Mathematics Colloquia, Fall 2019

SEPTEMBER 12, 2019

4:00 p.m., 245 Altgeld Hall
Speaker: Palle Jorgensen (University of Iowa)
Title: Non-Smooth Harmonic Analysis
Host: Florin Boca

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?”

SEPTEMBER 19, 2019

4:00 p.m., 245 Altgeld Hall
Speaker: Jozsef Balogh (University of Illinois at Urbana-Champaign)
Title: On the Container Method
Host: Chris Dodd (alternate: Alexandr Kostochka)

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.

OCTOBER 1, 2019

4:00 p.m., 245 Altgeld Hall
Speaker: Virginia Vassilevska-Williams (Massachusetts Institute of Technology)
Title: Limitations on All Known (and Some Unknown) Approaches to Matrix Multiplication
Host: Denka Kutzarova

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 ω, 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 ω, 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 ℓ>2 such that a large class of tensors generalizing the Coppersmith-Winograd tensor CWq cannot be used within the Galactic method to show a bound on ω better than ℓ, 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 ω than 2.16805 even using the Universal method.

OCTOBER 10, 2019

4:00 p.m., 245 Altgeld Hall
Speaker: Yanzhi Zhang (Missouri University of Science and Technology)
Title: Nonlocal Problems with the Fractional Laplacian and Their Applications
Host: Kay Kirkpatrick

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.

OCTOBER 17, 2019

4:00 p.m., 245 Altgeld Hall
Speaker: Hanfeng Li (University at Buffalo)
Title: Orbit Equivalence and Entropy
Host: Adam Dor On and Florin Boca

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.

OCTOBER 31, 2019

4:00 p.m., 245 Altgeld Hall
Speaker: TBA
Title: TBA

Abstract: To come.

NOVEMBER 7, 2019

4:00 p.m., 245 Altgeld Hall
Speaker: Richard Stanley (MIT and University of Miami)
Title: A survey of Sperner theory
Host: Alex Young

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.

NOVEMBER 21, 2019

4:00 p.m., 245 Altgeld Hall
Speaker: Pavlos Motakis (University of Illinois at Urbana-Champaign)
Title: Metric embeddings of graphs into Banach spaces
Host: Marius Junge

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.

DECEMBER 3 (Tuesday), 2019

4:00 p.m., 245 Altgeld Hall
Speaker: Dimitris Koukoulopoulos (University of Montreal)
Title: On the Duffin-Schaeffer conjecture 
Host: Kevin Ford

Abstract: Given any real number $\alpha$, Dirichlet proved that there are infinitely many reduced fractions $a/q$ such that $|\alpha-a/q|\le 1/q^2$. Can we get closer to $\alpha$ than that? For certain "quadratic irrationals" such as $\alpha=\sqrt{2}$ the answer is no. However, Khinchin proved that if we exclude such thin sets of numbers, then we can do much better. More precisely, let $(\Delta_q)_{q=1}^\infty$ be a sequence of error terms such that $q^2\Delta_q$ decreases. Khinchin showed that if the series $\sum_{q=1}^\infty q\Delta_q$ diverges, then almost all $\alpha$ (in the Lebesgue sense) admit infinitely many reduced rational approximations $a/q$ such that $|\alpha-a/q|\le \Delta_q$. Conversely, if the series $\sum_{q=1}^\infty q\Delta_q$ converges, then almost no real number is well-approximable with the above constraints. In 1941, Duffin and Schaeffer set out to understand what is the most general Khinchin-type theorem that is true, i.e., what happens if we remove the assumption that $q^2\Delta_q$ decreases. In particular, they were interested in choosing sequences $(\Delta_q)_{q=1}^\infty$ supported on sparse sets of integers. They came up with a general and simple criterion for the solubility of the inequality $|\alpha-a/q|\le\Delta_q$. In this talk, I will explain the conjecture of Duffin-Schaeffer as well as the key ideas in recent joint work with James Maynard that settles it.