Speaker: Tamas Kalman --------------------------------------------------------------------------------------------------------------------- Talk #1 title: Floer homology and the Homfly polynomial Abstract: All oriented links $L\subset S^3$ have special diagrams. Based on such a diagram we construct a $3$-manifold (a so called sutured handlebody) $M$ which embeds in the branched double cover of the link. From the sutured Floer homology of $M$ we recover the Alexander polynomial $\Delta$ of $L$ via a simple forgetful map. More surprisingly, in cases when the diagram is also positive (so that $L$ is a special alternating link), $\mathrm{SFH}(M)$ can be used to compute those coefficients of the Homfly polynomial of $L$ whose sum is the leading coefficient of $\Delta$. To extract this information algebraically, we need the notion of the interior polynomial of a bipartite graph, which I will discuss in my second talk. Geometrically, this entails the cutting of some handles of $M$ and identifying the resulting handlebody with a Seifert surface complement for another special alternating link. The talk involves joint results with A. Juh\'asz, H. Murakami, A. Postnikov, J. Rasmussen, and D. Thurston. --------------------------------------------------------------------------------------------------------------------- Talk #2 title: The Tutte polynomial for polymatroids Abstract: Inspired by the chromatic polynomial, Tutte introduced his two-variable polynomial for graphs and later Crapo generalized it to matroids. I will explain how to extend this notion in yet another direction: to hypergraphs and polymatroids. (The latter can be interpreted as M-convex sets, integer-valued submodular set functions, generalized permutohedra etc.) This first appeared in work of Cameron and Fink but our approach will be more direct. In particular, I will show natural counterparts to both the definition in terms of activities of bases, and the equivalent formulation as the corank-nullity generating function. One property is that Tutte's duality relation for planar graphs generalizes to hypergraphs. This is a purely discrete mathematical story but one application of it is that it provides the last step in the connection between Floer homology and the Homfly polynomial that is the topic of my first talk. This is joint work with O. Bernardi and A. Postnikov. ---------------------------------------------------------------------------------------------------------------------