4/1/22 Through 5/1/22

Graduate Student Homotopy Theory Seminar: Splitting $BP<2> ⋀ BP<2>$ at primes $p \ge 5$

In the 1980s, Mahowald and Kane used Brown-Gitler spectra to construct splittings of $bo ⋀ bo$ and $l ⋀ l$. These splittings helped make it feasible to do computations using the $bo$- and $l$-based Adams spectral sequences. In this talk, we will construct an analogous splitting for $BP<2> ⋀ BP<2>$ at primes $p \ge 5$.

Undergraduate Friday Seminar: Shapes of numbers

This week, on Friday Apr. 1st at 4pm in room 245 of Altgeld Hall, we'll be joined by Dr. Vesna Stojanoska! Dr. Stojanoska is an associate professor here at UIUC doing research in homotopy theory and some of its applications, and has managed an IGL project on the same topic for the past three semesters. On Friday, she'll be here to talk about a connection from partitions of natural numbers to structures in algebraic topology:

Take a natural number n, and look at all possible ways to break up the numbers from 1 to n into several groups. These are called partitions. Two partitions are linked if one can be obtained from the other by breaking up some groups.

Three partitions are linked if one is obtained from the other by breaking up, and the third is obtained by breaking up further. Similarly, we can define when any number of partitions are linked. Studying these linkage properties gives rise to a shape associated to the number n, whose symmetries know about deep structures in algebraic topology.

As usual, there will be free pizza courtesy of the IGL! Hope to see you there!

Symplectic and Poisson geometry seminar: A diffeological approach to solving the integration problem

In the theory of Lie groupoids and Lie algebroids, there is a procedure for differentiting a Lie groupoid to result in a Lie algebroid. This process is very much analogous to the construction of a Lie algebra from a Lie group. From this analogy, it is reasonable to ask whether or not it is possible to construct a Lie groupoid given the data of a Lie algebroid in much the same manner tha you do for Lie algebras. In fact, it turns out tha this is not possible due to the fact tha integration procedure results in something tha is not quite a manifold. In this talk I will discuss an approach to patching this problem using diffeological spaces which are a generalization of smooth manifolds.

PhD Defense: Dana Neidinger

Graduate Commutative Algebra and Algebraic Geometry Seminar: An Introduction to the New Improved Intersection Conjecture

Starting with Peskine and Szpiro's intersection theorem, we will introduce several homological conjectures/theorems and the connections between them. Peskine Szpiro's intersection theorem tells us about the codimension of the support of a coherent sheave of finite projective dimension. It also settled some earlier homological conjectures including Auslander’s Zero Divisor Conjecture and Bass’s Conjecture. We will introduce several later homological conjectures which are meant to generalize Peskine Szpiro intersection theorem, including the New Improved Intersection Conjecture. 

We followed the notes by Paul Roberts here: https://www.math.utah.edu/vigre/minicourses/algebra/roberts.pdf

Graph Theory and Combinatorics Seminar: Q-ary generalizations of set intersection and extremal graph theoretic problems

Characteristic vectors of subsets of an n-element ground set give a natural 1-to-1 correspondence between set systems and 0-1 vector systems. As the size of the intersection of two sets equals the scalar product of their characteristic vectors, this correspondence is often used in proofs of intersection theorems of finite sets. There exist several definitions of intersection for vectors of length n with entries from {0,1,...,q}. In this talk, we will propose a new one: the size of the s-sum intersection of two such vectors u,v is the number of coordinates where the entries have sum at least s, i.e. |{i: u_i+v_i\ge s}|. We address analogs of the following classical results in this setting: the Erdos-Ko-Rado theorem and the theorem of Bollob\'as on intersecting set pairs. We will also define an s-sum analog of graph Turan problems and survey results concerning them.

Joint work with Zsolt Tuza and Mate Vizer.

HADES Seminar: Uniform Asymptotic Stability for Convection-Reaction-Diffusion Equations in the Inviscid Limit Towards Riemann Shocks

Click here for abstract

Conversation Series: Chris Dodd

This is the next one in 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 are welcome!

(Email Vesna for zoom link.)

Special Colloquium for Students: Game-theoretic analysis of Guts Poker

We present the results of a summer 2021 REU project with students Luca Castronova and Yijia Chen on a game theoretic analysis  of the poker variant "Guts''.  This is interesting as a streamlined poker version that is widely played- also due to the feature of repeated play with increasing stakes, i.e., its nature as a noncontractive recursive game.  Also interesting is that we carry out the analysis for arbitrary numbers of players n, finding an explicit symmetric Nash equilibrium, but showing that it is not a strong equilibrium: that is, it can be beaten by a coalition of players 2-n.

Turing bifurcation in systems with conservation laws

Abstract: Generalizing results of Matthews-Cox/Sukhtayev for a model reaction-diffusion equation, we derive and rigorously justify weakly nonlinear amplitude equations governing general Turing bifurcation in the presence of conservation laws. In the nonconvective, reaction-diffusion case, this is seen similarly as in Matthews-Cox, Sukhtayev to be a real Ginsburg-Landau equation weakly coupled with a diffusion equation in a large-scale mean-mode vector comprising variables associated with conservation laws. In the general, convective case, by contrast, the amplitude equations consist of a complex Ginsburg-Landau equation weakly coupled with a singular convection-diffusion equation featuring rapidly-propagating modes with speed $\sim 1/\eps$ where $\eps$ measures amplitude of the wave as a disturbance from a background steady state. Applications are to biological morphogenesis, in particular vasculogenesis, as described by the Murray-Oster and other mechanochemical/hydrodynamical models. This work is joint with Kevin Zumbrun.

Graduate Student Homotopy Theory Seminar: Towards a universal property of the ∞-equipment of enriched (∞,1)-categories

One way or another, enriched 1-category theory has held an important spot in the study of homological and homotopical phenomena practically since the very start of ordinary category theory. For many purposes, enriched 1-categories or their model 1-categorical counterparts are simply too rigid, or they might not even exist at all. In recent years various models of enriched (∞,1)-categories have been introduced, and some comparisons at differing levels have been made e.g. the underlying parameterizing ∞-operads or their ∞-categories (with a closed left action over Cat_∞). We are interested in a universal property that can compare these theories at a level which can detect pointwise Kan extensions for example. Part of one approach to this involves upgrading the underlying machinery appearing in Gepner and Haugseng to the scaled simplicial setting. This talk will be heavily focused on examples and justifying why we would want such theories anyway. The only prerequisite is some knowledge of enriched 1-category theory and an appetite for homotopy theory. Time permitting, we may discuss the situation with enriched (∞,1)-operads and (∞,1)-properads, or other possible uses of intermediate results.

Symplectic and Poisson geometry seminar: Quasifold groupoids and diffeological quasifolds

A quasifold is a space that is locally modeled by quotients of R^n by countable group actions. These include orbifolds and manifolds. We approach quasifolds in two ways: by viewing them as diffeological spaces, we form the category of diffeological quasifolds, and by viewing them as Lie groupoids (with bibundles as morphisms), we form the category of quasifold groupoids. We show that, restricting to effictive groupoids, and locally invertible morphisms, these two categories are equivalent. In particular, an effective quasifold groupoid is determined by its diffeological orbit space. This is join work with Yael Karshon.

Number Theory Seminar: The sharp Eros-Turan inequality

Abstract: Erdos and Turan prove a classical inequality for complex polynomials, which says if the polynomial attains small value on the unit circle after normalization, then all zeros will cluster around the unit circle and moreover become equidistributed in angles. The optimal constant remained the only component that is not sharp for this inequality. We will explain how tools in potential theory and energy minimization enter this question, and how they help us in characterizing the extremal distribution of zeros and proving the optimal constant. This is a joint work with Ruiwen Shu. 

Graph Theory and Combinatorics Seminar: Chromatic polynomial and counting list and DP-colorings of graphs

Graduate Student Commutative Algebra and Algebraic Geometry Seminar: An Informal Introduction to Formal Groups

Infinitesimal objects in algebraic geometry have a rich structure, but they are difficult to study due to the failure of classical Lie theory in the algebraic context, especially in characteristic p. This failure can be attributed to the fact that Lie algebras only capture first-order infinitesimal behavior, a limitation which vanishes when we shift our focus to a new kind of infinitesimal object: formal groups. In this talk, I will describe the basic theory and examples of formal groups, as well as how they give rise to a surprising and deep connection between algebraic geometry and algebraic topology: chromatic homotopy theory.

Combinatorics Colloquium: Tiling in graphs

Abstract: Given two graphs H and G, an H-tiling in G is a collection of vertex disjoint copies of H in G. Thus, an H-tiling is simply a generalisation of the notion of a matching (which corresponds to the case when H is an edge). An H-tiling in G is perfect if every vertex of G is covered by the H-tiling.

Over the last 60 years there have been numerous results on perfect H-tilings. In this talk we give a high-level overview of some of the key ideas that permeate the topic. In particular, we will discuss some typical behaviour of extremal examples, and also some complexity questions.

Department Colloquium: Quantifying Lagrangian rigidity

Lagrangian submanifolds are ubiquitous in symplectic geometry, and Lagrangian intersection or isotopy results underlie much of symplectic rigidity. On the other hand, given enough space, intersections can be eliminated and Lagrangians unknotted. Quantitative symplectic topology aims to determine how much space is required.

We will describe results on which Lagrangians can be moved into a fixed region under a Hamiltonian isotopy, and whether corresponding embeddings are knotted. Finally we discuss the size of Lagrangian complements, which have full measure but may not admit symplectic embeddings of large balls. Most of this is joint works with Ely Kerman, Emmanuel Opshtein and Jun Zhang.

Candidate Presentation, Canary Professorship

Integrable probability: Random matrices at high and low temperatures

I will start by outlining what integrable probability is and then demonstrate its principles on a set of questions related to the random matrix theory.

The questions concern the dependence of eigenvalue distributions of random matrices on the parameter Beta, which takes values 1, 2, or 4, depending on whether we deal with real, complex, or quaternionic matrices.  In the terminology of statistical mechanics, Beta is inverse-proportional to the temperature in the system and, as I will explain, this parameter can also take arbitrary positive real values. In the talk we will discuss a rich asymptotic theory as Beta tends to zero or to infinity.

Graduate Student Homotopy Theory Seminar: You Already Care About ∞-Topoi

One of the most important roles played by topological spaces is being a base for geometry, i.e. “something to have sheaves on”. As is often the case, however, this classical notion falls short when it comes to describing homotopical geometry. The correct generalization is that of an ∞-topos. In this talk, I will describe the theory of ∞-topoi, how they generalize classical objects from topology and geometry, and several applications. No prior knowledge of 1-topoi or presentable ∞-categories will be assumed.

Actuarial Science and Financial Mathematics Seminar: Cyber risk management on power grids

Abstract: Cyber risks have been posing increasing concerns to the public and thus stimulate high demands on cyber insurance products. However, the development of the cyber insurance market has not been matching the demands, due to two main obstacles. First, the nature of cyber risks is still unknown to insurers because of the lack of data. Second, the potential dependence among cyber risks causes the concern of insolvency risk and calls for new pricing models. In this talk, we focus on cyber risks on power grids. Through simulations from physical models, we are able to generate loss data and uncover some basic characteristics of cyber risks. In particular, the simulation study confirms that dependence generally exists among cyber risks. In order to address insurer’s concern of insolvency raised by dependence, we propose three approaches: global pricing, issuing catastrophe bonds, and developing mutual insurance schemes. According to preliminary investigations, each approach exhibits its own advantages and disadvantages. In the long term, we hope to integrate the advantages of these approaches into a unified actuarial framework to advance the management of cyber risks on power grids as well as other general cyber risks.

About: Wei Wei is an Associate Professor in Actuarial Science at the University of Wisconsin-Milwaukee. He is an associate member of the Society of Actuaries. Before joining the University of Wisconsin-Milwaukee, he obtained his PhD degree in Actuarial Science from the University of Waterloo. His research interest lies in cyber risks management, dependence modeling, stochastic compassion, and decision making under risk aversion and ambiguity. His researches have been supported by the National Science Foundation and the Society of Actuaries.

Academic Program Review

Norms and Transfers in Motivic Homotopy Theory

Motivic homotopy theory is the study of homotopy-theoretic ideas in the setting of algebraic geometry. The basic categories of interest are those of motivic spaces $\mathcal{H}(S)$ and motivic spectra $\mathcal{SH}(S)$ over a base scheme $S$. In recent work of Bachmann--Hoyois, these categories were equipped with norm monoidal structures, variants of monoidal structures richer than what is usually the richest for homotopy theory (i.e. $\mathbb{E}_\infty$). In this talk, I will discuss norm monoidal structures on various extensions of motivic homotopy theory where the spaces/spectra are equipped with (generalized) transfers. The construction of norms for motivic spaces with framed transfers will allow us to prove a norm monoidal enhancement of the motivic infinite loop space recognition principle of Elmanto--Hoyois--Khan--Sosnilo--Yakerson. We'll also discuss the interactions of norms with other flavors of transfer.

Number Theory Seminar: Congruences for r-colored partitions

Click here for abstract (pdf)

Graph Theory and Combinatorics Seminar: r-cross t-intersecting families via necessary intersection points.

Given integers r\geq 2 and n,t\geq 1 we call families \mathcal{F}_1,...,\mathcal{F}_r\subseteq 2^[n] r-cross t-intersecting if for all F_i in \mathcal{F}_i, i in [r], we have \bigcap_{i in [r]}F_i\geq t. We obtain a strong generalisation of the classic Hilton-Milner theorem on cross intersecting families. In particular, we determine the maximum of \sum_{j in [r]} |\mathcal{F}_j| for r-cross t-intersecting non-empty families in the cases when these are k-uniform families (and n\geq 3k-t) or arbitrary subfamilies of 2^[n]. We obtain the aforementioned theorems as instances of a more general result that considers measures over the families. This also provides the maximum of \sum_{j in [r]}|\mathcal{F}_j| for families of possibly mixed uniformities k_1,...,k_r.

Graduate Commutative Algebra & Algebraic Geometry Seminar: Introduction to Geometric Invariant Theory

One of the classical ways to construct moduli space in algebraic geometry is to use group action. This theory was developed by Mumford. This talk is the first of a series of talks on this topic. These talks will basically follow Mumford’s book Geometric Invariant Theory, but more examples and details of construction will be provided. In the first talk, I will introduce group action on schemes, quotients, linearization of a group action, stability, and 1-parameter subgroups. Anyone who has taken Math512 should be comfortable understanding the contents. Cookies will be provided as usual.

Student Cluster Algebra Seminar: M-path on tagged arc

Every cluster variable on the surface can be expressed as cluster expansion of the labeled seed. M-path is one way to calculate the cluster expansion of given arc using the multiplication of 2 × 2 matrices. I will extend this idea of M-path to find the cluster expansion of the tagged arcs. We will define a loop graph, which can be understood as a combination of a generalized snake graph and a band graph, and then define M-path on this graph.

Candidate Presentation, Canary Professorship

Title: Geometry and the complexity of matrix multiplication

Abstract: In 1968 V. Strassen discovered that the usual row-column method for multiplying matrices is not optimal. After much work, it is now generally conjectured that as the size of the matrices grows large, it becomes nearly as easy to multiply two matrices as it is to add them! I will give a history of this astounding conjecture.

It has been approached using methods from combinatorics, probability, statistical mechanics, and other areas. I will primarily discuss how the conjecture is naturally approached as a problem in algebraic geometry and representation theory.

Actuarial Science and Financial Mathematics Seminar: Predictive modeling and what it means to you

Meeting ID: 792 221 9559
Password: 043910

Abstract: In this talk, we are going to on predictive modeling and its applications.  We are trying to answer some of the questions that you may have on predictive models:

1.       What is predictive modeling?

2.       What we can do with predictive models in business?

3.       What methods we often use to build predictive models?

4.       What kinds of data we often use?

5.       What kinds of job functions you can think of with data & analytics?

6.       What attributes and skills you can develop for your future career in industry?

About: Shawn Jin leads Data Science team to support rapid growth of the home product in Plymouth Rock Home Group (Bunker Hill Insurance) through advanced analytics with big data, supporting business strategies around product, underwriting, marketing, claims and renewals. Shawn has 25 years of experience in solving business problems using big data and analytics. Before came to Plymouth Rock, Shawn held various leadership positions in AIG, McKinsey, Targetbase, Merkle and CapitalOne.  Shawn received BS in Mathematical Statistics from University of Science and Technology of China and MS/PhD of Statistics from Purdue University.

Graduate Student Homotopy Theory Seminar: Rational Homotopy Theory

Rational homotopy theory is homotopy theory modulo torsion. This simplification reduces topology to algebra. More precisely, Quillen proved that the rational homotopy theory of 2-connected spaces is equivalent to that of (1) 1-connected dg Lie algebras (2) 2-connected dg cocommutative coalgebras. This is subsequently augmented by Sullivan, who provides a dg commutative algebra model of rational homotopy theory with computational strength. Time permitting, I will also discuss interesting applications to geometry and local algebra.

Symplectic and Poisson geometry seminar: Lifting Complexity-1 Spaces to Toric Manifolds

A toric manifold is a 2n dimensional compact connected symplectic manifold equipped with an n-dimensional torus acting effectively in a Hamiltonian manner. In 1980s, Delzant completely classified toric manifolds up to equivariant symplectomorphism by their moment images (Delzant polytopes). Given a toric manifold, we can take an (n-1)-dimensional subtorus and restrict our attention to the action of the subtorus. These spaces are important examples of complexity-1 space. A natural question to ask is: given a complexity-1 space, is there a way to lift it to a toric manifold? In this talk, I will first talk about complexity-1 spaces and present the explicit construction of lifting under certain assumptions. This is the joint work with Joey Palmer and Sue Tolman.

Operator Algebras Seminar: On the generalized Jung property for II_1 factors and Popa’s Factorial Commutant Embedding Problem

Abstract:  A landmark theorem of Jung is that the hyperfinite II_1 factor $\mathcal R$ is the unique separable factor with the property that any two embeddings of it into its ultrapower $\mathcal R^\mathcal{U}$ are conjugate by a unitary.  In 2020, Atkinson and Kunnawalkam Elayavalli observed that $\mathcal R$ is the unique separable $R^\{\mathcal{U}}$ embeddable II_1 factor $N$ with the property that any two embeddings of $N$ into $N^\mathcal{U}$ are conjugate by a unitary.  In the first half of this talk, I will discuss a recent result (joint with Atkinson and Kunnawalkam Elayavalli) showing that $\mathcal R$ is the unique separable II_1 factor $N$ with the property that any two embeddings of $N$ into $N^\mathcal{U}$ are conjugate by an arbitrary (not necessarily inner) automorphism.  The proof is a blend of operator algebraic and model theoretic techniques.  Along the way, we show that any separable II_1 factor elementarily equivalent to $\mathcal R$ admits an embedding into $\mathcal R^{\mathcal U}$ with factorial commutant, thus providing continuum many examples of factors satisfying the conclusion of a longstanding open problem of Popa, which we refer to as the Factorial Commutant Embedding Problem (FCEP).  In the second half of the talk, I will discuss a recent result showing that there is a separable II_1 factor M for which all property T factors admit an embedding into $M^\mathcal{U}$ with factorial commutant, thus providing a “poor man’s” resolution to the FCEP for property T factors.  We will also identify two barriers from extending this result to a full resolution of the FCEP for property T factors.  No knowledge of model theory will be assumed.

Number Theory Seminar: The Kohnen-Zagier formula and the partition function

Meeting ID: 840 2782 4197

Password: 271403

Abstract: The Hardy-Ramanujan-Rademacher formula for the partition function provides a formula for p(n) as a rapidly converging infinite series. Via the Kuznetsov trace formula, bounds for the tail of the HRR series are connected with estimates for coefficients of Maass cusp forms that transform like the Dedekind eta function. We give an improved bound for these coefficients using a new Kohnen-Zagier formula which allows us to relate the coefficients to central values of twisted Maass form L-functions. This is joint work with Han Wu.

Graph Theory and Combinatorics Seminar: Saturation for the 3-uniform loose 3-cycle

Graduate Commutative Algebra & Algebraic Geometry Seminar: Algebraic stacks and Deligne-Mumford stacks

The quotient of a scheme by a group action may not exist in the category of schemes. However, it can be described as an algebraic stack. An algebraic stack is a functor from the category of schemes to the (2-)category of groupoids which satisfies some gluing conditions.

In this introductory talk, I will follow the book <Algebraic spaces and stacks> and introduce the definition of algebraic stacks. I will also talk about the ‘orbifold like’ objects called Deligne-Mumford stacks and give some examples of them including the moduli stack of curves of genus g. I will try to cover the quasi-coherent sheaves if time permits.

Candidate Presentation, Canary Professorship

Please contact Aimo H. (aimo@illinois.edu) for Zoom link. 

Speaker: Christopher Bishop, Stony Brook University

Title:  Weil-Petersson curves, traveling salesman theorems, and minimal surfaces

Abstract: Weil-Petersson curves are a  class of rectifiable closed  curves in the plane, defined as the closure of the smooth curves with respect to the Weil-Petersson metric defined by Takhtajan and Teo in 2009. Their work solved a problem from string theory by making the space of closed loops into a Hilbert manifold, but the same class of curves also arises naturally in complex analysis, probability theory, knot theory,  applied mathematics, and other areas. No geometric description of Weil-Petersson curves was known until 2019, but there are now more than thirty equivalent conditions.   One involves inscribed polygons and can be explained to a calculus student. Another is a strengthening of Peter Jones's traveling salesman theorem characterizing rectifiable curves. A third  says a curve is Weil-Petersson iff it bounds a minimal surface in hyperbolic space that has finite total curvature. I will discuss these and several other characterizations, and sketch  why they are all equivalent to each other. 

Thesis Defense, Joseph Rennie: Quasicategorical Galois Theories

Topic: Joseph Rennie's Dissertation Defense
Time: Apr 28, 2022 09:00 AM Central Time (US and Canada)

Join Zoom Meeting
https://illinois.zoom.us/j/87224637693?pwd=M3lPNU5lL1hlZUI5NDJucE5VT2UxQT09

Meeting ID: 872 2463 7693
Password: 061093

Thesis Defense, Colleen Robichaux: Equivariant Schubert Calculus and Applications

In this thesis defense talk, I will discuss my research concerning Schubert combinatorics and its interplay with computational complexity, equivariant cohomology, and Castelnuovo-Mumford regularity. 

Thesis Defense, Elizabeth Tatum: On a Spectrum-level Splitting of the BP<2>-Cooperations Algebra

In the 1980s, Mahowald and Kane used Brown-Gitler spectra to construct splittings of bo ∧ bo and l ∧ l. These splittings helped make it feasible to do computations using the bo- and l-based Adams spectral sequences. In this talk, we will sketch the construction of an analogous splitting for BP<2> ∧ BP<2> at primes larger than 3.

Actuarial Science and Financial Mathematics Seminar: Enhancing Claims Triage with Dynamic Data

Abstract: In property insurance claims triage, insurers often use static information to assess the severity of a claim and identify the subsequent actions. We hypothesize that the pattern of weather conditions throughout the course of the loss event is predictive of insured losses and hence appropriate use of weather dynamics improve the operation of insurer's claim management. To test this hypothesis, we propose a deep learning method to incorporate the dynamic weather data in the predictive modeling of insured losses for reported claims. The proposed method features a hierarchical network architecture to address the challenges introduced into claims triage by weather dynamics.

In the empirical analysis, we examine a portfolio of hail damage property insurance claims obtained from a major U.S. insurance carrier. When supplemented by the dynamic weather information, the deep learning method exhibits substantial improvement in the hold-out predictive performance. Built upon the proposed deep learning method, we design a cost-sensitive decision strategy for triaging claims using the probabilistic forecasts of insurance claim amounts. We show that leveraging weather dynamics in claims triage leads to a substantial reduction in operational cost.

About: Peng Shi is on the faculty of the Risk and Insurance Department at the University of Wisconsin-Madison. He is also the Charles and Laura Albright Professor in Business and Finance. Professor Shi is an Associate of the Casualty Actuarial Society (ACAS) and a Fellow of the Society of Actuaries (FSA). Professor Shi's research interests are at the intersection of insurance and statistics. He has won various research awards in actuarial science, including the Charles A Hachemeister Prize, American Risk and Insurance Association Prize, Ronald Bornhuetter Loss Reserve Prize, and IAA Best Paper etc.  Current research focuses on longitudinal data, dependence models, insurance analytics, and actuarial data science

Candidate Presentation, Canary Professorship

Title: Descriptive Set Theory and generic measure preserving transformations

Abstract: The behavior of a measure preserving transformation, even a generic one, is highly non-uniform. In contrast to this observation, a different picture of a very uniform behavior of the closed group generated by a generic measure preserving transformation $T$ has emerged. This picture included substantial evidence that pointed to these groups being all topologically isomorphic to a single group, namely, $L^0$---the topological group of all Lebesgue measurable functions from $[0,1]$ to the circle. In fact, Glasner and Weiss asked if this was the case.

We will describe the background touched on above, including the relevant definitions and the connections with Descriptive Set Theory. Further, we will indicate a proof of the following theorem that answers the Glasner--Weiss question in the negative: for a generic measure preserving transformation $T$, the closed group generated by $T$ is {\bf not} topologically isomorphic to $L^0$. The proof rests on an analysis of unitary representations of the non-locally compact group $L^0$.

Zoom info: https://illinois.zoom.us/j/84458961980?pwd=MlZKQVZJbEJOd1R6OXFDbFhvYWgrQT09

Undergraduate Friday Seminar: IGL Info Session

This week, we'll be joined by Madie Faris! Madie is a PhD student here in the Math program and also the Research Manager for the Illinois Geometry Lab. On Friday, they'll be joining us to give some helpful info about the IGL - tips for applying, potential future research projects, and more:

IGL Info Session
The IGL aims to get undergraduates at U of I involved in math research and outreach. During this talk I will outline the different ways you can get involved with the IGL, tips and tricks for applications, give an overview of some past and future research projects, and answer any questions you might have about our programs. If you've ever been interested in math outreach or research, then this is a great opportunity to hear about ways to get started!

As usual, there will be free pizza courtesy of the IGL!

Hope to see you there!