Department of

# Mathematics

Seminar Calendar
for Math events the year of Thursday, July 2, 2020.

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events for the
events containing

Questions regarding events or the calendar should be directed to Tori Corkery.
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Tuesday, January 14, 2020

11:00 am in 241 Altgeld Hall,Tuesday, January 14, 2020

#### Superimposing theta structure on a generalized modular relation

###### Atul Dixit (Indian Institute of Technology in Gandhinagar)

Abstract: By a modular relation for a certain function $F$, we mean that which is governed by the map $z\to -1/z$ but not necessarily by $z\to z+1$. Equivalently, the relation can be written in the form $F(\alpha)=F(\beta)$, where $\alpha\beta=1$. There are many generalized modular relations in the literature such as the general theta transformation $F(w,\alpha)=F(iw, \beta)$ or the Ramanujan-Guinand formula $F(z, \alpha)=F(z, \beta)$ etc. The latter, equivalent to the functional equation of the non-holomorphic Eisenstein series on $\mathrm{SL}_{2}(\mathbb{Z})$, admits a beautiful generalization of the form $F(z, w,\alpha)=F(z, iw, \beta)$, that is, one can superimpose theta structure on it.

Recently, a modular relation involving infinite series of the Hurwitz zeta function $\zeta(z, a)$ was obtained. It generalizes a result of Ramanujan from the Lost Notebook. Can one superimpose theta structure on it? While answering this question affirmatively, we were led to a surprising new generalization of $\zeta(z, a)$. We show that this new zeta function, $\zeta_w(z, a)$, satisfies a beautiful theory. In particular, it is shown that $\zeta_w(z, a)$ can be analytically continued to the whole complex plane except $z=1$. Hurwitz's formula for $\zeta(z, a)$ is also generalized in this setting. We also prove a generalized modular relation involving infinite series of $\zeta_w(z, a)$, which is of the form $F(z, w,\alpha)=F(z, iw, \beta)$. This is joint work with Rahul Kumar.

Friday, January 17, 2020

5:30 pm in Mineral Hall F, Hyatt Regency Denver, Colorado Convention Center,Friday, January 17, 2020

#### 2020 Joint Math Meetings

Abstract: The Department of Mathematics will host a reception during the 2020 Joint Math Meetings (January 15018,2020, Colorado Convention Center). Everyone ever connected with the department is encouraged to get together for conversation and to hear about mathematics at the University of Illinois.

Tuesday, January 21, 2020

4:00 pm in 245 Altgeld Hall,Tuesday, January 21, 2020

#### The Helly geometry of some Garside and Artin groups

###### Jingyin Huang   [email] (Ohio State University)

Abstract: Artin groups emerged from the study of braid groups and complex hyperplane arrangements, and they are connected to Coxeter groups, 3-manifold groups, buildings and many others. Artin groups have very simple presentation, yet rather mysterious geometry with many basic questions widely open. I will present a way of understanding certain Artin groups and Garside groups by building geometric models on which they act. These geometric models are non-positively curved in an appropriate sense, and such curvature structure yields several new results on the algorithmic, topological and geometric aspects of these groups. No previous knowledge on Artin groups or Garside groups is required. This is joint work with D. Osajda.

Wednesday, January 22, 2020

12:00 pm in 141 Altgeld Hall,Wednesday, January 22, 2020

#### Predictive Actuarial Analystics Using Tree-Based Models

###### Zhiyu Quan (University of Connecticut)

Abstract: Because of its many advantages, the use of tree-based models has become an increasingly popular alternative predictive tool for building classification and regression models. Innovations to the original methods, such as random forests and gradient boosting, have further improved the capabilities of using tree-based models as a predictive model. Quan et al. (2018) examined the performance of tree-based models for the valuation of the guarantees embedded in variable annuities. We found that tree-based models are generally very efficient in producing more accurate predictions and the gradient boosting ensemble method is considered the most superior. Quan and Valdez (2018) applied multivariate tree-based models to multi-line insurance claims data with correlated responses drawn from the Wisconsin Local Government Property Insurance Fund (LGPIF). We were able to capture the inherent relationship among the response variables and improved marginal predictive accuracy. Quan et al. (2019) propose to use tree-based models with a hybrid structure as an alternative approach to the Tweedie Generalized Linear Model (GLM). This hybrid structure captures the benefits of tuning hyperparameters at each step of the algorithm thereby allowing for an improved prediction accuracy. We examined the performance of this model vis-\a-vis the Tweedie GLM using the LGPIF and simulated datasets. Our empirical results indicate that this hybrid tree-based model produces more accurate predictions without loss of intuitive interpretation.

2:00 pm in 447 Altgeld Hall,Wednesday, January 22, 2020

#### Organizational Meeting

###### Sungwoo Nam (Illinois Math)

Abstract: We will have an organizational meeting for this semester. This involves making a plan for this semester and possibly choose a topic for a reading seminar. If you want to speak this semester, or are interested in a reading seminar, please join us and make a suggestion.

3:00 pm in 243 Altgeld Hall,Wednesday, January 22, 2020

#### How do mathematicians believe?

###### Brian P Katz (Smith College)

Abstract: Love it or hate it, many people believe that mathematics gives humans access to a kind of truth that is more absolute and universal than other disciplines. If this claim is true, we must ask: what makes the origins and processes of mathematics special and how can our messy, biological brains connect to the absolute? If the claim is false, then what becomes of truth in mathematics? In this session, we will consider beliefs about truth and how they play out in the mathematics classroom, trying to understand a little about identity, authority, and the Liberal Arts.

4:00 pm in 245 Altgeld Hall,Wednesday, January 22, 2020

#### Statistical reduced models and rigorous analysis for uncertainty quantification of turbulent dynamical systems

###### Di Qi   [email] (Courant Institute of Mathematical Sciences)

Abstract: The capability of using imperfect statistical reduced-order models to capture crucial statistics in turbulent flows is investigated. Much simpler and more tractable block-diagonal models are proposed to approximate the complex and high-dimensional turbulent flow equations. A rigorous statistical bound for the total statistical uncertainty is derived based on a statistical energy conservation principle. The systematic framework of correcting model errors is introduced using statistical response and empirical information theory, and optimal model parameters under this unbiased information measure are achieved in a training phase before the prediction. It is demonstrated that crucial principal statistical quantities in the most important large scales can be captured efficiently with accuracy using the reduced-order model in various dynamical regimes with distinct statistical structures.

Thursday, January 23, 2020

11:00 am in 241 Altgeld Hall,Thursday, January 23, 2020

#### Heights and p-adic Hodge Theory

###### Lucia Mocz (University of Chicago)

Abstract: We discuss connections between p-adic Hodge theory and the Faltings height. Most namely, we show how new tools in p-adic Hodge theory can be used to prove new Northcott properties satisfied by the Faltings height, and demonstrate phenomenon which are otherwise predicted by various height conjectures. We will focus primarily on the Faltings height of CM abelian varieties where the theory can be made to be computational and explicit.

3:00 pm in 245 Altgeld Hall,Thursday, January 23, 2020

#### Application of Random Effects in Dependent Compound Risk Model

###### Himchan Jeong (University of Connecticut)

Abstract: In ratemaking for general insurance, the calculation of a pure premium has traditionally been based on modeling both frequency and severity in an aggregated claims model. Additionally for simplicity, it has been a standard practice to assume the independence of loss frequency and loss severity. However, in recent years, there has been sporadic interest in the actuarial literature exploring models that departs from this independence. Besides, usual property and casualty insurance enables us to explore the benefits of using random effects for predicting insurance claims observed longitudinally, or over a period of time. Thus, in this article, a research work is introduced with utilizes random effects in dependent two-part model for insurance ratemaking, testing the presence of random effects via Bayesian sensitivity analysis with its own theoretical development as well as empirical results and performance measures using out-of-sample validation procedures.

4:00 pm in 245 Altgeld Hall,Thursday, January 23, 2020

#### Semistable reduction in characteristic 0

###### Gaku Liu (Max Planck Institute for Mathematics in the Sciences)

Abstract: Semistable reduction is a relative generalization of the classical problem of resolution of singularities of varieties; the goal is, given a surjective morphism $f : X \to B$ of varieties in characteristic 0, to change $f$ so that it is "as nice as possible". The problem goes back to at least Kempf, Knudsen, Mumford, and Saint-Donat (1973), who proved a strongest possible version when $B$ is a curve. The key ingredient in the proof is the following combinatorial result: Given any $d$-dimensional polytope $P$ with vertices in $\mathbb{Z}^d$, there is a dilation of $P$ which can be triangulated into simplices each with vertices in $\mathbb{Z}^d$ and volume $1/d!$. In 2000, Abramovich and Karu proved, for any base $B$, that $f$ can be made into a weakly semistable morphism $f' : X' \to B'$. They conjectured further that $f'$ can be made semistable, which amounts to making $X'$ smooth. They explained why this is the best resolution of $f$ one might hope for. In this talk I will outline a proof of this conjecture. They key ingredient is a relative generalization of the above combinatorial result of KKMS. I will also discuss some other consequences in combinatorics of our constructions. This is joint work with Karim Adiprasito and Michael Temkin.

Friday, January 24, 2020

3:00 pm in 243 Altgeld Hall,Friday, January 24, 2020

#### Statistical inference for mortality models

###### Chen Ling (Georgia State University)

Abstract: Underwriters of annuity products and administrators of pension funds are under ﬁnancial obligation to their policyholder until the death of counterparty. Hence, the underwriters are subject to longevity risk when the average lifespan of the entire population increases, and yet, such risk can be managed through hedging practices based on parametric mortality models. As a benchmark mortality model in insurance industry is Lee-Carter model, we ﬁrst summarize some ﬂaws regarding the model and inference methods derived from it. Based on these understandings we propose a modiﬁed Lee-Carter model, accompanied by a rigorous statistical inference with asymptotic results and satisfactory numerical and simulation results derived from a small sample. Then we propose bias corrected estimator which is consistent and asymptotically normally distributed regardless of the mortality index being a unit root or stationary AR(1) time series. We further extend the model to accommodate AR(2) process for mortality index, and, a bivariate dataset of U.S. mortality rates. Finally, we conclude by a detailed model validation and some discussions of potential hedging practices based on our parametric model.

3:00 pm in 347 Altgeld Hall,Friday, January 24, 2020

#### Organizational Meeting

###### Kesav Krishnan (UIUC Math)

Abstract: This organizational meeting will be to decide on a schedule of speakers. All are welcome

Monday, January 27, 2020

2:00 pm in 241 Altgeld Hall,Monday, January 27, 2020

#### Two-Part D-Vine Copula Models for Longitudinal Insurance Claim Data

###### Lu Yang (University of Amsterdam)

Abstract: Insurance companies keep track of each policyholder's claims over time, resulting in longitudinal data. Efficient modeling of time dependence in longitudinal claim data can improve the prediction of future claims needed, for example, for ratemaking. Insurance claim data have their special complexity. They usually follow a two-part mixed distribution: a probability mass at zero corresponding to no claim and an otherwise positive claim from a skewed and long-tailed distribution. We propose a two-part D-vine copula model to study longitudinal mixed claim data. We build two stationary D-vine copulas. One is used to model the time dependence in binary outcomes resulting from whether or not a claim has occurred, and the other studies the dependence in the claim size given occurrence over time. The proposed model can predict the probability of making claims and the quantiles of severity given occurrence straightforwardly. We use our approach to investigate a dataset from the Local Government Property Insurance Fund in the state of Wisconsin.

3:00 pm in Altgeld Hall 441,Monday, January 27, 2020

#### Organizational meeting

###### William Balderrama (Illinois Math)

4:00 pm in 245 Altgeld Hall,Monday, January 27, 2020

#### A new approach to bounding L-functions

###### Jesse Thorner   [email] (University of Florida)

Abstract: Analytic number theory began with studying the distribution of prime numbers, but it has evolved and grown into a rich subject lying at the intersection of analysis, algebra, combinatorics, and representation theory. Part of its allure lies in its abundance of problems which are tantalizingly easy to state which quickly lead to deep mathematics, much of which revolves around the study of L-functions. These extensions of the elusive Riemann zeta function $\zeta(s)$ are generating functions with multiplicative structure arising from either arithmetic-geometric objects (like number fields or elliptic curves) or representation-theoretic objects (automorphic forms). Many equidistribution problems in number theory rely on one's ability to accurately bound the size of L-functions; optimal bounds arise from the (unproven!) Riemann Hypothesis for $\zeta(s)$ and its extensions to other L-functions. I will discuss some motivating problems along with recent work (joint with Kannan Soundararajan) which produces new bounds for L-functions by proving a suitable "statistical approximation" to the (extended) Riemann Hypothesis.

Tuesday, January 28, 2020

3:00 pm in 245 Altgeld Hall,Tuesday, January 28, 2020

#### Insuring longevity risk and long-term care: Bequest, housing and liquidity

###### Mengyi Xu (University of New South Wales)

Abstract: The demand for life annuities and long-term care insurance (LTCI) varies among retirees with different preferences and financial profiles. This paper shows that bequest motives can enhance the demand for LTCI unless the opportunity cost of self-insurance through precautionary savings is low. This typically occurs when retirees have sufficient liquid wealth. If retirees tend to liquidate housing assets in the event of becoming disabled that requires sizeable costs, housing wealth is likely to enhance the demand for annuities and to crowd out the demand for LTCI. Cash-poor-asset-rich retirees show little interest in annuities, but they may purchase LTCI to preserve their bequests.

Thursday, January 30, 2020

11:00 am in 241 Altgeld Hall,Thursday, January 30, 2020

#### Modularity of some $\mathrm{PGL}_2(\mathbb{F}_5)$ representations

###### Patrick Allen (Illinois)

Abstract: Serre's conjecture, proved by Khare and Wintenberger, states that every odd two dimensional mod p representation of the absolute Galois group of the rationals comes from a modular form. This admits a natural generalization to totally real fields, but even the real quadratic case seems completely out of reach. I'll discuss some of the difficulties one encounters and then discuss some new cases that can be proved when p = 5. This is joint work with Chandrashekhar Khare and Jack Thorne.

2:00 pm in 347 Altgeld Hall,Thursday, January 30, 2020

#### The Semicircle Law for Wigner Matrices

###### Kesav Krishnan (UIUC Math)

Abstract: I will introduce Wigner Matrices and their universal properties. I will then state the semi-circle law and sketch out three district proofs, in analogy to the proof of the usual central limit theorem. Talk 1 will sketch out the proof via the Stieltjes transform and via the energy entropy balance.

4:00 pm in 245 Altgeld Hall,Thursday, January 30, 2020

#### Local and global boundary rigidity

###### Plamen Stefanov   [email] (Purdue University)

Abstract: Abstract: The boundary rigidity problem consist of recovering a Riemannian metric in a domain, up to an isometry, from the distance between boundary points. We show that in dimensions three and higher, knowing the distance near a fixed strictly convex boundary point allows us to reconstruct the metric inside the domain near that point, and that this reconstruction is stable. We also prove semi-global and global results under certain an assumption of the existence of a strictly convex foliation. The problem can be reformulated as a recovery of the metric from the arrival times of waves between boundary points; which is known as travel-time tomography. The interest in this problem is motivated by imaging problems in seismology: to recover the sub-surface structure of the Earth given travel-times from the propagation of seismic waves. In oil exploration, the seismic signals are man-made and the problem is local in nature. In particular, we can recover locally the compressional and the shear wave speeds for the elastic Earth model, given local information. The talk is based on a joint work with G.Uhlmann (UW) and A.Vasy (Stanford). We will also present results for a recovery of a Lorentzian metric from red shifts motivated by the problem of observing cosmic strings. The methods are based on Melrose’s scattering calculus in particular but we will try to make the exposition accessible to a wider audience without going deep into the technicalities.

Friday, January 31, 2020

3:00 pm in 347 Altgeld Hall,Friday, January 31, 2020

#### Indroduction to Non Commutative Probability

###### Kesav Krishnan (UIUC Math)

Abstract: In this talk I will introduce Non Commutative Probability Theory, and highlight some of its uses in classical Probability, such as the study of random matrices. In particular, motivation of Wigner's semi-circle law as the non commutative analog of the Central Limit Theorem.

4:00 pm in 341 Altgeld Hall,Friday, January 31, 2020

#### What Is A Mathematics?

###### Robert Joseph Rennie   [email] (University of Illinois at Urbana-Champaign)

Abstract: In this talk, I will begin with a mathematization of the process of mathematization. We will then see how category theory and type theory provide a nice general framework for constructing and comparing systems of math. This discussion will motivate, without ever mentioning topological spaces, the study of higher toposes to anyone who cares about theoretical physics (not necessarily just those who study it). This talk requires only an interest in thinking about how math works.

Monday, February 3, 2020

3:00 pm in 441 Altgeld Hall,Monday, February 3, 2020

#### Exotic elements in Picard groups

###### Ningchuan Zhang (Illinois Math)

Abstract: In this talk, I will discuss the subgroup of exotic elements in the $K(h)$-local Picard groups. We will first show this subgroup is zero when $p\gg h$ and then focus on the $(h,p)=(1,2)$ and $(2,3)$ cases.

Tuesday, February 4, 2020

1:00 pm in 241 Altgeld Hall,Tuesday, February 4, 2020

#### Model-theoretic techniques in query learning

###### Hunter Chase (UIC)

Abstract: Several notions of complexity of set systems correspond both with model-theoretic dividing lines and notions of machine learning. We describe a new connection between query learning and stable formulas without the finite cover property.

1:00 pm in 347 Altgeld Hall,Tuesday, February 4, 2020

#### Local smoothing estimates for Fourier Integral Operators

###### David Beltran (U. Wisconsin, Madison)

Abstract: The sharp fixed-time Sobolev estimates for Fourier Integral Operators (and therefore solutions to wave equations in Euclidean space or compact manifolds) were established by Seeger, Sogge and Stein in the early 90s. Shortly after, Sogge observed that a local average in time leads to a regularity improvement with respect to the sharp fixed-time estimates. Establishing variable-coefficient counterparts of the Bourgain—Demeter decoupling inequalities, we improve the previous best known local smoothing estimates for FIOs. Moreover, we show that our results are sharp in both the Lebesgue and regularity exponent (up to the endpoint) in odd dimensions. This is joint work with Jonathan Hickman and Christopher D. Sogge.

Thursday, February 6, 2020

11:00 am in 241 Altgeld Hall,Thursday, February 6, 2020

#### The Kuznetsov formulas for GL(3)

###### Jack Buttcane (University of Maine)

Abstract: The Kuznetsov formulas for GL(2) connect the study of automorphic forms to the study of exponential sums. They are useful in a wide variety of seemingly unrelated problems in analytic number theory, and I will (briefly) illustrate this with a pair of examples: First, if we consider the roots v of a quadratic polynomial modulo a prime p, then the sequence of fractions v/p is uniformly distributed modulo 1; this is the “mod p equidistribution” theorem of Duke, Friedlander, Iwaniec and Toth. Second, the Random Wave Conjecture states that a sequence of automorphic forms should exhibit features of a random wave as their Laplacian eigenvalues tend to infinity. I will discuss their generalization to GL(3) and applications.

2:00 pm in 347 Altgeld Hall,Thursday, February 6, 2020

#### The Semicircle Law for Wigner Matrices Part 2

###### Kesav Krishnan (UIUC Math)

Abstract: I will introduce Wigner Matrices and their universal properties. I will then state the semi-circle law and sketch out three district proofs, in analogy to the proof of the usual central limit theorem. talk two will discuss the proof based on the method of moments and its relation to enumerative combinatorics.

4:00 pm in 245 Altgeld Hall,Thursday, February 6, 2020

#### Analytic Grothendieck Riemann Roch Theorem

###### Xiang Tang   [email] (Washington University St. Louis)

Abstract: Abstract: In this talk, we will introduce an interesting index problem naturally associated to the Arveson-Douglas conjecture in functional analysis. This index problem is a generalization of the classical Toeplitz index theorem, and connects to many different branches of Mathematics. In particular, it can be viewed as an analytic version of the Grothendieck Riemann Roch theorem. This is joint work with R. Douglas，M. Jabbari, and G. Yu.

Friday, February 7, 2020

3:00 pm in 347 Altgeld Hall,Friday, February 7, 2020

#### Indroduction to Non Commutative Probability Part 2

###### Kesav Krishnan (UIUC Math)

Abstract: I will continue the talk from last friday, on the Introduction to Non Commutative Probability. In this talk, I will focus on limit laws, in particular the non commutative CLT and the universality of the semi-circle law.

4:00 pm in Altgeld Hall,Friday, February 7, 2020

#### A Brief Introduction to Problem Writing

###### David Altizio   [email] (University of Illinois at Urbana-Champaign)

Abstract: Many of the best mathematics competition problems push the boundaries of pre-calculus math in unexpected ways. While these questions fuel the popularity of contests among middle and high school students, they also make competitions seem inherently unsustainable; constructing these questions appears to be a Herculean task. In this talk, I will shed some insight into how problems are made by exploring my eight-year-long journey through problem writing. In particular, I will discuss common writing philosophies, sources of inspiration, and the stories behind some of my favorite creations.

Monday, February 10, 2020

3:00 pm in 441 Altgeld Hall,Monday, February 10, 2020

#### Exotic elements in Picard groups (part 2)

###### Ningchuan Zhang (Illinois Math)

Abstract: In this talk, I will discuss the subgroup of exotic elements in the $K(h)$-local Picard groups. We will first show this subgroup is zero when $p\gg h$ and then focus on the $(h,p)=(1,2)$ and $(2,3)$ cases.

Tuesday, February 11, 2020

1:00 pm in 241 Altgeld Hall,Tuesday, February 11, 2020

#### The Borel complexity of quotient groups

###### Joshua Frisch (Caltech Math)

Abstract: The theory of Borel equivalence relations gives us rigorous methods to says when one classification problem/equivalence relation is more "complicated" than another. Given a countable group it's outer-automorphism group naturally has the structure of a borel equivalence relation. Motivated by this example, in this talk I will give a brief introduction to the theory of countable borel equivalence relations, describe some previously known connections with the theory of groups and, finally, describe a new new result explaining exactly how complicated the Borel complexity of quotient groups (which generalize outer-automorphism groups) can be. This is joint work with Forte Shinko.

Wednesday, February 12, 2020

3:30 pm in 341 Altgeld Hall,Wednesday, February 12, 2020

#### Strongly amenable groups

###### Joshua Frisch (Caltech Math)

Abstract: A topological dynamical system (i.e. a group acting by homeomorphisms on a compact Hausdorff space) is said to be proximal if for any two points $p$ and $q$ we can simultaneously "push them together" (rigorously, there is a net $g_n$ such that $\lim g_n(p) = \lim g_n(q)$). In his paper introducing the concept of proximality, Glasner noted that whenever $\mathbb{Z}$ acts proximally, that action will have a fixed point. He termed groups with this fixed point property "strongly amenable" and showed that non-amenable groups are not strongly amenable and virtually nilpotent groups are strongly amenable. In this talk I will discuss recent work precisely characterizing which (countable) groups are strongly amenable. This is joint work with Omer Tamuz and Pooya Vahidi Ferdowsi.

Thursday, February 13, 2020

11:00 am in 241 Altgeld Hall,Thursday, February 13, 2020

#### Divisors of integers, permutations and polynomials

###### Kevin Ford (Illinois Math)

Abstract: We describe a probabilistic model that describes the statistical behavior of the divisors of integers, divisors of permutations and divisors of polynomials over a finite field. We will discuss how this can be used to obtain new bounds on the concentration of divisors of integers, improving a result of Maier and Tenenbaum. This is joint work with Ben Green and Dimitris Koukoulopoulos.

2:00 pm in 347 Altgeld Hall,Thursday, February 13, 2020

#### Distribution of eigenvalues of random matrices (part I)

###### Peixue Wu (UIUC Math)

Abstract: Last time we proved a famous semicircular law for the limit distribution of the empirical measure of the eigenvalues of Wigner's matrix (i.i.d. under the symmetry restriction). When we go over the proof in detail, we find two essential ingredients to the proof: 1. Stochastic independence of the entries. 2. Most matrix entries are centered and have the same variance. Using the similar idea (methods of moments) we will show that semicircular law holds for a much larger class of random matrices. We will also talk about the joint distribution for the eigenvalues of the Gaussian Orthogonal (Unitary) Ensembles (GOE or GUE).

Friday, February 14, 2020

3:00 pm in 347 Altgeld Hall,Friday, February 14, 2020

#### An Introduction to $L^2$ Cohomology

###### Gayana Jayasinghe (UIUC Math)

Abstract: We'll see how we can construct quasi isometry invariants and some conformal invariants with function spaces and operators on manifolds (and some more general spaces), and how we can use analysis to study geometric structures

4:00 pm in 341 Altgeld Hall,Friday, February 14, 2020

#### Prime Number Conjectures

###### Raghavendra Bhat (University of Illinois at Urbana-Champaign)

Abstract: Freshman math major and author (Math -- A Subtle Language of the Universe) Raghavendra Bhat will present some of his prime number conjectures, which he has presented on many platforms across the world. His talk will focus on his recent conjectures and thoughts on number theory research and math in general.

Monday, February 17, 2020

2:00 pm in 241 Altgeld Hall,Monday, February 17, 2020

#### From Polaris Variable Annuities to Regression-based Monte Carlo

###### Zhiyi (Joey) Shen (University of Waterloo)

Abstract: In this talk, I will first discuss the no-arbitrage pricing of Polaris variable annuities (VAs), which were issued by the American International Group in recent years. Variable annuities are prevailing equity-linked insurance products that provide the policyholder with the flexibility of dynamic withdrawals, mortality protection, and guaranteed income payments against a market decline. The Polaris allows a shadow account to lock in the high watermark of the investment account over a monitoring period that depends on the policyholder’s choice of his/her first withdrawal time. This feature makes the insurer’s payouts depend on policyholder’s withdrawal behaviours and significantly complicates the pricing problem. By prudently introducing certain auxiliary state variables, we manage to formulate the pricing problem into solving a convoluted stochastic optimal control framework and developing a computationally efficient algorithm to approach the solution. Driven by the challenges from the pricing Polaris VAs, in the second part of the talk, I will introduce a regression-based Monte Carlo algorithm, which we propose to solve a class of general stochastic optimal control problems numerically. The algorithm has three pillars: a construction of auxiliary stochastic control model, an artificial simulation of the post-action value of state process, and a shape-preserving sieve estimation method. The algorithm enjoys many merits, including obviating forward simulation and control randomization, eliminating in-sample bias, evading extrapolating the value function, and alleviating the computational burden of the tuning parameter selection. This talk is based on two joint works with Chengguo Weng from the University of Waterloo.

3:00 pm in 441 Altgeld Hall,Monday, February 17, 2020

#### A geometric perspective on the foundations of modern homotopy theory

###### Brian Shin (Illinois Math)

Abstract: Homotopy theorists have always been interested in studying spaces. However, the meaning of the word space'' has evolved over the years. Whereas one used to say space to mean a topological space, it seems the modern stance is to view a space as an $\infty$-groupoid. In this expository talk, I would like to connect the modern stance back to geometry. In particular, I will demonstrate how the $\infty$-category of spaces can be built out of the category of manifolds. As an application, we will use this connection to give a geometric perspective on infinite loop space theory.

Tuesday, February 18, 2020

3:00 pm in 245 Altgeld Hall,Tuesday, February 18, 2020

#### Mixture of Experts Regression Models for Insurance Ratemaking and Reserving

###### Tsz Chai "Samson" Fung (University of Toronto)

Abstract: Understanding the effect of policyholders' risk profile on the number and the amount of claims, as well as the dependence among different types of claims, are critical to insurance ratemaking and IBNR-type reserving. To accurately quantify such features, it is essential to develop a regression model which is flexible, interpretable and statistically tractable. In this presentation, I will discuss a highly flexible nonlinear regression model we have recently developed, namely the logit-weighted reduced mixture of experts (LRMoE) models, for multivariate claim frequencies or severities distributions. The LRMoE model is interpretable as it has two components: Gating functions to classify policyholders into various latent sub-classes and Expert functions to govern the distributional properties of the claims. The model is also flexible to fit any types of claim data accurately and hence minimize the issue of model selection. Model implementation is illustrated in two ways using a real automobile insurance dataset from a major European insurance company. We first fit the multivariate claim frequencies using an Erlang count expert function. Apart from showing excellent fitting results, we can interpret the fitted model in an insurance perspective and visualize the relationship between policyholders' information and their risk level. We further demonstrate how the fitted model may be useful for insurance ratemaking. The second illustration deals with insurance loss severity data that often exhibits heavy-tail behavior. Using a Transformed Gamma expert function, our model is applicable to fit the severity and reporting delay components of the dataset, which is ultimately shown to be useful and crucial for an adequate prediction of IBNR reserve. This project is joint work with Andrei Badescu and Sheldon Lin.

4:00 pm in 341 Altgeld Hall,Tuesday, February 18, 2020

#### Julia Robinson and Hilbert's Tenth Problem (film)

Abstract: Julia Robinson was the first woman elected to the mathematical section of the National Academy of Sciences, and the first woman to become president of the American Mathematical Society. While tracing Robinson's contribution to the solution of Hilbert's tenth problem, the film illuminates how her work led to an unusual friendship between Russian and American colleagues at the height of the Cold War.

4:00 pm in 314 Altgeld Hall,Tuesday, February 18, 2020

#### In Transition - Mathematics and Art

###### Kirsi Peltonen   [email] (Aalto University, Finland)

Abstract: This is a talk for the general public and academic audience interested in possibilities for enhancing interaction between contemporary mathematics and arts. What are the needs for this dialogue in different levels of education, research and broader in the society? Recent multidisciplinary activities challenging the traditions and communication of mathematics and arts at Aalto University in Finland have given a new type of platform to share the beauty of mathematics systematically and open accessible layers to a useful interplay. Many outcomes and byproducts of our up-to-date experiments are perfect for applications in digital technologies such as programming, CAD, 3D printing, virtual and augmented reality. Some scenarios for the future development are presented.

Thursday, February 20, 2020

2:00 pm in 347 Altgeld Hall,Thursday, February 20, 2020

#### Distribution of eigenvalues of random matrices (part II)

###### Peixue Wu (UIUC Math)

Abstract: Last time we proved the classical Wigner's semicircular law for Wigner matrix. This time I will state a dynamical version of the semicircular law, which implies the classical Wigner's semicircular law. Our main tool will be stochastic analysis.

2:00 pm in 243 Altgeld Hall,Thursday, February 20, 2020

#### Around the Folds

###### Kirsi Peltonen (Aalto University, Finland)

Abstract: We will discuss about various ways to use origami and folding as a multidisciplinary tool in research and education. Some ongoing projects funded by Academy of Finland and Ministry of Education and Culture with theoretical and practical goals are described. The slides of the talk as a pdf-file can be obtained by clicking on the name of the speaker above.

4:00 pm in 245 Altgeld Hall,Thursday, February 20, 2020

#### Data-driven methods for model identification and parameter estimation of dynamical systems

###### Niall Mangan   [email] (Northwestern University)

Abstract: Inferring the structure and dynamical interactions of complex systems is critical to understanding and controlling their behavior. I am interested in discovering models from the time-series in order to understand biological systems, material behavior, and other dynamical systems. One can frame the problem as selecting which interactions, or model terms, are most likely responsible for the observed dynamics from a library of possible terms. Several challenges make model selection and parameter estimation difficult including nonlinearities, varying parameters or equations, and unmeasured state variables. I will discuss methods for reframing these problems so that sparse model selection is possible including implicit formulation and data clustering. I will also discuss preliminary results for parameter estimation and model selection for deterministic and chaotic systems with hidden or unmeasured variables. We use a variational annealing strategy that allows us to estimate both the unknown parameters and the unmeasured state variables.

Friday, February 21, 2020

4:00 pm in 341 Altgeld Hall,Friday, February 21, 2020

#### Julia Robinson and Hilbert's Tenth Problem

###### (UIUC Math)

Abstract: Julia Robinson was the first woman elected to the mathematical section of the National Academy of Sciences, and the first woman to become president of the American Mathematical Society. While tracing Robinson's contribution to the solution of Hilbert's tenth problem, the film illuminates how her work led to an unusual friendship between Russian and American colleagues at the height of the Cold War.

4:00 pm in 141 Altgeld Hall,Friday, February 21, 2020

#### Unifying Galois Theories with Categorification

###### Robert (Joseph) Rennie (UIUC)

Abstract: Since its inception nearly two centuries ago, what we call "Galois Theory" (say in an undergraduate algebra course) has led to many analogous results, and thus attained the status of a sort of metatheorem. In Galois' case, this concept was applied to fields, yielding an equivalence between some lattice of field extensions and a lattice of subgroups of a corresponding "galois group" ... under certain conditions. Later on, the same concept was shown to be present in Topology, with extensions being replaced by their dual notion of covering spaces, and the galois group being replaced by the fundamental group... again, under certain conditions. Even later, Galois' results for fields were generalized to arbitrary rings, introducing new associated data along the way. In this talk, we explore the process of formally unifying all of these "Galois Theories" into one Galois Principle, with the aim of developing an intuition for identifying some of its infinite use-cases in the wilds of Math (e.g. Algebra, Topology, and Logic). Along the way, I aim to discuss explicitly and to motivate categorification to the working mathematician using the results of this talk as concrete examples.

Monday, February 24, 2020

11:00 am in 464 Loomis ,Monday, February 24, 2020

#### Developments in the Bagger-Witten and Hodge line bundles

###### Eric Sharpe (Virginia Tech Physics)

Abstract: This talk will concern advances in understanding explicitly the Bagger-Witten line bundle appearing in four-dimensional N=1 supergravity, which is closely related to the Hodge line bundle on a moduli space of Calabi-Yaus. This has recently been a subject of interest, but explicit examples have proven elusive in the past. In this talk we will outline some recent advances, including (1) a description of the Bagger-Witten line bundle on a moduli space of Calabi-Yau's as a line bundle of covariantly constant spinors (resulting in a square root of the Hodge line bundle of holomorphic top-forms), (2) results suggesting that it (and the Hodge line bundle) is always flat, but possibly never trivial, over moduli spaces of Calabi-Yaus of maximal holonomy and dimension greater than two. We will propose its nontriviality as a new criterion for existence of UV completions of four-dimensional supergravity theories. If time permits, we will explicitly construct an example, to concretely display these properties, and outline results obtained with Ron Donagi and Mark Macerato for other cases.

3:00 pm in 441 Altgeld Hall,Monday, February 24, 2020

#### An introduction to motivic homotopy theory

###### Brian Shin (Illinois Math)

Abstract: Motivic homotopy is often thought of as the homotopy theory of algebraic varieties. In this expository talk, we'll see exactly what that means. In particular, we'll see how the construction of the category of motivic spaces is a direct algebro-geometric analog of that of the category of spaces. More interestingly, we'll also see how the analogy breaks down.

4:00 pm in 314 Altgeld Hall,Monday, February 24, 2020

#### Digits

###### Frank Calegari (University of Chicago)

Abstract: We discuss some results concerning the decimal expansion of 1/p for primes p, some due to Gauss, and some from the present day. This talk will be accessible to undergraduates.

Tuesday, February 25, 2020

11:00 am in 243 Altgeld Hall,Tuesday, February 25, 2020

#### \'Etale K-theory

###### Akhil Mathew (U Chicago)

Abstract: I will explain some general structural results about algebraic K-theory and its \'etale sheafification, in particular its approximation by Selmer K-theory. This is based on some recent advances in topological cyclic homology. Joint work with Dustin Clausen.

1:00 pm in 243 Altgeld Hall,Tuesday, February 25, 2020

#### Geometry of the Minimal Solutions of Linear Diophantine Equations

###### Papa A. Sissokho (Illinois State Univeristy)

Abstract: Let ${\bf a}=(a_1,\ldots,a_n)$ and ${\bf b}=(b_1,\ldots,b_m)$ be vectors with positive integer entries, and let $\mathcal{S}({\bf a},{\bf b})$ denote the set of all nonnegative solutions $({\bf x},{\bf y})$, where ${\bf x}=(x_1,\ldots,x_n)$ and ${\bf y}=(y_1,\ldots,y_m)$, of the linear Diophantine equation $x_1a_1+...+ x_na_n=y_1b_1+...+y_mb_m$. A solution is called minimal if it cannot be written as the sum of two nonzero solutions in $\mathcal{S}({\bf a},{\bf b})$. The set of all minimal solutions, denoted by $\mathcal{H}({\bf a},{\bf b})$, is called the Hilbert basis of $\mathcal{S}({\bf a},{\bf b})$. The solution ${\bf g}_{i,j}=(b_j{\bf e}_i,a_i{\bf e}_{n+j})$ of the above Diophantine equation, where ${\bf e}_k$ is the $k$th standard unit vector of $\mathbb{R}^{n+m}$, is called a generator. In this talk, we discuss a recent result which shows that every minimal solution in $\mathcal{H}({\bf a},{\bf b})$ is a convex combination of the generators and the zero-solution.

2:00 pm in 345 Altgeld Hall,Tuesday, February 25, 2020

#### Multiple SLE from a loop measure perspective

###### Vivian Healey (U Chicago Math)

Abstract: I will discuss the role of Brownian loop measure in the study of Schramm-Loewner evolution. This powerful perspective allows us to apply intuition from discrete models (in particular, the λ-SAW model) to the study of SLE while simultaneously reducing many SLE computations to problems of stochastic calculus. I will discuss recent work on multiple radial SLE that employs this method, including the construction of global multiple radial SLE and its links to locally independent SLE and Dyson Brownian motion. (Joint work with Gregory F. Lawler.)

4:00 pm in 245 Altgeld Hall,Tuesday, February 25, 2020

#### Counting

###### Frank Calegari (University of Chicago)

Abstract: What can one say about a system of polynomial equations with integer coefficients simply by counting the number of solutions to these equations modulo primes? We begin with the case of polynomials in one variable and relate this to how the polynomial factors and to Galois theory. We then discuss what happens in higher dimensions, and are led to a conjectural notion of the "Galois group" of an algebraic variety. This will be a colloquium style talk and will be independent of the first talk.

Wednesday, February 26, 2020

2:00 pm in 447 Altgeld Hall,Wednesday, February 26, 2020

#### Introduction to moduli spaces of sheaves

###### Sungwoo Nam (Illinois Math)

Abstract: This talk will be an introduction to moduli spaces of sheaves. We will see some motivating questions that lead to the study of moduli spaces of sheaves, and discuss examples telling us why the notion of stability is needed, even in the simplest case of vector bundles on curves. Then I will survey some results on moduli spaces of sheaves on surfaces, especially those of K3 and abelian surfaces and applications to holomorphic symplectic geometry.

4:00 pm in 245 Altgeld Hall,Wednesday, February 26, 2020

#### Coble

###### Frank Calegari (University of Chicago)

Abstract: Coble is known (in part) for his work on invariant theory and the geometry of certain of exceptional moduli spaces in low dimension. We discuss the quest to find explicit equations for one particular family of moduli spaces. An important role is played by a number of exceptional geometrical coincidences and also the theory of complex reflection groups. This will be a colloquium style talk and will be independent of the first two talks.

Thursday, February 27, 2020

11:00 am in 241 Altgeld Hall,Thursday, February 27, 2020

#### The shape of low degree number fields

###### Bob Hough (Stony Brook University)

Abstract: In his thesis, M. Bhargava proved parameterizations and identified local conditions which he used to give asymptotic counts for $S_4$ quartic and quintic number fields, ordered by discriminant. This talk will discuss results in an ongoing project to add detail to Bhargava's work by considering in addition to the field discriminant, the lattice shape of the ring of integers in the canonical embedding, and by giving strong rates with lower order terms in the asymptotics. These results build on earlier work of Taniguchi-Thorne, Bhargava-Shankar-Tsimerman and Bhargava Harron.

1:00 pm in 464 Loomis ,Thursday, February 27, 2020

#### Title: Probing heterotic/F-theory duality with a little string theory

###### Patrick Jefferson (MIT Physics)

Abstract: : The duality between heterotic string theory on a 2-torus and F-theory on an elliptically fibered K3 surface is one of the most groundbreaking results to emerge from the superstring revolution, being intimately related to all other known string dualities. Despite this, a precise map between the moduli spaces of the two theories is only known at special loci. In this talk I will propose a method to compute a general map between moduli spaces. Specifically, I will argue that applying Nekrasov’s instanton calculus to a torus-compactified probe little string theory permits an explicit construction of an elliptic fibration in terms of the Narain moduli of the heterotic string. I will also mention potential applications and future prospects for this work.

2:00 pm in 347 Altgeld Hall,Thursday, February 27, 2020

#### Tracy Widom Distribution and Spherical Spin Glass (Part I)

###### Qiang Wu (UIUC Math)

Abstract: We studied the global behavior of eigenvalues of random matrices in previous talks. This time we are going to zoom into the bulk to study some local behavior of eigenvalues. In particular, the edge scaling limit of largest eigenvalue is given by the Tracy-widom (TW) distribution, which as a universal object also appears in some other areas, like growth process, spin system and many other interacting particle systems. Taking GUE as our example, we will try to derive the TW distribution represented as a Fredholm determinant with Airy Kernel. Time permits, we will briefly go through the integral representation of TW, and some universality results even extended to the underlying integrable system for general beta ensembles.

Friday, February 28, 2020

3:00 pm in 347 Altgeld Hall,Friday, February 28, 2020

#### Eigenvalues on Forms

###### Xiaolong Han (UIUC Math)

Abstract: Recently there has been a growing interest in eigenvalues on forms. It is much more complicated than eigenvalues on functions but can detect finer geometry. It has applications in detecting length of axes of John ellipsoid of convex body, relating Monopole Floer homology to hyperbolic geometry, and commutator length in hyperbolic geometry. In this talk we will show some basic theory and definitions for eigenvalues on forms, and then provide some intuition for the geometry and applications.

4:00 pm in 143 Altgeld Hall,Friday, February 28, 2020

#### Re: Mathematical art and sculpture in connection with the Altgeld/Illini building project

Abstract: Meeting is scheduled for 4-5 p.m.

4:00 pm in 341 Altgeld Hall,Friday, February 28, 2020

#### How to Tile Your Bathroom: An Extremely Impractical Guide from a Mathematician

###### Prof. Sean English   [email] (UIUC Math)

Abstract: Tilings have been considered by mathematicians for centuries and by artists for millennia. The main question for tiling problems involves asking if a small number of shapes can be used to cover an entire geometric region without gaps or overlaps. We will briefly talk about some of the history behind tilings, then we will explore many interesting different directions these sorts of problems can take. We will explore some questions as simple as "which regular polygons can tile the plane?" to questions as obscure as "do chickens give rise to a periodic tiling?". Disclaimer: Unless your bathroom is infinite in size, follows spherical or hyperbolic geometry, or has a floor that is more than two dimensional, this talk may not actually be helpful for tiling your bathroom.

Sunday, March 1, 2020

12:30 pm in Salt Fork Center in the Homer Lake Forest Preserve, 2573 S. Homer Lake Road, Homer, IL,Sunday, March 1, 2020

#### Celebration of the Life of Richard L. Bishop

###### (Illinois)

Abstract: A celebration of the life of Richard L. Bishop will be held on Sunday, March 1, at the Salt Fork Center in the Homer Lake Forest Preserve, 2573 S. Homer Lake Road, Homer, IL 61849. The family is having an open house from 12:30-3 p.m. with speakers and sharing of memories at 1 p.m. Memorials may be made to the American Mathematical Society, 201 Charles St., Providence, RI 02904 or to the Sierra Club Prairie Group, P.O. Box 131, Urbana, IL 61803.

Monday, March 2, 2020

9:00 am in Altgeld Hall,Monday, March 2, 2020

Abstract: Visiting Day for students admitted to the Math PhD program and currently living in North America.

3:00 pm in 441 Altgeld Hall,Monday, March 2, 2020

#### Relations between Spectral Sequences

###### Liz Tatum (Illinois Math)

Abstract: Consider a ring spectrum E and a spectrum X. The E-based Adams Spectral Sequence is a tool for approximating the homotopy groups $\pi_{*}X$. Depending on the choice of ring spectrum E, the Adams spectral sequence might be easier to compute, but might give a weaker approximation to $\pi_{*}X$. One could ask “If A, B are two different ring spectra, what can an A-based Adams spectral sequence tells us about a B-based Adams spectral sequence”? In the paper “On Relations Between Adams Spectral Sequences, With an Application to the Stable Homotopy of a Moore Space”, Miller proves a theorem addressing this question. In this talk, I’ll introduce some of the tools Miller uses to formulate and prove this theorem, and outline the previously mentioned application.

3:00 pm in 243 Altgeld Hall,Monday, March 2, 2020

#### Analytic torsions associated with the Rumin complex on contact spheres

###### Akira Kitaoka (University of Tokyo)

Abstract: The Rumin complex, which is defined on contact manifolds, is a resolution of the constant sheaf of $\mathbb{R}$ given by a subquotient of the de Rham complex. In this talk, we explicitly write down all eigenvalues of the Rumin Laplacian on the standard contact spheres, and express the analytic torsion functions associated with the Rumin complex in terms of the Riemann zeta function. In particular, we find that the functions vanish at the origin and determine the analytic torsions.

Tuesday, March 3, 2020

2:00 pm in 243 Altgeld Hall,Tuesday, March 3, 2020

#### The avoidance density of (k, l)-sum-free sets

###### Yifan Jing (University of Illinois, Urbana-Champaign)

Abstract: Let $\mathscr{M}_{(2,1)}(N)$ be the infimum of the size of the largest sum-free subset of any set of $N$ positive integers. An old conjecture in additive combinatorics asserts that there is a constant $c=c(2,1)$ and a function $\omega(N)\to\infty$ as $N\to\infty$, such that $cN+\omega(N)<\mathscr{M}_{(2,1)}(N)<(c+\varepsilon)N$ for any $\varepsilon>0$. The constant $c(2, 1)$ is recently determined by Eberhard, Green, and Manners, while the existence of $\omega(N)$ is still open. In this talk, we consider the analogue conjecture for $(k,l)$-sum-free sets. We determine the constant $c(k,l)$ for every $(k,l)$, and prove the existence of the function $\omega(N)$ for infinitely many $(k,l)$. The proof uses tools from probabilistic combinatorics, fourier analysis, and nonstandard analysis.

Wednesday, March 4, 2020

2:00 pm in 447 Altgeld Hall,Wednesday, March 4, 2020

#### Deformation and Obstruction for moduli of sheaves

###### Lutian Zhao (Illinois Math)

Abstract: In this talk, I’ll introduce the deformation theory for moduli of sheaves,. I’ll give a proof for the description of tangent space of moduli of sheaves and the condition for which this moduli space is smooth. The final goal of this talk is to introduce the construction for Simpson’s moduli space of sheaves.

Thursday, March 5, 2020

11:00 am in 241 Altgeld Hall,Thursday, March 5, 2020

#### Potential automorphy of Galois representations into general spin groups

###### Shiang Tang (Illinois Math)

Abstract: Given a connected reductive group $G$ defined over a number field $F$, the Langlands program predicts a connection between suitable automorphic representations of $G(\mathbb A_F)$ and geometric $p$-adic Galois representations $\mathrm{Gal}(\overline{F}/F) \to {}^LG$ into the L-group of $G$. Striking work of Arno Kret and Sug Woo Shin constructs the automorphic-to-Galois direction when $G$ is the group $\mathrm{GSp}_{2n}$ over a totally real field $F$, and $\pi$ is a cuspidal automorphic representation of $\mathrm{GSp}_{2n}(\mathbb A_F)$ that is discrete series at all infinite places and is a twist of the Steinberg representation at some finite place: To such a $\pi$, they attach geometric $p$-adic Galois representations $\rho_{\pi}: \mathrm{Gal}(\overline{F}/F) \to \mathrm{GSpin}_{2n+1}$. In this work we establish a partial converse, proving a potential automorphy theorem, and some applications, for suitable $\mathrm{GSpin}_{2n+1}$-valued Galois representations. In this talk, I will explain the background materials and the known results in this direction before touching upon the main theorems of this work.

4:00 pm in 245 Altgeld Hall,Thursday, March 5, 2020

#### Plane Trees and Algebraic Numbers

###### George Shabat (Russian State University for the Humanities and Independent University of Moscow)

Abstract: The main part of the talk will be devoted to an elementary version of the deep relations between the combinatorial topology and the arithmetic geometry. Namely, an object defined over the field of algebraic numbers, a polynomial with algebraic coefficients and only two finite critical values, will be associated to an arbitrary plane tree. Some applications of this construction will be presented, including polynomial Pell equations and quasi-elliptic integrals (going back to N.-H. Abel). The relations with finite groups and Galois theory will be outlined. At the end of the talk the possible generalizations will be discussed, including the dessins d'enfants theory initiated by Grothendieck.

Friday, March 6, 2020

3:00 pm in 347 Altgeld Hall,Friday, March 6, 2020

#### Screened Sobolev Spaces

###### David Altizio (UIUC Math)

Abstract: TBA

4:00 pm in 341 Altgeld Hall,Friday, March 6, 2020

#### TBA

###### TBA (UIUC Math)

Monday, March 9, 2020

3:00 pm in 441 Altgeld Hall,Monday, March 9, 2020

#### Some applications of tangent categories

###### Tsutomu Okano (Illinois Math)

Abstract: The cotangent complex formalism is a useful framework for developing obstruction theoretic tools such as Andre-Quillen cohomology. I will present a theorem that identifies the tangent categories of Cat_S, where S is some symmetric monoidal infinity-category. Some more example applications of this formalism will follow.

Tuesday, March 10, 2020

11:00 am in 243 Altgeld Hall,Tuesday, March 10, 2020

#### $C_2$-equivariant homotopy groups of spheres

###### Mark Behrens (Notre Dame Math)

Abstract: I will explain how $RO(C_2)$-graded $C_2$-equivariant homotopy groups of spheres can be deduced from non-equivariant stable homotopy groups of stunted projective spaces, and the computation of Mahowald invariants.

Wednesday, March 11, 2020

3:30 pm in 341 Altgeld Hall,Wednesday, March 11, 2020

#### Random walks on graphs and spectral radius: part 1

###### Anush Tserunyan (UIUC Math)

Abstract: To motivate Kesten's theorem and its version for IRS, we will discuss random walks on graphs, the associated Markov operators, and the spectral radius. We will prove that the spectral radius is equal to the norm of the Markov operator.

Thursday, March 12, 2020

2:00 pm in 347 Altgeld Hall,Thursday, March 12, 2020

#### Tracy-Widom distribution and spherical spin glass (Part II)

###### Qiang Wu (UIUC Math)

Abstract: I will talk about the connection between spherical spin glass(SSK) and random matrices, in particular, the fluctuation of free energy in SSK on low temperatures regime is given by GOE Tracy-Widom distribution.

4:00 pm in 245 Altgeld Hall,Thursday, March 12, 2020

#### Stability of roll wave solutions in inclined shallow-water flow

###### Kevin Zumbrun   [email] (Indiana University Bloomington)

Abstract: We review recent developments in stability of periodic roll-wave solutions of the Saint Venant equations for inclined shallow-water flow. Such waves are well-known instances of hydrodynamic instability, playing an important role in hydraulic engineering, for example, flow in a channel or dam spillway. Until recently, the analysis of their stability has been mainly by formal analysis in the weakly unstable or near-onset'' regime. However, hydraulic engineering applications are mainly in the strongly unstable regime far from onset. We discuss here a unified framework developed together with Blake Barker, Mat Johnson, Pascal Noble, Miguel Rodrigues, and Zhao Yang for the study of roll wave stability across all parameter regimes, by a combination of rigorous analysis and numerical computation. The culmination of our analysis is a complete stability diagram, of which the low-frequency stability boundary is, remarkably, given explicitly as the solution of a a cubic equation in the parameters of the solution space.

Tuesday, March 24, 2020

11:00 am in 243 Altgeld Hall,Tuesday, March 24, 2020

#### CANCELLED

###### Christina Osborne (OSU Math)

1:00 pm in https://illinois.zoom.us/j/249415194,Tuesday, March 24, 2020

#### Borel structures on the space of left orderings

###### Filippo Calderoni (UIC Math)

Abstract: In this talk I will present some recent results on left-orderable groups and their interplay with descriptive set theory. We shall discuss how Borel classification can be used to analyze the space of left-orderings of a given countable group modulo the conjugacy action. In particular we shall see that if G is a countable nonabelian free group, then the conjugacy relation on its space of left orderings is a universal countable Borel equivalence relation. This is joint work with A. Clay.

Thursday, March 26, 2020

4:00 pm in 245 Altgeld Hall,Thursday, March 26, 2020

#### Trapping, resonances, and the decay of waves [to be rescheduled Fall 2020]

###### Jared Wunsch   [email] (Northwestern University)

Abstract: I will discuss some results, new and old, involving the influence of the geometry on the decay of waves. The quantum correspondence principle dictates that at high frequency, the dynamics of particle trajectories should be related to the rate at which the energy of a solution to the wave or Schrödinger equation decays. This relationship is mediated by the existence of resonances, which correspond to states with a finite (but possibly long) lifetime that ultimately decay owing to tunneling effects. I will discuss what we know about the existence and nonexistence of resonances, and focus on some recent results about resonances associated to the subtle effects of diffraction in classical and quantum problems that have singular structures in a metric or potential.

Friday, March 27, 2020

4:00 pm in 341 Altgeld Hall,Friday, March 27, 2020

#### TBA

###### TBA (UIUC Math)

Wednesday, April 1, 2020

3:30 pm in https://illinois.zoom.us/j/806582029 (email Anush Tserunyan for the password),Wednesday, April 1, 2020

#### Random walks on graphs and spectral radius: part 2

###### Anush Tserunyan (UIUC Math)

Abstract: We will continue discussing random walks on graphs and their spectral radius, computing that the spectral radius of $\mathbb{Z}^d$ is $1$, whereas it is less than $1$ for a $d$-regular tree with $d \ge 3$. We will then discuss the deep and general theorem of Kesten characterizing amenable normal subgroups and derive a couple of striking corollaries: a characterization of finitely generated amenable groups and a rigidity result for the free groups.

Thursday, April 2, 2020

11:00 amThursday, April 2, 2020

#### Poisson imitators and sieve theory

Abstract: I'll describe how sieve theory is actually a question about probability distributions whose low moments agree with the low moments of Poisson distributions. In particular, we can derive Selberg’s “parity problem” without using properties of the Möbius function or the Liouville function - instead, we use the fact that the alternating group forms a subgroup of the symmetric group.

4:00 pm in 245 Altgeld Hall,Thursday, April 2, 2020

#### To Be Rescheduled Fall 2020

###### Kevin Purbhoo   [email] (University of Waterloo)

Abstract: To come.

Friday, April 3, 2020

3:00 pm in https://illinois.zoom.us/j/521113604 (email Anush Tserunyan for the password),Friday, April 3, 2020

#### A dynamical obstruction for classification by actions of TSI Polish groups

###### Aristotelis Panagiotopoulos (Caltech Math)

Abstract: A big part of mathematical activity revolves around classification problems. However, not every classification problem has a satisfactory solution, and some classification problems are more complicated than others. Dynamical properties such as generic ergodicity and turbulence are crucial in the development of a rich complexity theory for classification problems. In this talk we will review some of the existing anti-classification techniques and we will introduce a new obstruction for classification by orbit equivalence relations of TSI Polish groups; a topological group is TSI if it admits a compatible two side invariant metric. We will then show that the Wreath product of any two non-compact subgroups of $S_{\infty}$ admits an action whose orbit equivalence relation is generically ergodic with respect to orbit equivalence relations of TSI group actions.
This is joint work with Shaun Allison.

4:00 pm in TBA,Friday, April 3, 2020

#### Integration Bee

###### MATRIX (UIUC Math)

Wednesday, April 8, 2020

3:30 pm in https://illinois.zoom.us/j/806582029 (email Anush Tserunyan for the password),Wednesday, April 8, 2020

#### Random walks on graphs and spectral radius: part 3

###### Anush Tserunyan (UIUC Math)

Abstract: We will continue discussing random walks on graphs. In this last talk of the series, we will consider recurrence/transience of random walks, proving that this is determined by whether or not the expectation of the number of visits to a fixed vertex is infinite. We will use it to deduce that the simple random walk on nonamenable Cayley graphs is transient. We will also show that the simple random walk on $\mathbb{Z}^d$ is recurrent if and only if $d \le 2$. As Kakutani put it, "A drunk man will find his way home, but a drunk bird may get lost forever."

Thursday, April 9, 2020

12:00 pm in The Zoom-verse,Thursday, April 9, 2020

#### Gaps of saddle connection directions for some branched covers of tori

###### Anthony Sanchez (U. Washington Math)

Abstract: Consider the class of translation surfaces given by gluing two identical tori along a slit. Every such surface has genus two and two cone-type singularities of angle $4\pi$. There is a distinguished set of geodesics called saddle connections that are the geodesics between cone points. We can recover a vector in the plane representing the saddle connection by keeping track of the amount that the saddle connection moves in the vertical and horizontal direction. How random is the set of saddle connections? We shed light to this question by considering the gap distribution of slopes of saddle connections. Zoom Meeting ID: 460 321 230. Email clein for password.

3:00 pm in 347 Altgeld Hall,Thursday, April 9, 2020

#### **Rescheduled due to COVID-19 campus-shutdown**

###### Benjamin Braun   [email] (University of Kentucky)

Abstract: TBA

Friday, April 10, 2020

4:00 pm in Zoom Meeting,Friday, April 10, 2020

#### Python Workshop

###### Kyle Begovich (University of Illinois at Urbana-Champaign)

Abstract: Kyle will be working at Google as a software engineer at the end of this semester. He'll be running a workshop on Python through Project Euler, a great way to get a jumpstart on learning some of the basics of solving problems through coding. No prior coding experience is needed for this workshop! Project Euler is an online platform for students in any discipline to work on “challenging mathematical/computer programming problems that will require more than just mathematical insights to solve”. In this time of remote work, a platform that is online and well-suited to developing problem-solving techniques can provide a good outlet for learning new skills, developing your analytic senses, and interacting with a community of mathematical thinkers. This seminar will help you set up an environment, discuss common approaches to work on these problems, and walk through some early problems to get you started. Please email undergradseminar@math.illinois.edu for Zoom link.

4:00 pm in 341 Altgeld Hall,Friday, April 10, 2020

#### TBA

###### TBA (UIUC Math)

Tuesday, April 14, 2020

2:00 pm in Zoom,Tuesday, April 14, 2020

#### On stability of triangle-free graphs

###### Felix Clemen (University of Illinois, Urbana-Champaign)

Abstract: In the first part of this talk we take a look at the structure of $K_{r+1}$-free graphs with number of edges slightly below the Turan number $\mathrm{ex}(n,K_{r+1})$. The Erdos-Simonovits stability theorem states that for all $\epsilon>0$ there exists $\alpha>0$ such that if $G$ is a $K_{r+1}$-free graph on $n$ vertices with $e(G) > \mathrm{ex}(n,K_{r+1}) - \alpha n^2$, then one can remove $\epsilon n^2$ edges from $G$ to obtain an $r$-partite graph. Furedi gave a short proof that one can choose $\alpha=\epsilon$. We give a bound for the relationship of $\alpha$ and $\epsilon$ which is asymptotically sharp as $\epsilon$ goes to $0$. This is joint work with Jozsef Balogh, Mikhail Lavrov, Bernard Lidicky and Florian Pfender.

In the second part of the talk we study graphs with number of edges slightly above the Turan number $\mathrm{ex}(n,K_{r+1})$. What is the minimum number of cliques of such graphs? Lovasz and Simonovits proved that an n-vertex graph with $e(G) > \mathrm{ex}(n,K_{3}) + t$ contains at least $t n/2$ triangles. Katona and Xiao considered the same problem under the additional condition that there are no $s$ vertices covering all triangles. They settled the case $t=1$ and $s=2$. Solving their conjecture, we determine the minimum number of triangles for general $s$ and $t$. Additionally, solving another conjecture of Katona and Xiao, we extend the theory for considering cliques instead of triangles. This is joint work with Jozsef Balogh.

Wednesday, April 15, 2020

3:30 pm in https://illinois.zoom.us/j/806582029 (email Anush Tserunyan for password),Wednesday, April 15, 2020

#### An obstruction for classification by actions of TSI Polish groups, part 2: proofs

###### Aristotelis Panagiotopoulos (Caltech Math)

Abstract: In this talk we will over the proof of my recent result (joint with Shaun Allison) that if a $G$-space $X$ is generically unbalanced then its orbit equivalence relation is not classifiable by actions of TSI Polish groups. I will also discuss how one can use this result to show that Morita equivalence between continuous-trace $C^*$ algebras, as well as isomorphism between Hermitian line bundles, are not classifiable by TSI group actions.

Thursday, April 16, 2020

11:00 am in Zoom (email Patrick Allen for the meeting ID and password),Thursday, April 16, 2020

#### The Wiles defect for Hecke algebras that are not complete intersections

###### Jeff Manning (University of California at Los Angeles)

Abstract: In his work on modularity theorems, Wiles proved a numerical criterion for a map of rings R->T to be an isomorphism of complete intersections. He used this to show that certain deformation rings and Hecke algebras associated to a mod p Galois representation at non-minimal level were isomorphic and complete intersections, provided the same was true at minimal level. In addition to proving modularity theorems, this numerical criterion also implies a connection between the order of a certain Selmer group and a special value of an L-function. In this talk I will consider the case of a Hecke algebra acting on the cohomology a Shimura curve associated to a quaternion algebra. In this case, one has an analogous map of ring R->T which is known to be an isomorphism, but in many cases the rings R and T fail to be complete intersections. This means that Wiles' numerical criterion will fail to hold. I will describe a method for precisely computing the extent to which the numerical criterion fails (i.e. the 'Wiles defect"), which will turn out to be determined entirely by local information at the primes dividing the discriminant of the quaternion algebra. This is joint work with Gebhard Bockle and Chandrashekhar Khare.

Friday, April 17, 2020

4:00 pm in 341 Altgeld Hall,Friday, April 17, 2020

#### TBA

###### TBA (UIUC Math)

Tuesday, April 21, 2020

1:00 pm in https://illinois.zoom.us/j/422077317 (email Anush Tserunyan for the password),Tuesday, April 21, 2020

#### The universal theory of random groups

###### Meng-Che (Turbo) Ho (Purdue University)

Abstract: Random groups are proposed by Gromov as a model to study the typical behavior of finitely presented groups. They share many properties of the free group, and Knight asked if they also have the same first-order theory as the free group. In this talk, we will discuss a positive result for the first step toward this question, namely the universal theory of random groups. The main tools we use are the machinery developed in Sela’s solution to the Tarski problem.
This is joint work with Remi Coulon and Alan Logan.

Wednesday, April 22, 2020

3:30 pm in https://illinois.zoom.us/j/806582029 (email Anush Tserunyan for password),Wednesday, April 22, 2020

#### Introduction to 𝓁2-Betti numbers for groups, part 1: Homology

###### Ruiyuan (Ronnie) Chen (UIUC Math)

Thursday, April 23, 2020

11:00 am in Zoom (email Patrick Allen for the meeting ID and password),Thursday, April 23, 2020

#### Log-free zero density estimates for automorphic L-functions

###### Chen An (Duke University)

Abstract: One of the most important topics in number theory is the study of zeros of L-functions. Near the edge of the critical strip, one may show that the number of zeros for certain L-functions is small; such a result is called a zero density estimate. For Dirichlet L-functions, this topic is well understood by the work of Gallagher, Selberg, Jutila, etc. For families of automorphic L-functions, Kowalski and Michel show that the number of zeros near the edge of the critical strip is small on average. The proof uses a large sieve inequality with key objects called pseudo-characters. I will present my recent progress on the refinement of Kowalski-Michel's large sieve inequality, which gives rise to a better zero density estimate for automorphic L-functions.

Friday, April 24, 2020

4:00 pm in Zoom Meeting,Friday, April 24, 2020

#### The Arithmetic of Quadratic Fields

###### Patrick Allen (UIUC Math)

Abstract: It is often said that the prime numbers are the building blocks of the integers, the precise statement of which is the fundamental theorem of arithmetic: any integer greater than one can be factored uniquely as a product of prime numbers. What if we move beyond the integers? The simplest cases to consider are the analogues of the integers in what are called quadratic fields, which are number systems obtained from adding to the rational numbers the square root of some fixed integer. Whether or not these quadratic integers satisfy the analogue of the fundamental theorem of arithmetic turns out to be very subtle and both what is known and what is not known are rather surprising. Please email drthoma2@illinois.edu for Zoom link.

Wednesday, April 29, 2020

3:30 pm in https://illinois.zoom.us/j/806582029 (email Anush Tserunyan for password),Wednesday, April 29, 2020

#### Descriptive combinatorics, distributed algorithms, and the Lovász Local Lemma: proofs

###### Anton Bernshteyn (CMU Math)

Abstract: Descriptive combinatorics is the study of combinatorial problems (such as graph coloring) under additional topological or measure-theoretic regularity restrictions. It turns out that there is a close relationship between descriptive combinatorics and distributed computing, i.e., the area of computer science concerned with problems that can be solved efficiently by a decentralized network of processors. At the heart of this relationship lies the Lovász Local Lemma—an important tool in probabilistic combinatorics—and its measurable versions. In this talk I will sketch the arguments behind this relationship.

Thursday, April 30, 2020

11:00 am in 241 Altgeld Hall,Thursday, April 30, 2020

#### Locally Split Galois Representations and Hilbert Modular Forms of Partial Weight One

###### Eric Stubley (University of Chicago)

Abstract: The Galois representation attached to a p-ordinary eigenform is upper triangular when restricted to a decomposition group at p. A natural question to ask is under what conditions this upper triangular decomposition splits as a direct sum. Ghate and Vatsal have shown that for Galois representations coming from families of p-ordinary eigenforms, the restriction to a decomposition group at p is split if and only if the family has complex multiplication; in their proof, the weight one members of the family play a key role. I'll talk about work in progress which aims to answer similar questions in the case of Galois representations for a totally real field which are split at only some of the primes above p. In this work Hilbert modular forms of partial weight one play a central role; I'll discuss what is known about them and to what extent the techniques of Ghate and Vatsal can be adapted to this situation.

3:00 pm in 347 Altgeld Hall,Thursday, April 30, 2020

#### **Rescheduled due to COVID-19 campus-shutdown**

###### Wai Ling Yee   [email] (University of Windsor)

Abstract: To Be Announced

Friday, May 1, 2020

4:00 pm in Zoom Meeting,Friday, May 1, 2020

#### IGL Fall 2020 Info Session

###### Alexi Taylor Block Gorman   [email] (UIUC Math)

Abstract: We will be joined by Alexi Block Gorman! She is a third-year PhD student here in the math program, and also the Illinois Geometry Lab's (IGL's) current research manager. She'll be joining us to to talk about applying to the IGL, as well as answering any questions you might have about how the IGL will be run during the next semester. If you're thinking about applying, or even just wondering what exactly the IGL is, this will be an important meeting to join! For a meeting link, please email drthoma2@illinois.edu.

Monday, May 4, 2020

2:00 pm in https://illinois.zoom.us/j/91278653762 (email Anush Tserunyan for password),Monday, May 4, 2020

#### Probabilistic limit theorems and percolation on Borel graphs

###### Grigory Terlov (UIUC Math)

Abstract: The first part of this preliminary examination talk I will dedicate to conditional central limit theorems, which, roughly speaking, are statements about sums of conditional random variables converging to the normal distribution. I will provide necessary background for the problem, discuss the approach of studying explicit rates of convergence in dependent settings via Stein's method, and mention current results in application to some examples.
In the second part, I will introduce a model of bond percolation on locally finite Borel graphs on a standard probability space. This model resembles classical Bernoulli percolation with an addition of some dependencies between the edges. The main motivation for this model is the fact that percolation theory on countable graphs often allows for a construction of subgraphs with desired properties and it is of strong interest in measured group theory and measured graph combinatorics to extend it to Borel graphs. I will discuss a spectacular example of this: a measured group theoretic approach to the Day–von Neumann question, known as the Gaboriau–Lyons theorem.

Wednesday, May 6, 2020

3:30 pm in https://illinois.zoom.us/j/806582029 (email Anush Tserunyan for password),Wednesday, May 6, 2020

#### Introduction to 𝓁2-Betti numbers for groups, part 2: Group homology

###### Ruiyuan (Ronnie) Chen (UIUC Math)

Abstract: We will discuss equivariant homology for spaces equipped with a group action, defined as the ordinary homology of the "homotopy quotient" of the action. Group homology is the special case for the trivial action on a point. Time permitting, we will then begin to discuss the theory of (Hattori-Stallings) traces, which provides a better-behaved equivariant analog of torsion-free rank.

Thursday, May 7, 2020

4:00 pm in 245 Altgeld Hall,Thursday, May 7, 2020

#### To Be Rescheduled Fall 2020

###### Sami Assaf   [email] (University of Southern California)

Abstract: To come.

5:00 pm in Online (see abstract for details),Thursday, May 7, 2020

#### Improving societal governance in the age of AI

###### Wendy K. Tam Cho   [email] (University of Illinois at Urbana-Champaign)

Abstract: Important insights into societal governance can be gained through an interdisciplinary approach that combines research from many fields, including statistics, operations research, computer science, high performance computing, math, law, and political science. My work integrates insights from all of these disciplines to create a novel approach for analyzing and reforming the redistricting process in the United States. While the development of these computational algorithms is important, understanding the role of this technology and managing its use is critical to improving societal governance in the digital age. To register: https://mailchi.mp/fields.utoronto.ca/2020keyfitzlecture?e=fa128d01a4

Tuesday, May 12, 2020

2:00 pm in Zoom,Tuesday, May 12, 2020

#### Lower bounds for difference bases

###### Anton Bernshteyn (Carnegie Mellon University)

Abstract: A difference basis with respect to $n$ is a subset $A \subseteq \mathbb{Z}$ such that $A - A \supseteq [n]$. Rédei and Rényi showed that the minimum size of a difference basis with respect to $n$ is $(c+o(1))\sqrt{n}$ for some positive constant $c$. The precise value of $c$ is not known, but some bounds are available, and I will discuss them in this talk. This is joint work with Michael Tait (Villanova University).

Wednesday, May 13, 2020

3:30 pm in https://illinois.zoom.us/j/806582029 (email Anush Tserunyan for password),Wednesday, May 13, 2020

#### Introduction to 𝓁2-Betti numbers for groups, part 3: 𝓁2-homology

###### Ruiyuan (Ronnie) Chen (UIUC Math)

Abstract: We will first finish our discussion of ordinary group homology by giving an alternative definition via equivariant chain complexes, completely bypassing topology. We will then discuss $\ell^2$-homology, defined by replacing (chain) homotopy quotients by $\ell^2$-completion to Hilbert $\Gamma$-modules. The $\ell^2$-Betti numbers are the von Neumann dimensions of the resulting homology Hilbert $\Gamma$-modules.

Tuesday, June 2, 2020

2:00 pm in Zoom,Tuesday, June 2, 2020

#### Turan numbers for a $4$-uniform hypergraph

###### Karen Gunderson (University of Manitoba)

Abstract: For any $r\geq 2$, an $r$-uniform hypergraph $\mathcal{H}$, and integer $n$, the Turan number for $\mathcal{H}$ is the maximum number of hyperedges in any $r$-uniform hypergraph on $n$ vertices containing no copy of $\mathcal{H}$. While the Turan numbers of graphs are well-understood and exact Turan numbers are known for some classes of graphs, few exact results are known for the cases $r \geq 3$. I will present a construction, using quadratic residues, for an infinite family of hypergraphs having no copy of the $4$-uniform hypergraph on $5$ vertices with $3$ hyperedges, with the maximum number of hyperedges subject to this condition. I will also describe a connection between this construction and a switching' operation on tournaments, with applications to finding new bounds on Turan numbers for other small hypergraphs.

Tuesday, June 23, 2020

2:00 pm in Zoom,Tuesday, June 23, 2020

#### Linear cycles of consecutive lengths in linear hypergraphs

###### Tao Jiang (Miami University)

Abstract: A well-known result of Verstraete states that for each integer $k\geq 2$ every graph $G$ with average degree at least $8k$ contains cycles of $k$ consecutive even lengths, the shortest of which is at most twice the radius of $G$. Besides being interesting on its own, Verstraete's result also immediately implies that the Turan number $\mathrm{ex}(n, C_{2k})$ of the even cycle of length $2k$ is at most $8kn^{1+1/k}$, hence retrieving the classic theorem of Erdos and of Bondy and Simonovits with improved coefficients.

In this talk, we establish two extensions of Verstraete's result for linear cycles in linear $r$-uniform hypergraphs, where $r\geq 3$. A hypergraph is linear if every two edges intersect in at most one vertex. An $r$-uniform linear cycle $C^r_m$ is an $r$-uniform hypergraph consisting a cyclic list of edges $e_1,\dots,e_m$ such that consecutive edges intersect in one vertex and otherwise they are disjoint.

We prove that for all fixed $r\geq 3$, $k\geq 3$, there exists a constant $c_1=c_1(r)$ such that every linear $r$-uniform hypergraph $G$ with average degree $d(G)\geq c_1k$ contains linear cycles of $k$ consecutive even lengths, the shortest of which is at most $2 \frac{ \log n}{\log (d(G)/k)}$. In particular, as an immediate corollary, we retrieve the current best known upper bound on the linear Turan number of $C^r_{2k}$ with improved coefficients.

Furthermore, we show that for any fixed integers $r,k\geq 3$, there exists a constant $c_2=c_2(r)$ such that every $n$-vertex linear $r$-uniform graph with average degree $d(G)\geq c_2 k$, contains linear cycles of $k$ consecutive lengths (even and odd together), the shortest of which has length at most $6\frac{\log n}{\log (d(G)/k)} +5$. Both the degree condition and the shortest length among the cycles guaranteed are tight up to a constant factor. As a corollary it follows that there exists a constant $c_3=c_3(r)$ such that every $r$-uniform linear hypergraph with average degree at least $c_3$ contains an even cycle and an odd cycle of lengths at most $O(\log n)$, which is interesting on its own.

This is joint work with Jie Ma and Liana Yepremyan.

Please email Sean at SEnglish (at) illinois (dot) edu for the Zoom ID.

Tuesday, July 7, 2020

2:00 pm in Zoom,Tuesday, July 7, 2020

#### A survey of Berge-Turan hypergraph problems

###### Cory Palmer (University of Montana)

Abstract: For a graph $F$, we say that a hypergraph $\mathcal{H}$ is a Berge-$F$ if there is an injection $f: V(F) \rightarrow V(\mathcal{H})$ and bijection $f':E(F)\rightarrow E(\mathcal{H})$ such that for every edge $uv\in E(F)$ we have $\{f(u),f(v)\}\subseteq f'(uv)$. Alternatively, $\mathcal{H}$ is Berge-$F$ if we can embed a distinct graph edge into each hyperedge of $\mathcal{H}$ to obtain a copy of $F$. Note that for a fixed $F$ there are many different hypergraphs that are a Berge-$F$ and a fixed hypergraph $\mathcal{H}$ can be a Berge-$F$ for more than one graph $F$.

A hypergraph is Berge-$F$-free if it contains no subhypergraph isomorphic to any Berge-$F$. The maximum number of edges in an Berge-$F$-free $n$-vertex $r$-graph is denoted $\mathrm{ex}_r(n,\textrm{Berge-}F)$. Observe that when $r=2$, then a Berge-$F$ is simply the graph $F$ and then we are investigating the classical Turan function $\mathrm{ex}(n,F)$.

Early work on $\mathrm{ex}_r(n,\textrm{Berge-}F)$ focused on the case when $F$ is a path or a cycle. Results of Gyori, Katona, and Lemons and Davoodi, Gyori, Methuku and Tompkins establish an analogue of the Erdos-Gallai theorem for Berge paths. Gyori and Lemons proved $\mathrm{ex}_r(n,\textrm{Berge-}C_{2k}) = O(n^{1+1/k})$ for $r \geq 3$. This matches the order of magnitude of the bound found in the graph case. They prove the same upper bound for $\textrm{Berge-}C_{2k+1}$-free hypergraphs which is significantly different from the graph case.

In this talk we will survey results on the function $\mathrm{ex}_r(n,\textrm{Berge-}F)$ for various graphs $F$. We will also discuss the connection to other extremal problems and give a number of interesting open problems.

Tuesday, July 14, 2020

2:00 pm in Zoom,Tuesday, July 14, 2020

#### On the graph homomorphism density functional

###### Alexander Sidorenko

Abstract: Let $H$ be a graph with vertices $1,2,\ldots,n$ and edge-set $E$. We associate with it a functional that acts on bounded measurable (symmetric) functions $F: \: [0,1]^2 \to \mathbb{R}$, namely $$t_H(F) \; = \; \int_{[0,1]^n} \prod_{\{i,j\} \in E} F(x_i,x_j) \: dx_1 dx_2 \cdots dx_n \; .$$ This notion arises from counting copies of $H$ in a large graph $F$.

We will review results and open problems in such areas as
Majorization ($H$ majorizes $G$ when $t_H(F) \geq t_G(F)$ for all $F$),
Positivity of $t_H$.
Convexity of $t_H$.

Thursday, August 20, 2020

3:00 pm in Zoom,Thursday, August 20, 2020

#### To Be Announced

###### Benjamin Braun   [email] (University of Kentucky)

Abstract: To Be Announced

Thursday, August 27, 2020

4:00 pm in 245 Altgeld Hall,Thursday, August 27, 2020

#### To Be Announced

###### Noga Alon   [email] (Princeton University)

Abstract: To come.

Thursday, September 3, 2020

4:00 pm in 245 Altgeld Hall,Thursday, September 3, 2020

#### To Be Announced

###### Marius Junge   [email] (University of Illinois at Urbana-Champaign)

Abstract: To come.

Thursday, September 10, 2020

6:00 pm in Zoom,Thursday, September 10, 2020

#### To Be Announced

###### Dominic Searles   [email] (University of Otago)

Abstract: To Be Announced

Thursday, October 22, 2020

12:00 pm in Silvercreek Restaurant, Garden Room,Thursday, October 22, 2020

#### 23rd Annual Math Dept. Retiree Luncheon

Abstract: An invitation will be sent in September, with more information posted here.

4:00 pm in 425 Altgeld Hall,Thursday, October 22, 2020

#### To Be Announced

###### Ralf Hiptmair   [email] (ETH Zurich)

Abstract: To come.

Thursday, November 5, 2020

4:00 pm in 245 Altgeld Hall,Thursday, November 5, 2020

#### To Be Announced

###### Mikhail Ostrovskii   [email] (St. John's University)

Abstract: To come