Variational methods can be used to create numerical methods that respect conservation laws. I will discuss applications to electromagnetism, the Yang-Mills equations, and mean curvature flow. I will also discuss some new ideas…

I will describe a homological algebra construction which is fundamental in algebraic topology, algebraic geometry, and related areas: the spectral sequence. Originally developed by Jean Leray in the 1940s, a…

Exact couples and the Bockstein spectral sequence.

Computing Tor with spectral sequences.

Knot homology theories

A Floer homology is an invariant of a closed, oriented 3-manifold Y that arises as the homology of a chain complex whose generators are either the set of solutions to…

Spectral sequences in Khovanov homology.

The cosmetic surgery conjecture states that no two surgeries on a given knot produce the same 3-manifold (up to orientation preserving diffeomorphism). Floer homology has proved to be a powerful…

For an un-oriented link K, let L(K) be the ropelength of K. It is known that in general L(K) is at least of the order O((Cr(K))3/4), and at most of the…

The original construction of the Khovanov homology of a link can be seen as a formal complex in the category of flat tangles and surfaces between them. There is a way to…

In his monograph “Systèmes Orthogonaux” (Leçons sur les systèmes orthogonaux et les coordonnées curvilignes, Gauthier-Villars, Paris, 1910), Darboux stated three theorems providing local existence and uniqueness of solutions to first order systems of…

One way to effectively show a group is non-trivial is to find a non-trivial representation.  A major open question in low-dimensional topology is whether the fundamental group of a closed…

In this talk I'll explain a surprising relationship between the objects in the title. Two n-dimensional polytopes, $P$, $Q$ are said to be scissors congruent if one can cut $P$…

The Penrose triangle, also known as the impossible tribar is an icon for cohomology.  It is literally the icon for Cech cohomology on Wikipedia.  The idea goes back to a…

Self-duality equations in gauge theory can be complexified in many inequivalent ways, but there are two obvious options: One can extend Hodge duality in either a complex linear fashion, or…

Topological Data Analysis (TDA)  is a relatively young field in both algebraic topology and machine learning.   Tools from TDA, in particular persistent homology, have proven successful in many scientific disciplines.  Persistence…