Mathematics

Random groups and surfaces

Moon Duchin (Tufts)

Abstract: I'll survey some of the beautiful history of the study of random objects in geometry, topology, and group theory. The focus will be the exchanges between work on random groups and work on random surfaces, including some very recent results and current research topics.

Probabilistic and combinatorial methods in the study of prime gaps

Kevin Ford (Illinois)

Abstract: We will describe how new bounds for the largest gaps between consecutive primes have utilized tools from several different areas, including number theory (efficient prime detecting sieves), probability (randomized congruence system coverings, concentration arguments) and probabilistic combinatorics (hypergraph covering). In particular, we will outline the recent breakthroughs of the speaker together with Ben Green, Sergei Konyain, James Maynard and Terence Tao. We will also describe new work with Konyagin, Maynard, Carl Pomerance and Tao which provides new bounds on consecutive composite values of integers in other sequences, e.g. polynomial sequences.

The Man Who Knew Infinity: the Movie, the Man, and the Mathematics

George Andrews (Penn State)

Abstract: In the spring of 2016, the motion picture, The Man Who Knew Infinity, was released. It is now available on DVD. The movie tells the life story of the Indian genius, Ramanujan. In this talk, I hope to start with the trailer from the movie. Then I shall provide a brief discussion of Ramanujan's life with a glimpse of the mathematics contained in his celebrated Lost Notebook (of which Bruce Berndt and I have just concluded preparing the fifth and final volume explicating the many assertion therein). The bulk of the talk will consider the path from the Rogers-Ramanujan identities to current open problems and how computer algebra assists in their study.

Modular forms, physics, and topology

Dan Berwick-Evans (Illinois)

Abstract: Modular forms appear in a wide variety of contexts in physics and mathematics. For example, they arise in two dimensional quantum field theories as certain observables. In algebraic topology, they emerge in the study of invariants called elliptic cohomology theories. A long-standing conjecture suggests that these two appearances of modular forms are intimately related. After explaining the ingredients, I’ll describe some recent progress. 

From Physics to Mathematics and back again: an exploration of generalized Kähler geometry

Marco Gualtieri (University of Toronto)

Abstract: In 1984, physicists Gates, Hull and Rocek, experts in the then-burgeoning field of supersymmetry, realized that their physical model required the existence of a unknown geometric structure, involving a Riemannian metric with a pair of compatible complex structures. Until relatively recently, we lacked a basic understanding of the features of this generalization of Kähler geometry. It was only after Hitchin's introduction of the concept of a generalized complex structure that we were able to prove the conjectures made by the physicists and provide various examples of what is now known as a generalized Kähler structure. I will explain the basic features of this fascinating geometric structure and outline the many relations we have discovered to other parts of geometry, including twistor theory, Poisson geometry, and Dirac geometry.

Lecture I. Set theory and trigonometric series

Alexander Kechris (Caltech)

Abstract: The Trjitzinsky Memorial Lectures will be held October 10-12, 2017. A reception will follow this first lecture from 5-6 pm in 239 Altgeld Hall. Alexander Kechris will present: "A descriptive set theoretic approach to problems in harmonic analysis, ergodic theory and combinatorics." Descriptive set theory is the study of definable sets and functions in Polish (complete, separable metric) spaces, like, e.g., the Euclidean spaces. It has been a central area of research in set theory for over 100 years. Over the past three decades, there has been extensive work on the interactions and applications of descriptive set theory to other areas of mathematics, including analysis, dynamical systems, and combinatorics. My goal in these lectures is to give a taste of this area of research, including an extensive historical background. These lectures require minimal background and should be understood by anyone familiar with the basics of measure theory and functional analysis. Also the three lectures are essentially independent of each other.

Lecture II. The complexity of classification problems in ergodic theory

Alexander Kechris (Caltech)

Abstract: The Trjitzinsky Memorial Lectures will be held October 10-12, 2017.Alexander Kechris will present "A descriptive set theoretic approach to problems in harmonic analysis, ergodic theory and combinatorics." Descriptive set theory is the study of definable sets and functions in Polish (complete, separable metric) spaces, like, e.g., the Euclidean spaces. It has been a central area of research in set theory for over 100 years. Over the past three decades, there has been extensive work on the interactions and applications of descriptive set theory to other areas of mathematics, including analysis, dynamical systems, and combinatorics. My goal in these lectures is to give a taste of this area of research, including an extensive historical background. These lectures require minimal background and should be understood by anyone familiar with the basics of measure theory and functional analysis. Also the three lectures are essentially independent of each other.

Lecture III: Descriptive graph combinatorics

Alexander Kechris (Caltech)

Abstract: The Trjitzinsky Memorial Lectures will be held October 10-12, 2017. Alexander Kechris will present "A descriptive set theoretic approach to problems in harmonic analysis, ergodic theory and combinatorics." Descriptive set theory is the study of definable sets and functions in Polish (complete, separable metric) spaces, like, e.g., the Euclidean spaces. It has been a central area of research in set theory for over 100 years. Over the past three decades, there has been extensive work on the interactions and applications of descriptive set theory to other areas of mathematics, including analysis, dynamical systems, and combinatorics. My goal in these lectures is to give a taste of this area of research, including an extensive historical background. These lectures require minimal background and should be understood by anyone familiar with the basics of measure theory and functional analysis. Also the three lectures are essentially independent of each other.