Department of

# Mathematics

Seminar Calendar
for Mathematics Colloquium events the year of Saturday, November 9, 2019.

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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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Friday, February 15, 2019

4:00 pm in 245 Altgeld Hall,Friday, February 15, 2019

#### Harry Potter's Cloak Via Transformation Optics

###### Gunther Uhlmann (University of Washington)

Abstract: Can we make objects invisible? This has been a subject of human fascination for millennia in Greek mythology, movies, science fiction, etc. including the legend of Perseus versus Medusa and the more recent Star Trek and Harry Potter. In the last fifteen years or so there have been several scientific proposals to achieve invisibility. We will introduce in a non-technical fashion one of them, the so-called "transformation optics" which has received a lot of attention in the scientific community.

Monday, February 18, 2019

4:00 pm in 245 Altgeld Hall,Monday, February 18, 2019

#### Cohomology of Shimura Varieties

###### Sug Woo Shin (University of California Berkeley)

Abstract: Shimura varieties are a certain class of algebraic varieties over number fields with lots of symmetries, introduced by Shimura and Deligne nearly half a century ago. They have been playing a central role in number theory and other areas. Langlands proposed a program to compute the L-functions and cohomology of Shimura varieites in 1970s; this was refined by Langlands-Rapoport and Kottwitz in 1980s. I will review some old and recent results in this direction.

Wednesday, February 20, 2019

4:00 pm in 245 Altgeld Hall,Wednesday, February 20, 2019

#### Some necessary uses of logic in mathematics

###### Ilijas Farah (York University)

Abstract: Every now and then, a difficult mathematical problem turns out to be difficult for a particularly objective reason: Provably, it cannot be solved by using 'conventional' means. Some classical examples are proving the Continuum Hypothesis, trisecting an angle, and solving the quintic equation. I’ll discuss more recent examples of such problems, giving some emphasis to the problems arising from the study of operator algebras.

Thursday, February 28, 2019

4:00 pm in 245 Altgeld Hall,Thursday, February 28, 2019

#### Quivers, representation theory and geometry

###### Kevin McGerty (University of Oxford and Visiting Fisher Professor, University of Illinois)

Abstract: A quiver is an oriented graph. It has a natural algebra associated to it called the path algebra, which as the name suggests has a basis given by paths in the quiver with multiplication given by concatenation. The representation theory of these algebras encompasses a number of classical problems in linear algebra, for example subspace arrangements and Jordan canonical form. A remarkable discovery of Gabriel however in the 1970s revealed a deep connection between these algebras and Lie theory, which has subsequently lead to a rich interaction between quivers, Lie theory and algebraic geometry. This talk will begin by outlining the elementary theory of representations of path algebras, explain Gabriel's result and survey some of the wonderful results which it has led to in Lie theory: the discovery of the canonical bases of quantum groups, the geometric realization of representations of affine quantum groups by Nakajima, and most recently deep connections between representations of symplectic reflection algebras and affine Lie algebras.

Thursday, March 7, 2019

4:00 pm in 245 Altgeld Hall,Thursday, March 7, 2019

#### New examples of Calabi-Yau metrics on a complex vector space

###### Frederic Rochon (University of Quebec in Montreal)

Abstract: After reviewing how the Riemann curvature tensor describes the local geometry of a space and how it may reflect some global aspects of its topology, we will focus on a special type of geometry: Calabi-Yau manifolds. By smoothing singular Calabi-Yau cones and using suitable compactifications by manifolds with corners, we will explain how to construct new examples of complete Calabi-Yau metrics on a complex vector space. Our examples are of Euclidean volume growth, but with tangent cone at infinity having a singular cross-section. This is a joint work with Ronan J. Conlon.

Thursday, March 28, 2019

4:00 pm in 245 Altgeld Hall,Thursday, March 28, 2019

#### Spherical conical metrics

###### Xuwen Zhu (University of California Berkeley)

Abstract: The problem of finding and classifying constant curvature metrics with conical singularities has a long history bringing together several different areas of mathematics. This talk will focus on the particularly difficult spherical case where many new phenomena appear. When some of the cone angles are bigger than $2\pi$, uniqueness fails and existence is not guaranteed; smooth deformation is not always possible and the moduli space is expected to have singular strata. I will give a survey of several recent results regarding this singular uniformization problem, connecting PDE techniques with complex analysis and synthetic geometry. Based on joint works with Rafe Mazzeo and Bin Xu.

Tuesday, April 9, 2019

4:00 pm in 314 Altgeld Hall,Tuesday, April 9, 2019

#### Recent progress on existence of minimal surfaces

###### André Neves (University of Chicago)

Abstract: The Tondeur Memorial Lectures will be given by Andre Neves (University of Chicago), April 9-11, 2019. Following this lecture, a reception will be held in 239 Altgeld Hall.

A long standing problem in geometry, conjectured by Yau in 1982, is that any any $3$-manifold admits an infinite number of distinct minimal surfaces. The analogous problem for geodesics on surfaces led to the discovery of deep interactions between dynamics, topology, and analysis. The last couple of years brought dramatic developments to Yau’s conjecture, which has now been settled due to the work of Marques-Neves and Song. In the first talk I will survey the history of the problem and the several contributions made. In the second talk I will talk about the Weyl law for the volume spectrum (Marques-Neves-Liokumovich) and how it can be used to prove denseness and equidistribution of minimal surfaces in the generic case (Irie-Marques-Neves and Marques-Neves-Song). In the third talk I will survey the recent breakthroughs due to Song, Zhou, and Mantoulidis-Chodosh.

Bio Note: André Neves is a leading figure in geometric analysis with important contributions ranging from the Yamabe problem to geometric flows. Jointly with Fernando Marques, he transformed the field by introducing new ideas and techniques that led to the solution of several open problems which were previously out of reach. Together or with coauthors, they solved the Willmore conjecture, the Freedman-He-Wang conjecture in knot theory and Yau’s conjecture on the existence of minimal surfaces in the generic case.

Neves received his PhD from Stanford University in 2005 under the supervision of Richard Schoen. He was a postdoctoral fellow and assistant professor at Princeton University, before joining the Imperial College of London in 2011, where he became a full professor. He joined the faculty of the University of Chicago in 2016. Among his many awards and recognitions, Neves was awarded the Philip Leverhulme Prize in 2012, the LMS Whitehead Prize in 2013, he was invited speaker at ICM in Seoul in 2014, received a New Horizons in Mathematics Prize in 2015, and the 2016 Oswald Veblen Prize in Geometry. In 2018, he received a Simons Investigator Award.

Wednesday, April 10, 2019

4:00 pm in 245 Altgeld Hall,Wednesday, April 10, 2019

#### Recent progress on existence of minimal surfaces

###### André Neves (University of Chicago)

Abstract: A long standing problem in geometry, conjectured by Yau in 1982, is that any any $3$-manifold admits an infinite number of distinct minimal surfaces. The analogous problem for geodesics on surfaces led to the discovery of deep interactions between dynamics, topology, and analysis. The last couple of years brought dramatic developments to Yau’s conjecture, which has now been settled due to the work of Marques-Neves and Song. In the first talk I will survey the history of the problem and the several contributions made. In the second talk I will talk about the Weyl law for the volume spectrum (Marques-Neves-Liokumovich) and how it can be used to prove denseness and equidistribution of minimal surfaces in the generic case (Irie-Marques-Neves and Marques-Neves-Song). In the third talk I will survey the recent breakthroughs due to Song, Zhou, and Mantoulidis-Chodosh.

Bio Note: André Neves is a leading figure in geometric analysis with important contributions ranging from the Yamabe problem to geometric flows. Jointly with Fernando Marques, he transformed the field by introducing new ideas and techniques that led to the solution of several open problems which were previously out of reach. Together or with coauthors, they solved the Willmore conjecture, the Freedman-He-Wang conjecture in knot theory and Yau’s conjecture on the existence of minimal surfaces in the generic case.

Neves received his PhD from Stanford University in 2005 under the supervision of Richard Schoen. He was a postdoctoral fellow and assistant professor at Princeton University, before joining the Imperial College of London in 2011, where he became a full professor. He joined the faculty of the University of Chicago in 2016. Among his many awards and recognitions, Neves was awarded the Philip Leverhulme Prize in 2012, the LMS Whitehead Prize in 2013, he was invited speaker at ICM in Seoul in 2014, received a New Horizons in Mathematics Prize in 2015, and the 2016 Oswald Veblen Prize in Geometry. In 2018, he received a Simons Investigator Award.

Thursday, April 11, 2019

4:00 pm in 245 Altgeld Hall,Thursday, April 11, 2019

#### Recent progress on existence of minimal surfaces

###### André Neves (University of Chicago)

Abstract: A long standing problem in geometry, conjectured by Yau in 1982, is that any any $3$-manifold admits an infinite number of distinct minimal surfaces. The analogous problem for geodesics on surfaces led to the discovery of deep interactions between dynamics, topology, and analysis. The last couple of years brought dramatic developments to Yau’s conjecture, which has now been settled due to the work of Marques-Neves and Song. In the first talk I will survey the history of the problem and the several contributions made. In the second talk I will talk about the Weyl law for the volume spectrum (Marques-Neves-Liokumovich) and how it can be used to prove denseness and equidistribution of minimal surfaces in the generic case (Irie-Marques-Neves and Marques-Neves-Song). In the third talk I will survey the recent breakthroughs due to Song, Zhou, and Mantoulidis-Chodosh.

Bio Note: André Neves is a leading figure in geometric analysis with important contributions ranging from the Yamabe problem to geometric flows. Jointly with Fernando Marques, he transformed the field by introducing new ideas and techniques that led to the solution of several open problems which were previously out of reach. Together or with coauthors, they solved the Willmore conjecture, the Freedman-He-Wang conjecture in knot theory and Yau’s conjecture on the existence of minimal surfaces in the generic case.

Neves received his PhD from Stanford University in 2005 under the supervision of Richard Schoen. He was a postdoctoral fellow and assistant professor at Princeton University, before joining the Imperial College of London in 2011, where he became a full professor. He joined the faculty of the University of Chicago in 2016. Among his many awards and recognitions, Neves was awarded the Philip Leverhulme Prize in 2012, the LMS Whitehead Prize in 2013, he was invited speaker at ICM in Seoul in 2014, received a New Horizons in Mathematics Prize in 2015, and the 2016 Oswald Veblen Prize in Geometry. In 2018, he received a Simons Investigator Award.

Thursday, April 18, 2019

4:00 pm in 245 Altgeld Hall,Thursday, April 18, 2019

#### The many aspects of Schubert polynomials

###### Karola Mészáros (Cornell University)

Abstract: Schubert polynomials, introduced by Lascoux and Schützenberger in 1982, represent cohomology classes of Schubert cycles in flag varieties. While there are a number of combinatorial formulas for Schubert polynomials, their supports have only recently been established and the values of their coefficients are not well understood. We show that the Newton polytope of a Schubert polynomial is a generalized permutahedron and explain how to obtain certain Schubert polynomials as projections of integer point transforms of polytopes. The latter generalizes the well-known relationship between Schur functions and Gelfand-Tsetlin polytopes. We will then turn to the study of the coefficients of Schubert polynomials and show that Schubert polynomials with all coefficients at most $k$, for any positive integer $k$, are closed under pattern containment. We also characterize zero-one Schubert polynomials by a list of twelve avoided patterns. This talk is based on joint works with Alex Fink, Ricky Liu and Avery St. Dizier.

Thursday, September 12, 2019

4:00 pm in 245 Altgeld Hall,Thursday, September 12, 2019

#### Non-Smooth Harmonic Analysis

###### Palle Jorgensen   [email] (University of Iowa)

Abstract: While the framework of the talk covers a wider view of harmonic analysis on fractals, it begins with a construction by the author of explicit orthogonal Fourier expansions for certain fractals. It has since branched off several directions, each one dealing with aspect of the wider subject. The results presented cover (among other papers) joint work with Steen Pedersen, then later, with Dorin Dutkay. Fractals. Intuitively, it is surprising that any selfsimilar fractals in fact do admit orthogonal Fourier series. And our initial result generated surprised among members of the harmonic analysts community. The theme of Fourier series on Fractals has by now taken off in a number of diverse directions; e.g., (i) wavelets on fractals, or frames; (ii) non-commutative analysis on graph limits, to mention only two. Two popular question are: “What kind of fractals admit Fourier series?” “If they don’t, then what alternative harmonic analysis might be feasible?”

Thursday, September 19, 2019

4:00 pm in 245 Altgeld Hall,Thursday, September 19, 2019

#### On the container method

###### Jozsef Balogh   [email] (University of Illinois at Urbana-Champaign)

Abstract: We will give a gentle introduction to a recently-developed technique, The Container Method’, for bounding the number (and controlling the typical structure) of finite objects with forbidden substructures. This technique exploits a subtle clustering phenomenon exhibited by the independent sets of uniform hypergraphs whose edges are sufficiently evenly distributed; more precisely, it provides a relatively small family of 'containers' for the independent sets, each of which contains few edges. The container method is very useful counting discrete structures with certain properties; transferring theorems into random environment; and proving the existence discrete structures satisfying some important properties. In the first half of the talk we will attempt to convey a general high-level overview of the method, in particular how independent sets in hypergraphs could be used to model various problems in combinatorics; in the second, we will describe a few illustrative applications in areas such as extremal graph theory, Ramsey theory, additive combinatorics, and discrete geometry.

Tuesday, October 1, 2019

4:00 pm in 245Altgeld Hall,Tuesday, October 1, 2019

#### Limitations on All Known (and Some Unknown) Approaches to Matrix Multiplication

###### Virginia Vassilevska-Williams   [email] (Massachusetts Institute of Technology)

Abstract: In this talk we will consider the known techniques for designing Matrix Multiplication algorithms. The two main approaches are the Laser method of Strassen, and the Group theoretic approach of Cohn and Umans. We define generalizations that subsume these two approaches: the Galactic and the Universal method; the latter is the most general method there is. We then design a suite of techniques for proving lower bounds on the value of $\omega$, the exponent of matrix multiplication, which can be achieved by algorithms using many tensors $T$ and the Galactic method. Some of our techniques exploit local' properties of $T$, like finding a sub-tensor of $T$ which is so weak' that $T$ itself couldn't be used to achieve a good bound on $\omega$, while others exploit global' properties, like $T$ being a monomial degeneration of the structural tensor of a group algebra. The main result is that there is a universal constant $\ell>2$ such that a large class of tensors generalizing the Coppersmith-Winograd tensor $CW_q$ cannot be used within the Galactic method to show a bound on $\omega$ better than $\ell$, for any $q$. We give evidence that previous lower-bounding techniques were not strong enough to show this. The talk is based on joint work with Josh Alman, which appeared in FOCS 2018. More recently, Alman showed how to extend our techniques so that they apply to the Universal method as well. In particular, Alman shows that the Coppersmith-Winograd tensor cannot yield a better bound on $\omega$ than 2.16805 even using the Universal method.

Thursday, October 10, 2019

4:00 pm in 235 Altgeld Hall,Thursday, October 10, 2019

#### Nonlocal Problems with the Fractional Laplacian and Their Applications

###### Yanzhi Zhang   [email] (Missouri University of Science and Technology)

Abstract: Recently, the fractional Laplacian has received great attention in modeling complex phenomena that involve long-range interactions. However, the nonlocality of the fractional Laplacian introduces considerable challenges in both analysis and simulations. In this talk, I will present numerical methods to discretize the fractional Laplacian as well as error estimates. Compared to other existing methods, our methods are more accurate and simpler to implement, and moreover they closely resembles the central difference scheme for the classical Laplace operator. Finally, I will show some applications of nonlocal problems involving the fractional Laplacian.

Thursday, October 17, 2019

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.

Thursday, November 7, 2019

4:00 pm in 245 Altgeld Hall,Thursday, November 7, 2019

#### A survey of Sperner theory

###### Richard Stanley   [email] (MIT and University of Miami)

Abstract: Let $X$ be a collection of subsets of an $n$-element set $S$ such that no element of $X$ is a subset of another. In 1927 Emanuel Sperner showed that the number of elements of $X$ is maximized by taking $X$ to consist of all subsets of $S$ with $\lfloor n/2\rfloor$ elements. This result started the subject of \emph{Sperner theory}, which is concerned with the largest subset $A$ of a finite partially ordered set $P$ that forms an \emph{antichain}, that is, no two elements of $A$ are comparable in $P$. We will give a survey of Sperner theory, focusing on some connections with linear algebra and algebraic geometry.

Thursday, November 21, 2019

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.

Tuesday, December 3, 2019

4:00 pm in 245 Altgeld Hall,Tuesday, December 3, 2019

#### To Be Announced

###### Dimitris Koukoulopoulos   [email] (University of Montreal)

Abstract: To come.