Department of

Mathematics


Seminar Calendar
for events the week of Saturday, February 29, 2020.

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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, 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 243 Altgeld Hall,Monday, February 24, 2020

On symplectic capacities and their blindspots

Ely Kerman (Illinois)

Abstract: Symplectic capacities are real-valued symplectic invariants which play an important role in embedding problems. Many fundamental questions concerning their properties remain unresolved in large part because they are difficult to compute. In this talk I will describe some new computations of the capacities defined for star-shaped domains by Gutt and Hutchings. The relevant class of examples is rich enough to yield several new insights into what these capacities can and cannot see. This is a report on joint work with Yuanpu Liang.

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.

5:00 pm in 241 Altgeld Hall,Monday, February 24, 2020

Connes' Embedding Problem

TBA (UIUC)

Abstract: CEB4

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 141 Altgeld Hall,Friday, February 28, 2020

Arnold-Liouville Theorem

Jungsoo (Ben) Park (UIUC)

Abstract: This talk will be an introduction to fundamental concepts of symplectic geometry. Furthermore, we will delve into the proof of Arnold-Liouville theorem: https://en.wikipedia.org/wiki/Liouville–Arnold_theorem.

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.