Noether's theorem tells us that if a system is invariant under a group of symmetries, then we have quantities that are conserved. For example, if a system is invariant under translation, then momentum…
Convex geometry is the study of convex bodies in Euclidean space. Despite the apparent simplicity of such objects, they are a source of many deep mathematical discoveries and mysteries. This…
Historically, the study of the collection of concordance classes of knots has focused on understanding its group structure while devoting relatively little attention to the natural metric induced by the 4-genus. Cochran…
We study applications of Heegaard Floer homology to homology cobordism. In particular, to a homology sphere Y, we associate a module HF_conn(Y), called the connected Heegaard Floer homology of Y,…
I will talk about recent developments on simplicial maps of complexes of curves on both orientable and nonorientable surfaces. I will also talk about joint work with Prof. Luis Paris. We…
We will describe an adaptation of the theory of trisections to the setting of properly embedded, smooth, compact surfaces in smooth, compact, orientable four-manifolds with boundary. The main result is…
In joint work with Rafal Komendarczyk and Ismar Volic, we study the space of braids, that is, the loopspace of the configuration space of points in a Euclidean space. We…
The configuration space of some $n$ points in a space $X$ is a well-studied object in topology, geometry, and combinatorics. We present a generalization, the simplicial configuration space $M(S,X)$, which takes as…
Using the knot filtration on the Heegaard Floer chain complex, Ozsváth and Szabó defined an invariant of knots in the three sphere called τ(K) and showed that it is a lower bound for…
The set of 3-manifolds with the same homology as the 3-dimensional sphere, modulo an equivalence relation called homology cobordance, forms a group. The additive structure of this group is given by…
We construct smooth compact 4-manifolds homotopy equivalent to S^2 which do not contain nicely embedded spheres realizing the homotopy equivalence.
Some physically significant solutions to Einstein's field equations are spacetimes which are foliated by a family of curves called a shear-free null geodesic congruence (SFNGC). Examples include models of gravitational…
Inspired by Bar-Natan's description of Khovanov homology, we discuss thin posets and their capacity to support homology and cohomology theories which categorify rank-statistic generating functions. Additionally, we present two main…
In this overview talk, I will first explain several mathematical approaches of mirror symmetry. Then I will focus on the mathematical theory of Witten's gauged linear sigma model based on the…
The smooth version of the 4-dimensional Poincare Conjecture (S4PC) states that every homotopy 4-sphere is diffeomorphic to the standard 4-sphere. One way to attack the S4PC is to examine a…