• Speaker: François Greer, Stony Brook University
• Title: Enumerative geometry and modular forms
• Date: 01/08/2020
• Time: 4:10 PM - 5:00 PM
• Place: C304 Wells Hall

Gromov-Witten invariants are counts of holomorphic curves on a smooth projective variety X. When assembled into a generating series, these invariants often produce special functions. A folklore conjecture predicts that when X admits an elliptic fibration, the Gromov-Witten generating functions are quasi-modular forms. I will discuss recent progress on this conjecture and a program to prove it in general.

• Speaker: Tetiana Shcherbyna, Princeton University
• Title: Random matrix theory and supersymmetry techniques
• Date: 01/10/2020
• Time: 4:10 PM - 5:00 PM
• Place: C304 Wells Hall

Starting from the works of Erdos, Yau, Schlein with coauthors, the significant progress in understanding the universal behavior of many random graph and random matrix models were achieved. However for the random matrices with a spacial structure our understanding is still very limited. In this talk I am going to overview applications of another approach to the study of the local eigenvalues statistics in random matrix theory based on so-called supersymmetry techniques (SUSY) . SUSY approach is based on the representation of the determinant as an integral over the Grassmann (anticommuting) variables. Combining this representation with the representation of an inverse determinant as an integral over the Gaussian complex field, SUSY allows to obtain an integral representation for the main spectral characteristics of random matrices such as limiting density, correlation functions, the resolvent's elements, etc. This method is widely (and successfully) used in the physics literature and is potentially very powerful but the rigorous control of the integral representations, which can be obtained by this method, is quite difficult, and it requires powerful analytic and statistical mechanics tools. In this talk we will discuss some recent progress in application of SUSY to the analysis of local spectral characteristics of the prominent ensemble of random band matrices, i.e. random matrices whose entries become negligible if their distance from the main diagonal exceeds a certain parameter called the band width.

• Speaker: Nathan Dowlin, Columbia University
• Title: Quantum and symplectic invariants in low-dimensional topology.
• Date: 01/15/2020
• Time: 4:10 PM - 5:00 PM
• Place: C304 Wells Hall

Khovanov homology and knot Floer homology are two powerful knot invariants developed around two decades ago. Knot Floer homology is defined using symplectic techniques, while Khovanov homology has its roots in the representation theory of quantum groups. Despite these differences, they seem to have many structural similarities. A well-known conjecture of Rasmussen from 2005 states that for any knot K, there is a spectral sequence from the Khovanov homology of K to the knot Floer homology of K. Using a new family of invariants defined using both quantum and symplectic techniques, I will give a proof of this conjecture and describe some topological applications.

• Speaker: Felix Janda, IAS, Princeton University
• Title: Enumerative geometry: old and new.
• Date: 01/27/2020
• Time: 4:10 PM - 5:00 PM
• Place: C304 Wells Hall

For as long as people have studied geometry, they have counted geometric objects. For example, Euclid's Elements starts with the postulate that there is exactly one line passing through two distinct points in the plane. Since then, the kinds of counting problems we are able to pose and to answer has grown. Today enumerative geometry is a rich subject with connections to many fields, including combinatorics, physics, representation theory, number theory and integrable systems. In this talk, I will show how to solve several classical counting questions. I will then move to a more modern problem with roots in string theory which has been the subject of intense study for the last three decades: The computation of the Gromov-Witten invariants of the quintic threefold, an example of a Calabi-Yau manifold

• Speaker: Jacob Tsimerman, University of Toronto
• Title: TBD
• Date: 02/20/2020
• Time: 4:10 PM - 5:00 PM
• Place: C304 Wells Hall

No abstract available.

• Speaker: Laure Saint-Raymond, École normale supérieure de Lyon
• Title: Disorder increases almost surely (First Phillips Lecture)
• Date: 03/09/2020
• Time: 6:30 PM - 7:30 PM
• Place: 105AB Kellogg Center

In the every day life, there are many examples of mixing phenomena : milk and water in a same container will not stay separate from each other, marbles in a bag will not line up spontaneously according to their color, ... In this first talk, we intend to study a simple mathematical model which explains why we can observe spontaneous mixing but not the reverse phenomenon.

• Speaker: Laure Saint-Raymond, École normale supérieure de Lyon
• Title: The structure of correlations (Third Philips Lecture)
• Date: 03/11/2020
• Time: 4:00 PM - 5:00 PM
• Place: C304 Wells Hall

Although the distribution of hard spheres remains essentially chaotic in this low density regime, collisions give birth to small correlations, which keep part of the information. The structure of these dynamical correlations is amazing, going through all scales. This analysis provides actually a characterization of small fluctuations (central limit theorem), and large deviations.

• Speaker: Dimitri Shlyakhtenko, UCLA
• Title: TBD
• Date: 03/19/2020
• Time: 4:10 PM - 5:00 PM
• Place: C304 Wells Hall

No abstract available.

• Speaker: Rafe Mazzeo, Stanford University
• Title: TBD
• Date: 04/23/2020
• Time: 4:10 PM - 5:00 PM
• Place: C304 Wells Hall

No abstract available.