I will describe joint work with Tyler Kelly and Ran Tessler. FJRW (Fan-Jarvis-Ruan-Witten) theory is an enumerative theory of quasi-homogeneous singularities, or alternatively, of Landau-Ginzburg models. It associates to a potential W:C^n -> C given by a quasi-homogeneous polynomial moduli spaces of (orbi-)curves of some genus and marked points along with some extra structure, and these moduli spaces carry virtual fundamental classes as constructed by Fan-Jarvis-Ruan. Here we specialize to the case W=x^r+y^s and construct an analogous enumerative theory for disks. We show that these open invariants provide perturbations of the potential W in such a way that mirror symmetry becomes manifest. Further, these invariants are dependent on certain choices of boundary conditions, but satisfy a beautiful wall-crossing formalism.

The integral of mean curvature squared is a conformal invariant of surfaces reintroduced by Willmore in 1965 whose study exercised a tremendous influence on geometric analysis and most notably on minimal surfaces in the last years.

On the other hand, the Loewner energy is a conformal invariant of planar curves introduced by Yilin Wang in 2015 which is notably linked to SLE processes and the Weil-Petersson class of (universal) Teichmüller theory.

In this presentation, after a brief historical introduction, we will discuss some recent developments linking the Willmore energy to the Loewner energy and mention several open problems.

Joint work with Yilin Wang (MIT/MSRI)

Most Ricci flow theory takes the short-time existence of solutions as a starting point and ends up concerned with understanding the long-time limiting behaviour and the structure of any finite-time singularities that may develop along the way. In this talk I will look at what you can think of as singularities at time zero. I will describe some of the situations in which one would like to start a  Ricci flow with a space that is rougher than a smooth bounded curvature Riemannian manifold, and some of the situations in which one considers smooth initial data that is only achieved in a non-smooth way. A particularly interesting and useful case is the problem of starting a Ricci flow on a Riemann surface equipped with a measure. I will not be assuming expertise in Ricci flow theory. Parts of the talk are joint with either Hao Yin (USTC) or ManChun Lee (CUHK).

I will discuss a string theory way to study G_2 instanton moduli space and explain how to compute the instanton partition function for a certain G_2 manifold. An important insight comes from the twisted M-theory on the G_2 manifold. Building on the example, I will explain a possibility to extend the story to a large set of conjectural G_2 manifolds and a possible connection to 4d N=1 SCFT via geometric engineering. This talk is based on https://arxiv.org/abs/2109.01110 and a series of works in progress with Michele Del Zotto and Yehao Zhou.