Abstracts of upcoming talks at the MPIM. For an overview see also the calendar.

## Seminar on Kac-Moody algebras and related topics

## The local version of the Lorentzian index theorem

The Atiyah-Singer index theorem for Dirac operators $D$ on compact Riemannian spin $n$-manifolds can be proved using the heat kernels of $D^*D$ and of $DD^*$.

Namely, one easily sees that

$$

\mathrm{ind}(D) = \mathrm{Tr}(e^{-tD^*D}) - \mathrm{Tr}(e^{-tDD^*})

$$

for any $t>0$. Inserting the short time asymptotics

$$

\mathrm{Tr}(e^{-tD^*D}) \quad\stackrel{t\searrow 0}{\sim}\quad (4\pi t)^{-n/2} \sum_{j=0}^\infty t^j \int_M a_j^{D^*D}(x)

$$

and similarly for $DD^*$, yields

$$

## Uniqueness of weak solutions to the Ricci flow - part 2

In his resolution of the Poincaré and Geometrization Conjectures, Perelman constructed Ricci flows in which singularities are removed by a surgery process. His construction depended on various auxiliary parameters, such as the scale at which surgeries are performed. At the same time, Perelman conjectured that there must be a canonical flow that automatically "flows through its surgeries”, at an infinitesimal scale.

## Variations of semi-infinite Hodge structures, quantum master equation on cyclic cochains, Feynman transforms and cohomological field theories

## Lower sectional curvature bounds via optimal transport

We give an optimal transport characterization of lower sectional curvature bounds for smooth Riemannian manifolds. More generally we characterize lower (and, in some cases, upper) bounds on the so-called p-Ricci curvature which corresponds to taking the trace of the Riemann curvature tensor on p-dimensional planes. Such characterization roughly consists on a convexity condition of the p-Reny entropy along Wasserstein geodesics, where the role of the reference measure is played by the p-dimensional Hausdorff measure.

## On the regularity of Alexandrov spaces

Alexandrov spaces with curvature bounded from below, finite Hausdorff dimension, and equipped with the corresponding Hausdorff measure are particular instances of metric measure spaces with Ricci curvature bounded from below (namely RCD spaces). In this talk, I will give results on the regularity of an Alexandrov space considered as a singular Riemannian manifold. These results are based on joint work with Luigi Ambrosio.

## Sub-Riemannian geometric inequalities

Sub-Riemannian structures can be described as limits of Riemannian ones with $\mathrm{Ric}(g_n) \to -\infty$ and they represent, in a certain sense, the most singular case among the three great classes of geometries (Riemannian, Finlser, and sub-Riemannian ones). In this talk, we discuss how, under generic assumptions, these structures support interpolation inequalities \`a la Cordero\--Erasquin\--McCann\--Schmuckenschl\"ager. As a byproduct, we characterize the sub-Riemannian cut locus as the set of points where the squared sub-Riemannian distance fails to be semiconvex.

## Regularity theory for Type I Ricci flows

A Ricci flow exhibits a Type I singularity if the curvature blows up at a certain rate near the singular time. Type I singularities are abundant and in fact it is conjectured that they are the generic singular behaviour for the Ricci flow on closed manifolds.

In this talk, I will describe some new integral curvature estimates for Type I flows, valid up to the singular time. These estimates partially extend to higher dimensions an estimate that was recently shown to hold in dimension three by Kleiner-Lott, using Ricci flow with surgery.

## Ricci flow beyond closed manifolds - part 2

Ricci flow theory has a remarkable track record of settling big problems in other areas, and has developed into a beautiful body of work in its own right. However, most applications and much of the theory applies only when the underlying manifold is closed, or in highly restricted noncompact settings. In these lectures I will describe some of the major developments that have occurred over the past few years geared at extending the theory to cases where the underlying manifold can be noncompact, and the flow and initial metric can be of unbounded curvature.

## Comparison between cuspidal generators on the depth-graded motivic Lie algebra and on the linearized double shuffle Lie algebra

According to the Broadhurst-Kreimer conjecture, there are modular phenomena in the ring of multiple zeta values

with the depth filtration. The goal of this project is to give a substantial solution to the Broadhurst-Kreimer conjecture in depth 4. In this talk, I will describe a conjectural connection, which involves a ratio of critical values of the L-function