The generalised Ricci flow, first studied in the physics literature by Callan, Friedan, Martinec, and Perry and later by J. Streets, is a coupling of the Ricci flow with the heat flow for closed 3-forms. In the setting of Hitchin's generalised geometry, it is the natural analogue of the Ricci flow for generalised metrics on exact Courant algebroids. As well as generalising Ricci flow, in the setting of non-Kähler complex geometry it is equivalent to the pluriclosed flow, introduced by Streets–Tian.

In this talk, we discuss recent results on generalised Ricci flows on Lie groups. Our main result is a formula for the generalised Ricci curvature of a left-invariant generalised metric in terms of the classical Ricci curvature of a certain abelian extension of the underlying Lie group, which arises naturally from generalised geometry. Using this, we prove invariant dynamical stability of Bismut-flat metrics (certain fixed points of the flow) on compact semisimple Lie groups. We also exploit the formula to prove long-time existence and convergence to solitons for large classes of solvable Lie groups. The main tool in this context is an adaptation of Lauret's “bracket flow” to the setting of generalised geometry. For the pluriclosed flow, we prove long-time existence for not-necessarily-invariant solutions on compact nilmanifolds and certain classes of solvmanifolds. The results are based on upcoming joint work with Elia Fusi, Ramiro Lafuente, and Luigi Vezzoni.

A joint seminar with the Geometry & Topology series. 

Ricci solitons are generalizations of the Einstein manifolds and are self similar solutions to the Ricci flow. In particular, steady Ricci solitons are eternal solutions to the Ricci flow. In this talk, we will discuss the flying wing construction of some Kahler and Riemannian steady Ricci solitons of nonnegative curvature. This is based on joint work with Ronan Conlon and Yi Lai, as well as with Yi Lai and Man-Chun Lee.

We will present the enumerative min-max theory, which relates the number of genus g minimal surfaces in 3-manifolds to topological properties of the set of all embedded surfaces of genus ≤g. As a consequence, we can show that in every 3-sphere of positive Ricci curvature, there exist ≥5 minimal tori (confirming a conjecture by B. White (1989) in the Ricci-positive case), ≥4 minimal surfaces of genus 2, and ≥1 minimal surface of genus g for all g. This is based on a joint work with Yangyang Li and Zhihan Wang.

This is joint work with Minghao Wang. In this talk, I will explain the convergence of Feynman graph integrals on closed real-analytic K\"ahler manifolds: Using Getzler’s rescaling technique, the graph integrands extend naturally to the Fulton–MacPherson compactification as forms with divisorial singularities, allowing a rigorous definition as Cauchy principal value integrals. As an application, this yields a mathematical construction of the higher-genus B-model invariants on Calabi–Yau threefolds in the sense of Bershadsky–Cecotti–Ooguri–Vafa (BCOV).