcome! 

Graduate Student Homotopy Theory Seminar: Applications of Dualizing Complex in Commutative Algebra
Starting with a finitely generated module of finite projective dimension over the completion of a Noetherian local ring A, a natural question is when does this module descend, i.e. when is this module the completion of a finite Amodule of finite projective dimension? We will need the theorem of local duality to show it happens over some “good” rings. We will introduce dualizing complex both in the language of derived categories, and in an explicit form for commutative Noetherian ring. We will apply them to study some descent problems for module of finite projective dimension. The techniques employed also allow one to recover a theorem of Horrocks about vector bundles over a punctured spectrum of a local ring. We follow Section I.5 in Dimension projective finie et cohomologie locale by Peskine and Szpiro. 

Spring 2022 Actuarial Science and Financial Mathematics Seminar: Hedging with Automatic Liquidation and Leverage Selection on Bitcoin Futures
Abstract: Bitcoin derivatives positions are maintained with a selfselected margin, often too low to avoid liquidation by the exchange, without notice, during periods of excessive volatility. Recently, the size and scale of such liquidations precipitated extreme discontent among traders and numerous lawsuits against exchanges. Clearly, hedgers of bitcoin should account for the possibility of automatic liquidation. That is the mathematical and operational problem that we address, deriving a semiclosed form for an optimal hedging strategy with dual objectives – to minimize both the variance of the hedged portfolio and the probability of liquidation due to insufficient collateral. An empirical analysis based on minutelevel data compares the performance of major direct and inverse bitcoin hedging instruments traded on five major exchanges. The products have markedly different speculative trading scores according to new metrics introduced here. Instruments having similar hedging effectiveness can exhibit marked differences in speculative activity. Inverse perpetuals offer greater effectiveness than direct perpetuals, which also exhibit more speculation. We model hedgers with different levels of loss aversion that select their own level of leverage and collateral in the margin account. By following the optimal strategy, the hedger can reduce the liquidation probability to less than 1% and control leverage to a reasonable level, mostly below 5X. Based on a joint work with Carol Alexander (University of Sussex) and Jun Deng (University of International Business and Economics). About: Bin Zou is currently an assistant professor in the Department of Mathematics at the University of Connecticut. Prior to joining UConn in 2017, he was an acting assistant professor at the University of Washington from 2016/9 to 2017/8 and a postdoctoral fellow at the Technical University of Munich from 2015/5 to 2016/8. He obtained his PhD in Mathematical Finance from the University of Alberta in 2015. His main research interests are stochastic control and optimization, with applications in actuarial science and financial mathematics, and recent interests include cryptocurrency markets and sports betting. Please visit his website for more information: https://sites.google.com/site/zoubin019/. 

Symplectic and Poisson geometry seminar: Integration of Courant algebroids
In this talk I will introduce 2shifted symplectic Lie 2groupoids and construct explicit examples that integrate some particular kind of Courant algebroids. This is joint work with Chenchang Zhu. 

Number Theory Seminar: A spectral theory approach to the Apollonian counting problem
Abstract: A classic problem in number theory and group theory is to count the number of points in a group orbit satisfying a particular cutoff. In 1982, Lax and Phillips used spectral theory to obtain accurate counts of such orbits in a particular context. In 2009 this method was generalized by Kontorovich to a wider class of counting problems, however this included a restriction on the groups considered. In this talk I will present this spectral approach to counting, how to remove the restriction, and how to generalize these problems. In doing so we will see how to apply these methods to the Apollonian counting problem. This is joint work in progress with Alex Kontorovich. 

Graph Theory and Combinatorics Seminar: On the Maximum F_5free Subhypergraphs of G^3(n,p)
Denote by F_5 the 3uniform hypergraph on vertex set {1,2,3,4,5} with hyperedges {123,124,345}. Balogh, Butterfield, Hu, and Lenz proved that if p > K log n/n for some large constant K, then every maximum F_5free subhypergraph of G^3(n,p) is tripartite with high probability, and showed if p_0 = 0.1\sqrt{log n}/n, then with high probability there exists a maximum F_5free subhypergraph of G^3(n,p_0) that is not tripartite. We sharpen the upper bound to be best possible up to a constant factor. We prove that when p > C\sqrt{log n}/n for some large constant C, every maximum F_5free subhypergraph of G^3(n,p) is tripartite with high probability. In this talk, I will introduce the main technique we use to improve this bound. This is a joint work with Igor Araujo and Jozsef Balogh. 

Algebra, Geometry & Combinatorics: Minimal equations for matrix Schubert varieties
Matrix Schubert varieties are affine varieties that arise in the study of the complete flag varieties. They are the orbit closures of permutation matrices under B_{}*B action and are the vanishing sets of the Schubert determinantal ideals. We gave a minimal list of equations that define matrix Schubert varieties along with some implications. This is based on joint work with Alexander Yong. 

AWM Graduate Colloquium: The Graph Reconstruction Problem
This talk is an introduction to, as well as an overview of, the reconstruction problem in graphs. This problem seeks to gain information about various properties of a graph we do not know, by observing a multiset of some of its subgraphs. In addition to the history of the problem, some new developments will be mentioned, including joint works with Professors Kostochka and West, and Dara Zirlin, a former math graduate student at UIUC. No prior knowledge of graph theory is required, and the talk is accessible to undergraduates. 

Zomback Thesis Defense: Pointwise ergodic theorems via descriptive set theory


GGD/GEAR Seminar: PSL(2,C) representations of knot groups
I will discuss a method of producing defining equations for representation varieties of the canonical component of a knot group into PSL2(C). This method uses only a knot diagram satisfying a mild restriction and is based upon the underlying geometry of the knot complement. In particular, it does not involve any polyhedral decomposition or triangulation of the link complement. This is joint work with Anastasiia Tsvietkova. 

Conversation Series: Bruce Reznick
This is a series of informal interviews with faculty in the math department, discussing things like one's career path, challenges, professional inspiration, advice for mathematicians at various career stages, etc. All department members and visitors are welcome! 

Analysis Seminar: An operator system approach to quantum correlations
In this talk, I will explain a novel approach to Tsirelson's problem using the theory of operator systems. Tsirelson's problem relates to whether the commuting operator model of quantum mechanics produces different statistics than the tensor product model of quantum mechanics in nonlocal measurement scenarios. These questions have been shown to be equivalent to Connes' embedding problem from the theory of Von Neumann algebras. After tremendous effort by physicists, mathematicians, and computer scientists, Tsirelson's problem was finally resolved in a recent paper. Nevertheless, interest in understanding Tsirelson's problem in greater detail remains. After exploring some background in the theory of operator systems, I will explain how to characterize quantum correlations using only abstract operator system theory, building upon existing C*algebraic and operator theoretic characterizations in the literature. This new characterization yields an equivalent restatement of Tsirelson's problem in the language of abstract operator systems. 

Algebra, Geometry & Combinatorics: A Polytopal View of Schubert Polynomials
Schubert polynomials are polynomial representatives of cohomology classes of subvarieties of the flag manifold. Despite the many combinatorial formulas developed for them over the years, there remain many mysteries surrounding these polynomials. I will describe Schubert (and related) polynomials with a focus on discrete geometry. From this perspective, I will address questions such as vanishing of Schubert coefficients, relative size of coefficients, and interesting properties of the support of a Schubert polynomial. I will discuss some extensions to Grothendieck polynomials, Ktheory analogues of Schubert polynomials. 

Risk Analytics Symposium 2022: Actuarial Perspectives on Climate Change
Registration is now available for the Risk Analytics Symposium 2022: Actuarial Perspectives on Climate Change! Register today for this opportunity to learn about how professionals from the insurance industry and the academic community are working together to better understand the risk presented by climate change. This event will be held on March 25, 2022 from 10:00 AM to 2:00 PM (CT) at the University of Illinois in Urbana, Champaign in Huff Hall Room 112. There will be a virtual option available via Zoom to attend live panel discussions and network with expert speakers, peers, and future actuaries!
Any practicing actuaries with their ASA or FSA who attend will have the opportunity to earn up to 4.8 SOA and AAA CPD credits!


Graduate Student Homotopy Theory Seminar: An introduction to 2categories
This talk will be a survey of some of the basic notions and facts about 2categories. After introducing 2categories and various types of functors between them, we will consider constructions that map 2categories to more familiar objects. On one hand, viewing a 2category as a “poor” higher category, we can “reduce” it to an ordinary category, via a homotopy construction. On the other hand, viewing a 2category as a “rich” ordinary category, we can “uplift” it to a simplicial set, via the duskin nerve. I will talk about these and related ideas, indicate some quirks of the theory, and provide examples along the way. 

Symplectic and Poisson geometry seminar: On the symplectic equivalences for parabolic orbits of integrable Hamiltonian systems
Parabolic orbits are the simplest examples of degenerate singularities of integrable two degree of freedom Hamiltonian systems. Yet until recently, their symplectic (and even C^\infty smooth) classification was not known. We fill in this gap and show that the action variables corresponding to such an orbit form a complete set of symplectic invariants (up to the fiberwise symplectic equivalence). This generalises an earlier result by A.V. Bolsinov, L. Guglielmi and E.A. Kudryavtseva proving this in the analytic category. The smooth case that we present is more complicated and has useful consequences for more global symplectic classification problems. We shall also discuss a new classification result for parabolic orbits in the analytic category; specifically, we shall give a simple normal form for such orbits up to the right symplectic equivalence. This type of equivalence is different, but closely related to the usual fiberwise (or rightleft) symplectic equivalence and interestingly enough, the normal form that we obtain is not given in terms (of the asymptotics) of the action variables.
This talk is based on a recent work with Prof. E.A. Kudryavtseva.


Thesis Defense: Yuji Yang


Number Theory Seminar: On the RankinSelberg problem
Abstract: In this talk, I will introduce a method to solve the RankinSelberg problem on the second moment of Fourier coefficients of a GL(2) Hecke eigenform. This improves the classical result 3/5 of Rankin and Selberg (in 1939/1940). If time permit, a sketch of proof and some other results will also be presented. 

Graph Theory and Combinatorics Seminar: Strong complete minors in digraphs
Kostochka and Thomason independently showed that any graph with average degree Omega (r sqrt{log r}) contains a K_r minor. We consider strong minors in digraphs and ways to force them. A strong →K_r minor is a digraph whose vertex set is partitioned into r parts such that each part induces a stronglyconnected subdigraph, and there is at least one edge in each direction between any two distinct parts. We show that any tournament with dichromatic number at least 2r contains a strong →K_r minor, and any tournament with minimum outdegree Omega(r sqrt{log r}) also contains a strong →K_r minor. The latter result is tight up to the implied constant, and may be viewed as a strongminor analogue to the classical result of Kostochka and Thomason. Lastly, we show that there is no function f: N to N such that any digraph with minimum outdegree at least f(r) contains a strong →K_r minor, but such a function exists when considering dichromatic number. This is a joint work with Antonio Girao, Richard Snyder, and Lea Weber. 

AWM Teaching and Diversity Seminar
Note: this talk is open to current and prospective graduate and undergraduate students only. 

GGD/GEAR Seminar: Hyperbolic manifolds and their embedded totally geodesic submanifolds
The study of embedded surfaces in hyperbolic 3manifolds has led to several major advances in the fields of geometry, topology, and geometric group theory. In this talk we address the higher dimensional analogue of embedded codimension1 submanifolds in finite volume hyperbolic manifolds, with a focus on the existence of totally geodesic 3manifolds in small volume hyperbolic 4manifolds. 

Student Cluster Algebra Seminar: Enumeration of tilings and Kuo condensation
Enumeration of tilings is the mathematical study concerning the total number of ways to cover a certain region by similar pieces so that there are no gaps or overlaps. The study dates back to the early 1900s when MacMahon's proved his classical theorem about boxed plane partitions, which are in bijection with lozenge tilings of a hexagon. Enumeration of tilings has become a vibrant subfield in algebraic and enumeration combinatorics with applications and connections to various areas, including symmetric functions, statistical mechanics, cluster algebra, and probability. We also talk about a powerful method of enumerating tilings, namely Kuo's graphical condensation. 

Algebra, Geometry & Combinatorics: An SOS counterexample to an inequality of symmetric polynomials
Sum of squares (SOS) relaxations are often used to certify nonnegativity of polynomials and are equivalent to solving a semidefinite program (SDP). The feasible region of the SDP for a given polynomial is the Gram Spectrahedron. For symmetric polynomials, there are reductions to the problem size that can be done using tools from representation theory. This gives rise to a smaller, more manageable spectrahedron, the Symmetry Adapted Gram Spectrahedron. With this machinery, we disprove a 2011 conjecture about the complete homogeneous symmetric polynomials. Specifically, we find an SOS counterexample to the claim that H_lambda <= H_mu if and only if mu dominates lambda in partition dominance order. 
