In this talk I will discuss some new properties of an invariant for 4-manifold with boundary which was originally defined by Nobuo Iida. As one of the applications of this new invariant I will demonstrate how one can obstruct a knot from being h-slice (i.e bound a homologically trivial disk) in 4-manifolds. Also, this invariant can be useful to detect exotic smooth structures of 4-manifolds. This a joint work with Nobuo Iida and Masaki Taniguchi.

Algebraic cycles on a variety tell us a lot about the structure and behavior of that variety. They give rise to many important invariants such as Chow groups, Picard groups, and cohomology. As usual, there is a tropical analogue to algebraic cycles in tropical geometry, which are called tropical cycles. Unlike in the classical world, tropical cycles are less understood. This talk will discuss work by Allermann and Rau who laid the groundwork for tropical cycles as well as outline their proof that the bounded tropical Chow group of is the group of fan cycles.

I'll attempt to discuss some aspects of the speculative idea that quantum gravity can be thought of as an emergent topological phase of matter.

Abstract: Mostow Rigidity says that a closed hyperbolic 3 manifold has only one hyperbolic structure. What happens if our hyperbolic 3 manifold has flaring ends with negative euler characteristic? I found in Chapter 11 of Thurston's notes that hyperbolic structures on such a 3 manifold are parametrized by the product of Teichmuller spaces of its boundary components. I will attempt to explain how this works

The talk will focus on the following results: a finitely generated group quasi-isometric to the fundamental group of a compact 3–manifold or to a finitely generated Kleinian group contains a finite index subgroup isomorphic to the fundamental group of a compact 3–manifold or to a finitely generated Kleinian group. This talk is based on joint work with Cyril Lecuire.

(Joint with Nick Sheridan.) Fukaya categories can be constructed for arbitrary closed symplectic manifolds, but at considerable technical cost. I’ll explain a comparatively simple construction that works for Calabi-Yau manifolds, leveraging the complex geometry of a system of divisors close to a specified ample normal crossing divisor. An unusual feature is that the coefficients lie in a multivariable power series ring with integer coefficients. The construction is used in the proof of homological mirror symmetry (HMS) for “generalized Greene-Plesser mirrors” by Sheridan-Smith - the current cutting-edge for HMS.

In this talk, I hope to go over the basics of gradient flows. Starting from gradient flows on Euclidean space, I will motivate the various notions of gradient flows on general metric spaces. If time permits, I will give a few examples of PDEs which appear (sometimes naturally, sometimes not) as gradient flows on metric spaces.