Boston Graduate Topology Seminar
Saturday, September 30, 9AM-12:15PM
We will provide coffee and bagels, starting at 9AM.
9:30-10:15: Cliff Taubes (Harvard)
Title: What are Z/2 harmonic forms?
Abstract: Z/2 harmonic forms are closed and coclosed differential forms with values in a real line bundle that is defined only on the complement of a subset of the zero locus of the form itself. Their topological significance is not understood at present--but they do have some significance because they characterize (in part) the non-compactness of the moduli spaces of solutions to certain generalizations of the self-dual Yang-Mills equations. (Very little is known about these objects.)
10:30-11:15: Melissa Zhang (BC)
Title: Annular Khovanov homology of 2-periodic links
Abstract: In the late 1970's, Murasugi studied knots exhibiting p-fold symmetry and produced formulas relating the Alexander and Jones polynomials of these periodic knots with those of their quotient knots. Today, we are armed with richer, categorified link invariants such as knot Floer homology (HFK) and Khovanov homology (Kh). Their homological nature allows us to apply algebraic methods to obtain spectral sequences, which in turn produce rank inequalities and Murasugi-type formulas.
By the Smith conjecture, the data of a periodic link specifies an unknotted braid axis in the 3-sphere. Since deleting the braid axis places the link inside a solid torus, or a thickened annulus, it is natural to study its annular Khovanov homology (AKh). In this talk, we exhibit a spectral sequence relating the AKh of a 2-periodic link with that of its quotient, and derive various decategorified consequences. Curiously, our method also hints at a similar spectral sequence from Kh of the periodic link to the AKh of the quotient link; we discuss partial results and open questions on this fron t.
11:30-12:15: Yu Pan (MIT)
Title: Augmentations and Exact Lagrangian cobordisms
Abstract: To a Legendrian knot, one can associate an A∞ category, the augmentation category. An exact Lagrangian cobordism between two Legendrian knots gives a functor of the augmentation categories of the two knots. We study the functor and establish a long exact sequence relating the corresponding cohomology of morphisms of the two ends. As applications, we prove that the functor between augmentation categories is injective on the level of equivalence classes of objects and find new obstructions to the existence of exact Lagrangian cobordisms in terms of linearized contact homology and ruling polynomials.