Department of

Mathematics


Seminar Calendar
for events the week of Saturday, October 19, 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, October 14, 2019

3:00 pm in 441 Altgeld Hall,Monday, October 14, 2019

Motivating Higher Toposes: Higher Bundle Theory

Joseph Rennie (UIUC Math)

Abstract: In this (self-contained) talk, I will begin with a quick recap of the motivation for higher bundle theory from the first talk. I will then say a few words about Toposes, and proceed to spend the majority of the talk attempting to develop a general theory of higher bundles. Along the way, we will see how the necessary properties for this development (almost) force higher topos structure. (Technical details will be sacrificed for intuitive clarity. No particular model of higher categories will be imposed.)

3:00 pm in 243 Altgeld Hall,Monday, October 14, 2019

Supersymmetric localization and the Witten genus

Dan Berwick-Evans (Illinois)

Abstract: Equivariant localization arguments generalize the Duistermaat–Heckman formula, allowing one to express an integral on a manifold in terms of integrals over fixed point sets of a torus action. Supersymmetric localization seeks to apply these formulas to path integrals in quantum field theory. In fortuitous cases, this affords a rigorous definition of the path integral. I will explain one such example in a 2-dimensional quantum field theory built on a classical theory of maps from elliptic curves to a smooth manifold. Up to a certain choice of orientation (which may be obstructed), the path integral is well-defined. The volume of the mapping space (i.e., the path integral of 1) turns out to be the Witten genus, an invariant of smooth manifolds valued in modular forms.

5:00 pm in 241 Altgeld Hall,Monday, October 14, 2019

The Dixmier Trace

Haojian Li (UIUC)

Abstract: We will construct so-called exotic traces on the space of bounded operators on a Hilbert space, the so called Dixmier traces. In the long run we are interested in geometric applications of Connes' Dixmier trace calculus.

Tuesday, October 15, 2019

1:00 pm in 347 Altgeld Hall,Tuesday, October 15, 2019

Angled crested type water waves

Siddhant Agrawal (U Mass Amherst)

Abstract: We consider the two-dimensional water wave equation which is a model of ocean waves. The water wave equation is a free boundary problem for the Euler equation where we assume that the fluid is inviscid, incompressible, irrotational and the air density is zero. In the case of zero surface tension, we show that the singular solutions recently constructed by Wu (19) are rigid. In the case of non-zero surface tension, we construct an energy functional and prove a local wellposedness result without assuming the Taylor sign condition. This energy reduces to the energy obtained by Kinsey and Wu (18) in the zero surface tension case and allows angled crest interfaces. For non zero surface tension, the energy does not allow singularities in the interface but allows interfaces with large curvature. We show that in an appropriate regime, the zero surface tension limit of our solutions is a solution of the gravity water wave equation which includes waves with angled crests.

2:00 pm in 243 Altgeld Hall,Tuesday, October 15, 2019

Equiangular lines with a fixed angle

Zilin Jiang (MIT Math)

Abstract: An equiangular set of lines is a family of lines (through the origin) such that they are pairwise separated by the same angle. A central question in Algebraic Graph Theory is to determine the maximum cardinality of an equiangular set of lines in n-dimensional Euclidean space. In this talk, we will prove the key spectral result on the multiplicity of the second largest eigenvalue of a connected graph, and we will then connect it to the question on equiangular lines. Joint work with Jonathan Tidor, Yuan Yao, Shengtong Zhang and Yufei Zhao.

2:00 pm in 347 Altgeld Hall,Tuesday, October 15, 2019

On the spectral heat content for subordinate killed Brownian motions with respect to a wide class of subordinators

Hyunchul Park (SUNY New Paltz)

Abstract: In this talk, we study the asymptotic behavior of the spectral heat content for subordinate killed Brownian motions (SKBM) with respect to a wide class of subordinators. Previously, the spectral heat content for SKBM via stable subordinators was studied by the author and R. Song. This result gives an upper bound for the heat loss for the spectral heat content for killed Levy processes, whose asymptotic limit is not available for $\mathbb{R}^d$, $d\ge 2$, even for killed $\alpha$-stable processes when $\alpha\in [1; 2)$. This is a joint work with R. Song and is in progress.

3:00 pm in 243 Altgeld Hall,Tuesday, October 15, 2019

Koszul Modules and Green’s Conjecture

Claudiu Raicu (University of Notre Dame)

Abstract: Formulated in 1984, Green’s Conjecture predicts that one can recognize the intrinsic complexity of an algebraic curve from the syzygies of its canonical embedding. Green’s Conjecture for a general curve has been resolved in two landmark papers by Voisin in the early 00s. I will explain how the recent theory of Koszul modules provides more elementary solutions to this problem, by relating it to the study of the syzygies of some very concrete surfaces. Joint work with M. Aprodu, G. Farkas, S. Papadima, S. Sam and J. Weyman.

Thursday, October 17, 2019

11:00 am in 241 Altgeld Hall,Thursday, October 17, 2019

A new approach to bounds for L-functions

Jesse Thorner (University of Florida)

Abstract: Let $L(s)$ be the $L$-function of a cuspidal automorphic representation of $GL(n)$ with analytic conductor $C$. The Phragmen-Lindelof principle implies the convexity bound $|L(1/2)| \ll C^{1/4+\epsilon}$ for all fixed $\epsilon>0$, while the generalized Riemann hypothesis for $L(s)$ implies that $|L(1/2)|\ll C^{\epsilon}$. A major theme in modern number theory is the pursuit of subconvexity bounds of the shape $|L(1/2)| \ll C^{1/4-\delta}$ for some fixed constant $\delta>0$. I will describe how to achieve (i) an unconditional nontrivial improvement over the convexity bound for all automorphic $L$-functions (joint work with Kannan Soundararajan), and (ii) an unconditional subconvexity bound for almost all automorphic $L$-functions (joint work with Asif Zaman).

1:00 pm in 464 Loomis Laboratory ,Thursday, October 17, 2019

Sphere packing, modular bootstrap and extremal functionals

Dalimil Mazac

Abstract: I will prove a new theorem about 2D CFTs: Every unitary 2D CFT must contain a non-trivial Virasoro primary of scaling dimension at most c/8 + 1/2, where c is the central charge. At large c, this is an improvement of the Hellerman bound c/6 + O(1), and is relevant for constraining the spectrum of gravitational theories in AdS3. The proof follows from the modular bootstrap and uses analytic extremal functionals, originally developed in the context of four-point SL(2) conformal bootstrap. In the second part of the talk, I will discuss a surprising connection between modular bootstrap and the sphere-packing problem from discrete geometry. In particular, the above bound on the gap becomes a bound on the sphere-packing density. In 8 and 24 dimensions, this bound is sharp and leads to a solution of the sphere-packing problem in these dimensions, as originally proved by Viazovska et al. The talk will be based on arXiv:1905.01319.

2:00 pm in 347 Altgeld Hall,Thursday, October 17, 2019

Branching Processes Part 1

Peixue Wu (UIUC Math)

Abstract: The first part of this talk is very introductory, I will talk about the basic ideas of branching mechanism (originated from random walk) and some generalizations of the simple branching process, e.g., age-dependent processes, multi-type branching process. ​I will focus on the limit theorem of branching processes.​ ​In the second part, I will talk about the superprocess (which is measure-valued branching processes) and some recent works about it.

4:00 pm in 245 Altgeld Hall,Thursday, October 17, 2019

Orbit Equivalence and Entropy

Hanfeng Li   [email] (University at Buffalo)

Abstract: Entropy is one of the most important numerical invariants for probability-measure-preserving (pmp) actions of countable infinite groups. Orbit equivalence is a fairly weak equivalence relation between pmp actions. In general orbit equivalence may not preserve entropy. A few years ago Tim Austin showed that integrable orbit equivalence between pmp actions of finitely generated amenable groups does preserve entropy. I will introduce a notion of Shannon orbit equivalence, weaker than integrable orbit equivalence, and a property SC for pmp actions. The Shannon orbit equivalence between pmp actions of sofic groups with the property SC preserves the maximal sofic entropy. If a group G has a w-normal subgroup H such that H is amenable and neither locally finite nor virtually cyclic, then every pmp action of G has the property SC. In particular, if two Bernoulli shifts of such a sofic group are Shannon orbit equivalent, then they are conjugate. This is joint work with David Kerr.

Friday, October 18, 2019

2:00 pm in 245 Altgeld Hall,Friday, October 18, 2019

Careers for Math Students in the Life Sciences and Medicine

Howard Aizenstein, Tandy Warnow, James O'Dwyer, Olgica Milenkovic

Abstract: Are you interested in learning more about the role of mathematics in the fields of biology, biochemistry, or medicine? Come hear from a distinguished panel of applied mathematicians whose research addresses societally relevant problems in the biological sciences. Panelists: Howard Aizenstein, Charles F. Reynolds III and Ellen G. Detlefsen Endowed Chair in Geriatric Psychiatry and Professor of Bioengineering and Clinical and Translational Science at the University of Pittsburgh; Tandy Warnow, Founder Professor of Computer Science and Associate Head for Computer Science, UIUC;; James O'Dwyer, Associate Professor, Department of Plant Biology, UIUC; and Olgica Milenkovic, Professor and Donald Biggar Willett Scholar, Department of Electrical and Computer Engineering, UIUC

2:00 pm in 347 Altgeld Hall,Friday, October 18, 2019

Brown-Goodearl conjecture for PI weak Hopf algebras

James Zhang (University of Washington, Seattle)

Abstract: Brown and Goodearl conjectured that every noetherian Hopf algebra is Artin-Schelter Gorenstein. This conjecture is known to be true for many cases, in particular, for affine polynomial identity Hopf algebras. Weak Hopf algebras are an important generalization of Hopf algebras, and the category of modules over a weak Hopf algebra has a monoidal structure. Let $W$ be a weak Hopf algebra that is a finitely generated module over its affine center. We prove that $W$ has finite self-injective dimension and is a direct sum of Artin-Schelter Gorenstein algebras. Therefore Brown-Goodearl conjecture holds in this special weak Hopf setting. We will also give some motivations and consequences of Brown-Goodearl conjecture. This is joint work with Dan Rogalski and Robert Won.

3:00 pm in 341 Altgeld Hall,Friday, October 18, 2019

The geometry of real hypersurfaces in complex space

Martino Fassina (UIUC Math)

Abstract: Real hypersurfaces are the boundaries of complex domains, and their geometry is therefore crucial in understanding the theory of functions of several complex variables. I will focus in particular on the following question: does the hypersurface contain some local analytic structure? More generally, how closely ambient complex analytic varieties can contact the hypersurface? The talk will be elementary, and with plenty of pictures.

4:00 pm in 141 Altgeld Hall,Friday, October 18, 2019

Lines in Space

Brian Shin (UIUC)

Abstract: Consider four lines in three-dimensional space. How many lines intersect these given lines? In this expository talk, I'd like to discuss this classical problem of enumerative geometry. Resolving this problem will give us a chance to see some interesting algebraic geometry and algebraic topology. If time permits, I'll discuss connections to motivic homotopy theory.

4:00 pm in 347 Altgeld Hall,Friday, October 18, 2019

Where Automata Theory Meets Metric Geometry

Alexi Block Gorman (UIUC Math)

Abstract: The results in this talk illustrate and expand on connections between automata theory and metric geometry. We will begin by defining automata, Buchi automata, fractals, and iterated function systems. We say that a function is regular if there is a Buchi automaton that accepts precisely the set of base n representations of points in the graph of the function. We show that a continuous regular function (with closed and bounded domain) "looks linear" almost everywhere, if you zoom in enough. As a result, we show that every differentiable regular function is a shift of linear function (or hyperplane, in higher dimensions).