The study of dynamics of energy critical wave equations has seen remarkable progresses in recent years, resulting in deeper understanding of the singularity formation, soliton dynamics, and global large data theory. I will firstly review some of the landmark results, with emphasis on the channel of energy inequalities discovered by Duyckaerts, Kenig and Merle. Applications in the study of global dynamics of defocusing energy critical wave equation with a trapping potential in the radial case will be presented in some detail. We remark that the channel of energy argument provides crucial control on the global dynamics of the solution, and seems to be the only tool currently available to measure dispersion in this context, when we do not assume any smallness condition. The channel of energy argument is however sensitive to dimensions, and in higher dimensions, it is less powerful. We will mention a new approach to eliminate the dispersive energy when the channel of energy argument fails. Lastly, a new Morawetz estimate in the context of focusing energy critical wave equations will be discussed. This estimate allows us to study the singularity formation in more details in the non-radial case, without size restriction. As a result, we can characterize the solution along a sequence of times approaching the singular time, up to every nontrivial scale, as modulated solitons. 

Part of the talk is based on joint works with C. Kenig, and with B.P. Liu, W. Schlag, G.X. Xu.
 

The group of rational points is an important but subtle invariant of an abelian variety defined over a number field.  In the case of an elliptic curve over Q, a celebrated theorem of Mazur asserts that there are only finitely many possibilities for the torsion part; the same is conjectured to be true for all abelian varieties over number fields though very little has been proven in higher dimensions.  The natural geometric analog, known as the geometric torsion conjecture, asks for a bound on the torsion sections of a family of abelian varieties over a complex curve, and can be interpreted as the nonexistence of low genus curves in congruence towers of Siegel modular varieties.  We will discuss a general method for bounding the genus of curves in locally symmetric varieties using hyperbolic geometry and apply it to some special cases of the torsion conjecture as well as some related problems.  Along the way we will also deduce some results about the global geometry of these moduli spaces.  This is joint work with J. Tsimerman.

In the 1960s, the KdV equation was discovered to have infinitely many conserved quantities, explained by a "Lax pair" formalism. Due to this, the KdV equation is often described as completely integrable. Similar features were soon found in other nonlinear equations, spurring the field of integrable PDEs in which the KdV equation continues to be one of the flagship models.

These ideas were originally implemented for sufficiently fast decaying initial data and, in the 1970s, for periodic initial data. In this talk, we will describe recent progress for almost periodic initial data, centered around a conjecture of Percy Deift that the solution is almost periodic in time. The case of almost periodic initial data is strongly motivated by the periodic case but carries significant challenges so, beyond a class of algebro-geometric solutions, rigorous results have remained scarce. We will discuss the proof of existence, uniqueness, and almost periodicity in time, in the regime of absolutely continuous and sufficiently "thick" spectrum (in a sense made precise in the talk), and in particular, the proof of Deift's conjecture for small analytic quasiperiodic initial data.

Online lectures and online classes are changing the landscape of math education. At New York University, we are creating a hybrid Calculus 1 course which will combine interactive online content with an in-class component involving lectures and problem sessions. We relate the structure of this course to that of other non-traditional calculus classes, and we discuss some potential advantages of this class over the standard lecture format.

The Alexandrov-Bakel'man-Pucci (ABP) estimate is one of
the most beautiful applications of geometric ideas in PDE and it is the
backbone of the regularity theory of fully nonlinear elliptic PDE. I will
start from the classical ABP estimate and then talk about its general-
ization on Riemannian manifolds, obtained in joint work with Yu Wang.
As applications, I will present results about the Harnack inequalities for
non-divergent PDE on manifolds and also an ABP approach to the clas-
sical Minkowski and Heintze-Karcher inequalities. In the second part of
the talk, I will give a brief overview of the classical Minkowski integral
formulae which are related to the divergence structure of some elliptic
operators. I will present the spacetime analogue of this type formula
I obtained with co-authors. Motivated by the problems from general
relativity, we consider the co-dimension two submanifolds in Lorentzian
spacetimes and establish some new Minkowski formulae in this setting.