1. Phạm Tuấn Huy, Stanford University, USA 

Title: When are structures robust under randomness?

Abstract: Given a collection of substructures of a large discrete system, when are they robust under random subsampling of the system? For instance, among the substructures in the collection, we may require at least one complete substructure to survive the subsampling; or alternatively, we may ask for ‘’most’’ of a substructure, for appropriate notions of ‘’most’’, to survive. This theme entails a number of important questions in probabilistic combinatorics; including in particular questions about thresholds in random graphs and hypergraphs. The Kahn-Kalai conjecture predicts that a natural necessary condition to guarantee a complete substructure to survive the random subsampling is also roughly sufficient. Interestingly, this condition on the collection of substructures — so-called ‘’being not p-small’’ — also appears in other contexts. In particular, it arises in several important conjectures of Michel Talagrand on suprema of positive stochastic processes, which are intimately linked to appropriate notions of robustness. I will discuss recent developments around robustness of structures under randomness, as well as new understanding on how to verify conditions for robustness in applications of interest.

2. Tom Mrowka, Massachusetts Institute of Technology, USA 
TBA

3. Gigliola Staffilani, Massachusetts Institute of Technology, USA 

Title: The Schrödinger equations as inspiration of beautiful mathematics.

Abstract: In the last two decades great progress has been made in the study of dispersive and wave equations. Over the years the toolbox used in order to attack highly nontrivial problems related to these equations has developed to include a collection of techniques: Fourier and harmonic analysis, analytic number theory, math physics, dynamical systems, probability and symplectic geometry. In this talk I will introduce a variety of results using as model problem mainly the periodic 2D cubic nonlinear Schrödinger equation. I will start by giving a physical derivation of the equation from a quantum many-particles system, I will introduce periodic Strichartz estimates along with some remarkable connections to analytic number theory, I will move on the concept of energy transfer and its connection to dynamical systems, and I will end with some results following from viewing the periodic nonlinear Schrödinger equation as an infinite dimensional Hamiltonian system.

4. Richard Taylor, Stanford University, USA 

Title: Counting solutions to polynomial congruences: a survey.

Abstract: Given a collection of polynomials with integer coefficients one can wonder how many solutions these equations have modulo a varying prime number. This information can be encoded in a Dirichlet series, called an arithmetic L-function, whose behaviour is closely connected with the arithmetic of the polynomial equations. Similar Dirichlet series also arise in the theory of discrete subgroups of Lie groups and Langlands conjectured that every arithmetic L-function also arises in this way. This provides a powerful tool to study arithmetic questions. Very little was known about Langlands' conjecture until Wiles work on Fermat's Last Theorem, in which the key step was to prove Langlands' conjecture for certain cubic equations in two variables. (In this case Langlands' conjecture coincides with the earlier Shimura-Taniyama conjecture.) In the last 30 years much progress has been made on these questions, but many important questions still seem completely out of reach. In this talk I will survey some of the progress that has (and has not) been made.

5. Akshay Venkatesh, Institute of Advanced Study, USA 
TBA