since the seminal work of yudovich in 1963, it has been known that for a given uniformly bounded and compactly supported initial vorticity profile, there exists a unique global solution to the 2d incompressible euler equation. a special class of yudovich solutions are so-called vortex patch solutions where the vorticity profile is the characteristic function of an (evolving) bounded set in $\mathbb{r}^2.$ in 1993 chemin and bertozzi-constantin proved that sufficiently high regularity of the boundary is propagated for all time. since then, there have been numerous numerical and rigorous works on understanding the long-time dynamics of smooth vortex patches as well as the short time dynamics of vortex patches with corners. in this work, we consider two regimes; one where we prove well-posedness and the other where we prove ill-posedness. first, for vortex patches with corners enjoying a certain symmetry property at the corners, we prove global propagation of the corners; we also give examples where these vortex patches cusp in infinite time. second, we prove that vortex patches with a single corner (which do not satisfy the symmetry condition) immediately cease to have a corner. this is joint work with i. jeong.

with respect to the numerical solution of partial differential equations, we explore the simple idea of treating time
as a space variable, and not employing the usual
method of lines time stepping approach. while this increases the space
dimension of a given pde problem by one, it introduces a
static convection term that can be treated by a variety of
techniques. this approach can be especially beneficial
in the setting of parallel adaptive finite element
computations.

in a series of recent works blomer, fouvry, kowalski, milicevic, sawin and myself have been able to solve the vexing problem of evaluating asymptotically the second moment of the central l-values of character twists (of large prime conductor) of a fixed modular form; the solution combines the spectral theory of modular forms, bounds for bilinear sums of kloosterman sums and advanced methods in l-adic cohomology. we will describe the proof and especially the second and third ingredient which is joint work with e. kowalski an w. sawin.
references: https://arxiv.org/abs/1411.4467 and https://arxiv.org/abs/1511.01636.

with the increasing interest in applying the methodology of difference-of-convex (dc) optimization to diverse problems in engineering and statistics, we show that many well-known functions arising therein can be represented as the difference of two convex functions. these include a univariate folded concave function commonly employed in statistical learning, the value function of a copositive recourse function in two-stage stochastic programming, and many composite statistical functions in risk analysis, such as the value-at-risk (var), conditional value-at-risk (cvar), expectation-based, var-based, and cvar-based random deviation functionals. we also discuss decomposition methods for computing directional stationary points of a class of nonsmooth, nonconvex dc programs that combined the gauss-seidel idea, the alternating direction method of multipliers, and a special technique to handle the negative of a pointwise max function.

i will talk about joint work with raymond heitmann that gives a construction of big cohen-macaulay algebra in mixed characteristics following the recent breakthroughs on the direct summand conjecture by andr\'{e} and bhatt. in fact, we prove a weakly functorial version for certain surjective ring homomorphism that leads to the solution of the vanishing conjecture for maps of tor in mixed characteristic. our work also gives a simplified proof of the direct summand conjecture, and that direct summand of regular rings are cohen-macaulay.

on a compact riemannian manifold with boundary having positive mean
curvature, a fundamental result of shi and tam states that, if the
manifold has nonnegative scalar curvature and if the boundary is
isometric to a strictly convex hypersurface in the euclidean space,
then the total mean curvature of the boundary is no greater than the
total mean curvature of the corresponding euclidean hypersurface. in
3-dimension, shi-tam's result is known to be equivalent to the
riemannian positive mass theorem.

in this talk, we will discuss a supplement to shi-tam's theorem
by including the effect of minimal hypersurfaces on a chosen boundary
component. more precisely, we consider a compact manifold with
nonnegative scalar curvature, whose boundary consists of two parts,
the outer boundary and the horizon boundary. here the horizon
boundary is the union of all closed minimal hypersurfaces in the
manifold and the outer boundary is assumed to be a topological
sphere. in a relativistic context, such a manifold represents a body
surrounding apparent horizon of black holes in a time symmetric
initial data set. by assuming the outer boundary is isometric to a
suitable 2-convex hypersurface in a schwarzschild manifold of
positive mass m, we establish an inequality relating m, the area of
the horizon boundary, and two weighted total mean curvatures of the
outer boundary and the hypersurface in the schwarzschild manifold. in
3-dimension, our result is equivalent to the riemannian penrose
inequality. this is joint work with siyuan lu.

i will present a recent work (joint with sung-jin oh) on the strong
cosmic censorship conjecture for the
einstein-maxwell-(real)-scalar-field system in spherical symmetry for
two-ended asymptotically flat data. for this model, it was previously
proved (by m. dafermos and i. rodnianski) that a certain formulation
of the strong cosmic censorship conjecture is false, namely, the
maximal globally hyperbolic development of a data set in this class
is extendible as a lorentzian manifold with a c0 metric. our main
result is that, nevertheless, a weaker formulation of the conjecture
is true for this model, i.e., for a generic (possibly large) data set
in this class, the maximal globally hyperbolic development is
inextendible as a lorentzian manifold with a c2 metric.