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 recent joint work with Gang Tian, which settles the mathematical foundation of the approach of Hori and Vafa.
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 restricted class of 4-manifolds. For example, Gabai's proof of Property R implies that every homotopy 4-sphere built with one 2-handle and one 3-handle is standard. …
The main goal of geometric topology is the classification of manifolds within a certain framework (topological, piecewise linear, smooth, simply-connected, symplectic, etc.). Dimension four is special, as it is the only dimension in which a manifold can admit infinitely many non-equivalent smooth structures, and the only dimension in which there exist manifolds homeomorphic but not…
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 about finite element spaces of differential forms.
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 spectral sequence is a simultaneous higher-dimensional generalization of homology and long exact sequences. I will discuss a few examples of spectral sequences and their applications.
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 a differential equation or the intersection points between Lagrangian manifold, and its differential arises as the count of solutions of a differential equation on Y…
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 tool for approaching this problem; I will survey partial results that are known and then show that these results can be improved significantly. If a…
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 order O(Cr(K) ln5 (Cr(K)) where Cr(K) is the minimum crossing number of K. Furthermore, it is known that there exist families of (infinitely many) links with the property…
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 associate a chain map with a link cobordism, but only up to a sign. Blanchet has fixed this by introducing the category of gl(2)-foams, certain singular cobordisms…
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 PDEs, where for each unknown function a certain subset of partial derivatives is prescribed and the values of the unknown functions are prescribed along the corresponding transversal coordinate…
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 three-manifold other than S^3 has a non-trivial SU(2) representation, and this is a strategy for an alternate proof of the three-dimensional Poincare conjecture. We will…
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 $P$ along a finite number of hyperplanes, and re-assemble it into $Q$. The scissors congruence problem asks: when can we do this? what obstructs this? In…