Department of

Mathematics


Seminar Calendar
for events the week of Thursday, November 21, 2019.

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More information on this calendar program is available.
Questions regarding events or the calendar should be directed to Tori Corkery.
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Monday, November 18, 2019

3:00 pm in 441 Altgeld Hall,Monday, November 18, 2019

Splitting induced from transfer

Abhra Abir Kundu (UIUC Math)

Abstract: Associated to a finite covering map is a map in homology and cohomology, known as transfer, which goes in the opposite direction. This map usually doesn’t come from a map at the level of spaces but interestingly enough it does when we look at its version in spectra. This often leads to interesting splittings in spectra after appropriate localisation. In this talk, I will explain how one gets this map. Then I will concentrate on one such transfer and the splitting induced from it. I will show a few pictures of algebraic objects associated to this transfer which will illustrate this splitting and then end by observing some facts about the splitting from those pictures.

3:00 pm in 243 Altgeld Hall,Monday, November 18, 2019

Local symplectic groupoids, their generating functions and quantization

Alejandro Cabrera (Universidade Federal do Rio de Janeiro)

Abstract: We will first review how Poisson brackets can be seen as the infinitesimal data underlying certain types of transformations which form a 'local symplectic groupoid'. The composition map in this structure defines a lagrangian submanifold and this, in turn, can be locally described by a generating function. We show how to obtain an explicit analytic formula for this generating function for any Poisson bracket on a coordinate space and how this formula reproduces, upon suitable Taylor expansion, a certain tree-level part of a formula given by Kontsevich in the context of quantization. We will also discuss further the link between the generating function, symplectic realizations, and general quantizations, as well as their connection to a related 2d field theory (the 'Poisson-sigma model'). The first part of this talk is based on joint work with B. Dherin.

5:00 pm in 241 Altgeld Hall,Monday, November 18, 2019

Graph Hormanders

Marius Junge (University of Illinois at Urbana-Champaign)

Abstract: I would talk about the link between graph Laplacians and Hormander systems of orthogonal groups.

Tuesday, November 19, 2019

11:00 am in 347 Altgeld Hall,Tuesday, November 19, 2019

Two theories of real cyclotomic spectra

Jay Shah

Abstract: The topological Hochschild homology $THH(R)$ constitutes a powerful and well-studied invariant of an associative ring $R$. As originally shown by Bokstedt, Hsiang and Madsen, $THH(R)$ admits the elaborate structure of a cyclotomic spectrum, whose formulation depends upon equivariant stable homotopy theory. More recently, inspired by considerations in p-adic Hodge theory, Nikolaus and Scholze demonstrated (under a bounded-below assumption) that the data of a cyclotomic spectrum is entirely captured by a system of circle-equivariant Frobenius maps, one for each prime p. They also give a formula for the topological cyclic homology $TC(R)$ directly from these maps. The purpose of this talk is to extend the work of Nikolaus and Scholze in order to accommodate the study of real topological Hochschild homology $THR$, which is a $C_2$-equivariant refinement of $THH$ defined for an associative ring with an anti-involution, or more generally an $E_\sigma$-algebra in $C_2$-spectra. The key idea is to make use of the $C_2$-parametrized Tate construction. This is joint work with J.D. Quigley and is based on the arXiv preprint 1909.03920.

1:00 pm in 347 Altgeld Hall,Tuesday, November 19, 2019

Global Strichartz estimates for the semiperiodic Schrodinger equation.

Alex Barron (illinois Math)

Abstract: We will discuss some recent results related to space-time estimates for solutions to the linear Schrodinger equation on manifolds which are products of tori and Euclidean space (e.g. a cylinder embedded in R^3 ). On these manifolds it is possible to prove certain analogues of the classical Euclidean Strichartz estimates which are scale-invariant and global-in-time. These estimates are strong enough to prove small-data scattering for solutions to the critical quintic NLS on R × T and the critical cubic NLS on R^2 × T (where T is the one-dimensional torus).

2:00 pm in 243 Altgeld Hall,Tuesday, November 19, 2019

An overview of Erdős-Rothschild problems and their rainbow variants

Lina Li (Illinois Math)

Abstract: In 1974, Erdős and Rothschild conjectured that the complete bipartite graph has the maximum number of two-edge-colorings without monochromatic triangles over all n-vertex graphs. Since then, a new class of colored extremal problems has been extensively studied by many researchers on various discrete structures, such as graphs, hypergraphs, Boolean lattices and sets.

In this talk, I will first give an overview of some previous results on this topic. The second half of this talk is to explore the rainbow variants of the Erdős-Rothschild problem. With Jozsef Balogh, we confirm conjectures of Benevides, Hoppen and Sampaio, and Hoppen, Lefmann, and Odermann, and completes the characterization of the extremal graphs for the edge-colorings without rainbow triangles. Next, we study a similar question on sum-free sets, where we describe the extremal configurations for integer colorings with forbidden rainbow sums. The latter is joint work with Yangyang Cheng, Yifan Jing, Wenling Zhou and Guanghui Wang.

2:00 pm in 347 Altgeld Hall,Tuesday, November 19, 2019

Absolute continuity and singularity for probability measures induced by a drift transform of the independent sum of Brownian motion and symmetric stable process

Ruili Song (Nanjing University of Finance and Economics, and UIUC Math)

Abstract: We consider a Levy process $X$ which is the independent sum of a Brownian motion and a symmetric $\alpha$-stable process in $R^d$. The probability measure $P^b$ is induced by the drift transform of $X$ via the vector valued function $b$. We study mutual absolute continuity and singularity of $P$ and $P^b$ on the path space. We also investigate the problem of finiteness of the relative entropy of these measures on $R^d$ ($d\ge 3$).

Wednesday, November 20, 2019

4:00 pm in 447 Altgeld Hall,Wednesday, November 20, 2019

Geometry and arithmetic of curves over finite fields

Ravi Donepudi (Illinois Math)

Abstract: The theory of algebraic curves over a finite field runs entirely parallel to the classical theory of number fields (finite extensions of the rational numbers). Analogues of many results that are long standing open conjectures in the number field case are theorems in the case of curves over finite fields. We will introduce the key concepts in this area, survey important results and (time permitting) state some original results. This talk assumes only a passing familiarity with finite fields.

Thursday, November 21, 2019

11:00 am in 241 Altgeld Hall,Thursday, November 21, 2019

Sums with the Mobius function twisted by characters with powerful moduli

William Banks (University of Missouri)

Abstract: In the talk, I will describe some recent joint work with Igor Shparlinski, in which we have combined classical ideas of Postnikov and Korobov to derive new bounds on short character sums for certain nonprincipal characters of powerful moduli. Our results are used to bound sums with the Mobius function twisted by such characters, and we obtain new results on the size and zero-free region of Dirichlet L-functions attached to the same class of moduli.

1:00 pm in 347 Altgeld Hall,Thursday, November 21, 2019

Robust Analysis of Metabolic Pathways

Al Holder (Rose-Hulman Institute of Technology)

Abstract: Flux balance analysis (FBA) is a widely adopted computational model in the study of whole-cell metabolisms, often used to identify drug targets, to study cancer, and to engineer cells for targeted purposes. The most common model is a linear program that maximizes cellular growth rate subject to achieving steady metabolic state and to satisfying environmental bounds. Quadratic and integer modifications are also common. Standard stoichiometry decides the preponderance of data in most instances, and hence, the majority of information defining an optimization model is certain. However, several key parts of a model rely on inferred science and are less certain; indeed, the method of deciding several of these values is opaque in the literature. This prompts the question of how the resulting science might depend on our lack of knowledge. We suggest a robust extension of FBA called Robust Analysis of Metabolic Pathways (RAMP) that accounts for uncertain information. We show that RAMP has several mathematical properties concomitant with our biological understanding, that RAMP performs like a relaxation of FBA in practice, and that RAMP requires special numerical awareness to solve.

3:00 pm in 347 Altgeld Hall,Thursday, November 21, 2019

Degree of Grothendieck polynomials and Castelnuovo-Mumford regularity

Colleen Robichaux   [email] (UIUC)

Abstract: Matrix Schubert varieties are sets of matrices satisfying certain rank conditions. We explore the connection between the degree of the Grothendieck polynomial and the Castelnuovo-Mumford regularity of coordinate rings of matrix Schubert varieties. Further we give an easily computable formula for this degree for the case of $w$ grassmannian. We end with a connection to a conjecture of Kummini-Lakshmibai-Sastry-Seshadri on regularity of standard open patches of Grassmannian Schubert varieties. This is joint work with Jenna Rajchgot, Yi Ren, Avery St. Dizier, and Anna Weigandt.

4:00 pm in 245 Altgeld Hall,Thursday, November 21, 2019

Metric embeddings of graphs into Banach spaces

Pavlos Motakis   [email] (University of Illinois at Urbana-Champaign)

Abstract: The embedding of a graph into a Banach space can be used to study either object by exploiting the properties of the other. The type of information that can be retrieved depends on the type of graph, the type of Banach space, and the type of metric embedding at hand. Various cases in which this approach has been useful will be explored. Particular weight will be given to finite lamplighter graphs and infinite Hamming graphs and their relation to local properties of Banach spaces and asymptotic properties of Banach spaces respectively.

Friday, November 22, 2019

2:00 pm in 347 Altgeld Hall,Friday, November 22, 2019

A category interpolating Yetter-Drinfeld modules over symmetric groups

Robert Laugwitz (University of Nottingham)

Abstract: P. Deligne introduced a remarkable tensor category Rep(St) interpolating the representation theory of symmetric groups, allowing for the natural number of permuted letters to be replaced by any complex number t. This category is defined using diagrams. We compute objects in the monoidal center of this category to obtain a ribbon category that interpolates the category of Yetter-Drinfeld modules over symmetric groups. As an application, interpolations of untwisted Dijkgraaf-Witten invariants. These are polynomial invariants of framed links. I will motivate Deligne's category Rep(St) and the monoidal center construction before describing how objects of the center or Rep(St) are obtained. This talk is based on joint work with Johannes Flake, RWTH Aachen University.

3:00 pm in 341 Altgeld Hall,Friday, November 22, 2019

Eigenvalues in Euclidean and Hyperbolic Geometry

Xiaolong Hans (UIUC Math)

Abstract: In this talk, we will explore the beauty and power of eigenvalues Laplacian operator. We start by talking about basic properties of Laplacian, Rayleigh quotient, monotonicity, how the lowest eigenvalues detect the difference of a geometric object from being a ball, Cheeger's inequality and how it distinguishes two different classes of infinite volume hyperbolic manifolds. The talk will emphasize intuition with no background in Laplacian assumed.

5:00 pm in 241 Altgeld Hall,Friday, November 22, 2019

Stochastic Differential Equations and Unitaries (Part 3)

Marius Junge (UIUC)

Abstract: Applications of Brownian motion and SDE. For example, show that an operator is completely positive by using SDE.