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.

Joyce gave a framework of wall-crossing formulae for enumerative invariants in \C-linear abelian categories. We investigate implications for generating functions of Pandharipande-Thomas invariants for smooth projective Fano 3-folds. This is joint work-in-progress with Dominic Joyce.

It has long been known that homeomorphic 4-manifolds may admit inequivalent smooth structures. Analogous behavior holds for embedded surfaces. Some surfaces are topologically isotopic without being smoothly isotopic. Such surfaces are said to be smoothly knotted. 

It turns out that it is easier to construct inequivalent smooth structures on larger 4-manifolds. Similarly, it is easier to construct closed, smoothly knotted surfaces in large 4-manifolds. 

In this talk, we will explain how to construct smoothly knotted surfaces in a small 4-manifold. This talk will have many pictures.

This is joint work with Konno, Mukherjee, Ruberman, and Taniguchi. 

Algebraic K-theory of smooth compact manifolds provides a homotopical lift of the classical h-cobordism theorem and serves as a critical link in the chain of homotopy theoretic constructions that show up in the classification of manifolds and their diffeomorphisms. I will give an overview of this story and recent progress on an equivariant homotopical lift of the h-cobordism theorem developed in joint work with Goodwillie, Igusa, and Malkiewich. 

For two thousand years geometry was synonymous with the perfectly flat expanse imagined by Euclid. But nineteenth‑century investigations into the parallel postulate lifted a veil from our eyes, revealing the richer realms charted by Gauss and Riemann. In this talk we’ll take an “insider’s tour” of those curved landscapes.

We begin by thinking carefully about what it means to see, and use this to step inside new geometries, by tracing light rays along their geodesics. Modern computing affords us the ability to make this thought experiment a reality, with interactive ray-traced demos and experiments.  Using these, we will explore curved spaces important to modern mathematics, and physics, including the curved spacetime we live in.