I will describe some recent work on the incompressible Euler equations and related partial differential equations specifically related to "Critical Phenomena". It is, by now, known that the incompressible Euler equation is ill-posed in most "critical classes" such as the class of Lipschitz continuous or C^1 velocity fields (even when the data is taken to be smooth away a single point). Despite this, we prove well-posedness (global in 2d and local in 3d) for merely Lipschitz data which is smooth away from the origin and satisfies a mild symmetry assumption. To do this requires a deep understanding of the nature of unboundedness of singular integrals on $L^\infty$. Through this understanding, we define a well-adapted class of critical function spaces in which we prove well-posedness. After this, we extract a simplified equation which is satisfied by "scale invariant" solutions which lie within the setting of our local/global well-posedness theory. These scale-invariant solutions, in the 2d Euler setting, can be shown to have very interesting dynamical properties such as time-quasiperiodic behavior. Moreover, these scale-invariant solutions (while having infinite energy) can be used to prove the existence of finite-energy solutions with the "same" dynamical properties.This is joint work with In-Jee Jeong.

Large amounts of data now stream from daily life; data analytics has been helping to discover hidden patterns, correlations and other insights. This talk introduces the mode decomposition problem in the analysis of oscillatory data. This problem aims at identifying and separating pre-assumed data patterns from their superposition. It has motivated new mathematical theory and scientific computing tools in applied harmonic analysis. These methods are already leading to interesting and useful results, e.g., electronic health record analysis, microscopy image analysis in materials science, art and history.

The Birch and Swinnerton-Dyer conjecture, one of the millenium problems, is a bridge between algebraic invariants of an elliptic curve and its (complex analytic) L-function. In the case of low ranks, we prove this conjecture up to the finitely many bad primes and the prime 2, by proving the Iwasawa main conjecture in full generality. The ideas in the proof and formulation also lead us to new and mysterious phenomena. This talk assumes no specialized background in number theory.

 From the revolutionary work of Thurston and Perelman, we know the topology of 3-manifolds is deeply intertwined with their geometry. In particular, hyperbolic geometry, the non-Euclidean geometry of constant negative curvature, plays a central role. In turn, hyperbolic geometry opens the door to applying tools from number theory, specifically automorphic forms, to what might seem like purely topological questions.

After a passing wave at the recent breakthrough results of Agol, I will focus on exciting new questions about the geometric and arithmetic meaning of torsion in the homology of finite covers of hyperbolic 3-manifolds, motivated by the recent work of Bergeron, Venkatesh, Le, and others. I will include some of my own results in this area that are joint work with F. Calegari and J. Brock.

In this talk I will introduce the moduli space of curves, and a class of vector bundles “of conformal blocks” on the moduli space of curves.   I’ll give a nonspecialist definition of these bundles, which have connections to algebraic geometry, representation theory and mathematical physics.  I’ll talk about how by studying the bundles we can learn about the moduli space of curves, and vice versa, focusing on just a few recent results, and open problems.