Department of

Mathematics


Seminar Calendar
for events the week of Tuesday, April 25, 2017.

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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, April 24, 2017

3:00 pm in 243 Altgeld Hall,Monday, April 24, 2017

The Relative Compactification of the Universal Centralizer(CANCELED)

Ana Balibanu (U Chicago)

Abstract: Let $G$ be a semisimple algebraic group of adjoint type. The universal centralizer $\mathcal X$ is the family of centralizers in $G$ of regular elements in Lie($G$). This algebraic variety has a natural symplectic structure, obtained by Hamiltonian reduction from the cotangent bundle $T^ ∗G$. We introduce a relative compactification of $\mathcal X$ , in which every centralizer fiber is replaced by its closure in the wonderful compactification of $G$. We show that the symplectic structure extends to a log-symplectic structure on the boundary, using the logarithmic cotangent bundle of the wonderful compactification.

4:00 pm in 239 Altgeld Hall,Monday, April 24, 2017

Retirement Reception

Abstract: A farewell open house for Chimesmaster Sue Woods will be held from 4-5 pm in 239 Altgeld Hall to recognize her many contributions to campus. Program at 4:30 pm.

4:00 pm in 245 Altgeld Hall,Monday, April 24, 2017

Some algorithmic problems for presentations of groups

Sergei Ivanov (Department of Mathematics, University of Illinois)

5:00 pm in 241Altgeld Hall,Monday, April 24, 2017

Locally Compact Groups and Duality

Terence Harris (UIUC&NSW)

Abstract: I will talk about locally compact groups and duality, both in the commutative and then the noncommutative case.

Tuesday, April 25, 2017

11:00 am in 345 Altgeld Hall,Tuesday, April 25, 2017

The signature modulo 8 of a fiber bundle

Carmen Rovi (Indiana)

Abstract: In this talk we shall be concerned with the residues modulo 4 and modulo 8 of the signature of a 4k-dimensional geometric Poincare complex. I will explain the relation between the signature modulo 8 and two other invariants: the Brown-Kervaire invariant and the Arf invariant. In my thesis I applied the relation between these invariants to the study of the signature modulo 8 of a fiber bundle, showing in particular that the non-multiplicativity of the signature modulo 8 is detected by an Arf invariant. In 1973 Werner Meyer used group cohomology to show that a surface bundle has signature divisible by 4. I will discuss current work with David Benson, Caterina Campagnolo and Andrew Ranicki where we are using group cohomology and representation theory of finite groups to detect non-trivial signatures modulo 8 of surface bundles.​

1:00 pm in 347 Altgeld Hall,Tuesday, April 25, 2017

Strichartz estimates for linear wave equations with moving potentials

Gong Chen (U Chicago)

Abstract: We will discuss Strichartz estimates for linear wave equations with several moving potentials in $\mathbb{R}^{3}$ (a.k.a. charge transfer Hamiltonians) which appear naturally in the study of nonlinear multisoliton systems. We show that local decay estimates systematically imply Strichartz estimates. To study local decay estimates, we introduce novel reversed Strichartz estimates along slanted lines and energy comparison under Lorentz transformations. As applications, we will also discuss related scattering problems and a construction of multisoliton in $\mathbb{R}^{3}$ with strong interactions.

1:00 pm in 345 Altgeld Hall,Tuesday, April 25, 2017

Differential-henselian extensions

Nigel Pynn-Coates (UIUC)

Abstract: The general motivating question is: What aspects of valuation theory can be adapted to the setting of valued differential fields, and under what assumptions? In valuation theory, henselian fields play an important role. I will concentrate on differential-henselian fields, introduced by Scanlon and developed in a more general setting by Aschenbrenner, van den Dries, and van der Hoeven. What do we know about uniqueness of differential-henselian extensions? Do differential-henselizations exist? After reviewing what is known, I will discuss my ongoing work towards answering these questions, and sketch a proof of the answers when the value group has finite archimedean rank. This talk is part of my preliminary examination.

2:00 pm in 347 Altgeld Hall,Tuesday, April 25, 2017

A new family of random sup-measures

Yizao Wang (University of Cincinnati)

Abstract: Random sup-measures are natural objects when investigating extremes of stochastic processes. A new family of stationary and self-similar random sup-measures are introduced. The representation of this family of random sup-measures is based on intersections of independent stable regenerative sets. These random sup-measures arise in limit theorems for extremes of a family of stationary infinitely-divisible processes with long-range dependence. The talk will first review random sup-measures in extremal limit theorems, and then focus on the representation of the new family of random sup-measures. Joint work with Gennady Samorodnitsky.

3:00 pm in Altgeld Hall,Tuesday, April 25, 2017

Small $k$-certificate in hypergraphs and representing all min-cuts

Chao Xu (Illinois Computer Science)

Abstract: For a hypergraph $H=(V,E)$, a hypergraph $H'=(V,E')$ is called a $k$-certificate of $H$ if it preserves all the cut values up to $k$. That is, $|\delta_{H'}(S)| \geq \min(|\delta_H(S)|,k)$ for all $S\subset V$. Nagamochi and Ibaraki showed there exists a $k$-certificate with $O(kn)$ edges for graphs, where $n$ is the number of vertices. A similar argument shows this extends to hypergraphs. We show a stronger result of hypergraphs: there is a $k$-certificate with size (sum of degrees) $O(kn)$, and it can be obtained by removing vertices from edges in $H$. We also devise an algorithm that finds a representation of all min-cuts in a hypergraph in the same running time as finding a single min-cut. The algorithm uses Cunningham's decomposition framework, and different generalizations of maximum adjacency ordering. This is joint work with Chandra Chekuri.

4:00 pm in 314 Altgeld Hall,Tuesday, April 25, 2017

Cobordisms: old and new

Ulrike Tillmann (Oxford University)

Abstract: Cobordims have played an important part in the classification of manifolds since their invention in the 1950s. In a different way, they are fundamental to the axiomatic approach to Topological Quantum Field Theory. In this colloquium style talk I will explain how recent results have shed new light on both of them.

The Tondeur Lectures in Mathematics will be held April 25-27, 2017. A reception will be held following the first lecture from 5-6 pm April 25 in 239 Altgeld Hall.

Wednesday, April 26, 2017

3:00 pm in 243 Altgeld Hall,Wednesday, April 26, 2017

Merit Program for Emerging Scholars

Jennifer McNeilly (UIUC)

Abstract: The Merit Program at the University of Illinois has helped with the retention and recruitment of underrepresented students in STEM fields for nearly three decades. With this talk, I will briefly review the history of the program and describe how it is structured in our department. I will also describe the teaching methods used (based on Dr. Uri Treisman’s collaborative learning model) and share some examples/data to demonstrate the program’s success. The goals will be to both introduce the Merit Program to those who are unfamiliar with it and also share a few teaching and TA training ideas which could be useful in a variety of settings.

4:00 pm in 245 Altgeld Hall,Wednesday, April 26, 2017

Classifying spaces of bordism categories and a filtration of Thom's theory

Ulrike Tillmann (Oxford University)

Abstract: We describe a refinement of a theorem with Galatius, Madsen and Weiss which describes the classifying space of bordism categories. In particular this can be interpreted to give evidence for the cobordism hypothesis for invertible TQFTs.

The Tondeur Lectures in Mathematics will be held April 25-27, 2017. A reception will be held following the first lecture from 5-6 pm April 25 in 239 Altgeld Hall.

Thursday, April 27, 2017

11:00 am in 241 Altgeld Hall,Thursday, April 27, 2017

A formula for the partition function that "counts"

Andrew Sills (Georgia Southern University)

Abstract: A partition of an integer n is a representation of n as a sum of positive integers where the order of the summands is considered irrelevant. Thus we see that there are five partitions of the integer 4, namely 4, 3+1, 2+2, 2+1+1, 1+1+1+1. The partition function p(n) denotes the number of partitions of n. Thus p(4) = 5. The first exact formula for p(n) was given by Hardy and Ramanujan in 1918. Twenty years later, Hans Rademacher improved the Hardy-Ramanujan formula to give an infinite series that converges to p(n). The Hardy-Ramanujan-Rademacher series is revered as one of the truly great accomplishments in the field of analytic number theory. In 2011, Ken Ono and Jan Bruinier surprised everyone by announcing a new formula which attains p(n) by summing a finite number of complex numbers which arise in connection with the multiset of algebraic numbers that are the union of Galois orbits for the discriminant -24n + 1 ring class field. Thus the known formulas for p(n) involve deep mathematics, and are by no means "combinatorial" in the sense that they involve summing a finite or infinite number of complex numbers to obtain the correct (positive integer) value. In this talk, I will show a new formula for the partition function as a multisum of positive integers, each term of which actually counts a certain class of partitions, and thus appears to be the first truly combinatorial formula for p(n). The idea behind the formula is due to Yuriy Choliy, and the work was completed in collaboration with him. We will further examine a new way to approximate p(n) using a class of polynomials with rational coefficients, and observe this approximation is very close to that of using the initial term of the Rademacher series. The talk will be accessible to students as well as faculty, and anyone interested is encouraged to attend!

12:30 pm in 464 Loomis Laboratory,Thursday, April 27, 2017

Entanglement branes in a two-dimensional string theory

Gabriel Wong (Virginia Physics)

Abstract: There is an emerging viewpoint that classical spacetime emerges from highly entangled states of more fundamental constituents. In the context of AdS/CFT, these fundamental constituents are strings, with a dual description as a large-N gauge theory. To understand entanglement in string theory, we consider the simpler context of two-dimensional large-N Yang-Mills theory, and its dual string theory description due to Gross and Taylor. We will show how entanglement in the gauge theory is described in terms of the string theory as thermal entropy of open strings whose endpoints are anchored on a stretched entangling surface which we call an entanglement brane.

1:00 pm in 243 Altgeld Hall,Thursday, April 27, 2017

Compressed Learning II

Marius Junge (UIUC)

Abstract: Part II

2:00 pm in 241 Altgeld Hall,Thursday, April 27, 2017

MacMahon's partial fractions

Andrew Sills (Georgia Southern University)

Abstract: A. Cayley used ordinary partial fractions decompositions of $1/[(1-x)(1-x^2)\ldots(1-x^m)]$ to obtain direct formulas for the number of partitions of $n$ into at most $m$ parts for several small values of $m$. No pattern for general m can be discerned from these, and in particular the rational coefficients that appear in the partial fraction decomposition become quite cumbersome for even moderate sized $m.$ Later, MacMahon gave a decomposition of $1/[(1-x)(1-x^2). . .(1-x^m)]$ into what he called "partial fractions of a new and special kind" in which the coefficients are "easily calculable numbers" and the sum is indexed by the partitions of $m$. While MacMahon's derived his "new and special" partial fractions using "combinatory analysis," the aim of this talk is to give a fully combinatorial explanation of MacMahon's decomposition. In particular, we will observe a natural interplay between partitions of $n$ into at most $m$ parts and weak compositions of $n$ with $m$ parts.

4:00 pm in 245 Altgeld Hall,Thursday, April 27, 2017

Operads from TQFTs

Ulrike Tillmann (Oxford University)

Abstract: Manifolds give rise to interesting operads, and in particular TQFTs define algebras over these operads. In the case of Atiyah's 1+1 dimensional theories these algebras are well-known to correspond to certain algebras. Surprisingly, independent of the dimension of the underlying manifolds, in the topologically enriched setting the manifold operads detect infinite loop spaces. We will report on joint work with Basterra, Bobkova, Ponto, Yeakel.

The Tondeur Lectures in Mathematics will be held April 25-27, 2017. A reception will be held following the first lecture from 5-6 pm April 25 in 239 Altgeld Hall.

Friday, April 28, 2017

1:00 pm in 347 Altgeld Hall,Friday, April 28, 2017

Estimates of invariant metrics and applications

Siqi Fu (Rutgers Univ. -Camden)

Abstract: We will discuss various estimates of invariant metrics (such as the Kobayashi and Bergman metrics) on a smooth bounded pseudoconvex domain, in particular, near a piece of the boundary whose Levi form has constant rank. We will also discuss how these estimates of the invariant metric can be used to characterize geometric properties on the boundary.

3:00 pm in 343 Altgeld Hall,Friday, April 28, 2017

On decomposition of the product of Demazure atom and Demazure characters

Anna Pun (Drexel University)

Abstract: t is an open problem to prove the Schubert positivity property combinatorially. Recently Haglund, Mason, Remmel, van Willigenburg et al. have studied the skyline fillings (a tableau-combinatorial object giving a combinatorial description to nonsymmetric MacDonald polynomials , proved by Haglund, Haiman and Loehr) specifically for the case of Demazure atoms (atoms) and key polynomials (keys). This suggests a new approach to a combinatorial proof of Schubert positivity property. In this talk, I will introduce Demazure atoms and key polynomials using skyline fillings called semi-standard augmented fillings (SSAFs) and define generalized Demazure atoms by some modifications on SSAF defining atoms and keys. I will illustrate the insertion algorithm on Demazure atoms proved by Mason and describe refinements of Littlewood-Richardson rule proved by Haglund, Mason and Willigenburg. Then I will describe an algorithm to prove the atom positivity property of the product of a monomial and a Demazure atom. The last result gives a positive support to the approach of the combinatorial proof of Schubert positivity property. If time allows, I will show some connection with polytopes and discuss some conjectures.

4:00 pm in 345 Altgeld Hall,Friday, April 28, 2017

On definability and interpretability in model theory

Ward Henson (UIUC, UC Berkeley)

Abstract: This will be an expository talk. Topics covered/mentioned will include: extension by definition and expansion/extension by interpretation (the eq-construction), Beth's Theorem (characterizing definability), and Makkai's Theorem (characterizing the eq-expansion). The setting will be model theory of classical (discrete) structures and then of metric (real-valued) structures. In the metric setting, definability of sets/relations has some subtleties that do not arise in discrete structures.