## An Introduction to Liouville Theory

## Percolation of sign clusters for the Gaussian free field II

## Percolation of sign clusters for the Gaussian free field I

## The Cool Alter-Ego of a Black Hole

## A converse to a theorem of Gross--Zagier, Kolyvagin and Rubin, II

Let $E$ be a CM elliptic curve over a totally real number field $F$ and $p$ an odd ordinary prime. If the ${p^{\infty}\mbox{-}\mathrm{Selmer}}$ group of $E$ over $F$ has ${\mathbb{Z}_{p}\mbox{-}\mathrm{corank}}$ one, we show that the analytic rank of $E$ over $F$ is also one (joint with Chris Skinner and Ye Tian). We plan to discuss the setup and strategy.

## Birational Calabi-Yau manifolds have the same small quantum products.

We show that any two birational projective Calabi-Yau manifolds have isomorphic small quantum cohomology algebras after a certain change of Novikov rings. The key tool used is a version of an algebra called symplectic cohomology, which is constructed using Hamiltonian Floer cohomology. Morally, the idea of the proof is to show that both small quantum products are identical deformations of symplectic cohomology of some common open affine subspace.