We will discuss new estimates for the size and structure of the nodal set $\{u=0\}$ and the singular set $\{u=|\nabla u|=0\}$ of solutions to parabolic inequalities with parabolic Lipschitz coefficients. In particular, we show that almost all of these sets are covered by regular parabolic Lipschitz graphs, with quantitative control, and that both satisfy parabolic Minkowski bounds depending only on a doubling quantity at a point. Many of these results are new even in the case of the heat equation on $\mathbb{R}^n\times \mathbb{R}$. This is joint work with Max Hallgren and Zilu Ma.

The classification of ancient solutions of CSF under some geometric conditions is a parabolic version of Liouville-type theorem. We will present that any ancient smooth embedded finite-entropy curve shortening flow is one of the following: a static line, a shrinking circle, a paper clip, a translating grim reaper, or a graphical ancient trombone constructed by Angenent-You. In particular, our result implies that any compact ancient smooth embedded finite-entropy flow is convex. Moreover, any non-compact ancient smooth embedded finite-entropy flow is either a static line or a complete graph over a fixed open interval. This is based on the joint work with Kyeongsu Choi, Dong-Hwi Seo, and Wei-Bo Su.

Act I: Convexity, Duality, and Volume
Act II: Convex meets Differential
Act III: Convex meets Complex
Once convexity, duality and volume appear on stage, the Mahler
Conjectures are inevitable. These conjectures, originating in Number Theory,
predict the extremizers of the volume of a convex body times the volume of its dual.
They date from the 1930's and are still largely open. This "mathematical opera"
mostly aimed at a broad mathematical audience will attempt to recount the
over-a-century-old story of these beautiful conjectures, and their profound impact
on modern Geometric and Functional Analysis.

On convex domains in R^n and S^n, the first Dirichlet eigenfunction is known to be log concave, a fact that is crucial to estimate the spectral gap, which is the difference between the second and first Dirichlet eigenvalue. It is known that the first Dirichlet eigenfunction is in general not log-concave for convex domains in H^n. I will discuss concavity estimates on horoconvex domains in hyperbolic space - which are domains whose boundaries second fundamental form is greater than 1 -  which yield new spectral-gap bounds in H^n. In doing so, we resolve a conjecture by Nguyen, Stancu and Wei. This is based on joint work with G. Khan and on joint work with S. Saha and G. Khan.

A celebrated result of Sui-Yau says that manifolds with positive bisectional curvature are biholomorphic to complex projective space. In this talk we will introduce new curvature conditions that provide characterizations of cohomology complex projective spaces. For example, the curvature tensor of a Kaehler manifold induces an operator on symmetric holomorphic 2-tensors, called Calabi operator. This operator is the identity for complex projective space with the Fubini Study metric. We show that a compact n-dimensional Kaehler manifold with n/2-positive Calabi curvature operator has the rational cohomology of complex projective space. The complex quadric shows that this result is sharp if n is even. This talk is based on joint work with K. Broder, J. Nienhaus, P. Petersen, J. Stanfield.