I will explain a new proof of the non-linear stability of the Minkowski spacetime as a solution of the Einstein vacuum equation. The proof relies on an iteration scheme at each step of which one solves a linear wave-type equation globally. The analysis takes place on a suitable compactification of $\mathbb{R}^4$ to a manifold with corners whose boundary hypersurfaces correspond to spacelike, null, and timelike infinity; I will describe how the asymptotic behavior of the metric can be deduced from the structure of simple model operators at these boundaries. This talk is based on joint work with Andras Vasy.

We study variations of the Bergman kernel and their asymptotic behaviors at degeneration. For a holomorphic family of hyperelliptic nodal or cuspidal curves and their Jacobians, we announce our results on the Bergman kernel asymptotics near various singularities. For genus-two curves particularly, asymptotic formulas with precise coefficients involving the complex structure information are written down explicitly. Time permitting, we would like to talk about the equality part of the Suita conjecture as an application.

Entanglement is a core concept in quantum mechanics and quantum information theory. Put simply: a tensor is entangled if is not a product state. Measuring precisely how much entanglement a given tensor has is a big question with competing answers in the physics community. One natural measure is the {\bf geometric measure of entanglement}, which is a version of the Hilbert--Schmidt distance of the given tensor from the set of product states. It can also be described as the log of the spectral norm.

In 2009, Gross, Flammia, and Eisert showed that, as the mode of the tensor grows, the geometric measure of entanglement of a random tensor is, with high probability, very close to the theoretical maximum. In this talk, I will describe my joint work with Shmuel Friedland on the analogous situation for symmetric tensors. While symmetric tensors are inherently entangled, it turns out their maximum geometric measure of entanglement is exponentially smaller than for generic tensors. Using tools from representation theory and random matrix theory, we prove that, nevertheless, random symmetric tensors are, with high probability, very close to maximally entangled.

We identify the linear space spanned by the real-valued excessive functions of a Markov process with the set of those functions which are quasimartingales when we compose them with the process. Applications to semi-Dirichlet forms are given. We provide a unifying result which clarifies the relations between harmonic, co-harmonic, invariant, co-invariant, martingale and co-martingale functions, showing that in the conservative case they are all the same. The talk is based on joint works with Iulian Cimpean (Bucharest) and Michael Roeckner (Bielefeld).

We provide two asymptotic continued fraction expansions of the
prime counting function. We also develop a ``degree'' calculus that
enables us to strengthen the connections between various reformulations
and extensions of the Riemann hypothesis.