Abstract:
Many applications of Heegaard-Floer theory in knot theory stem from its structure as a Topological Quantum Field Theory (TQFT). Among these applications, bounds on the unknotting number are particularly intriguing, as such results can often be described without recourse to advanced mathematics. In addition to fascinating results that address some long-standing open questions, there are also numerous conjectures with strikingly simple statements that remain unresolved. These topics will be explored in the talk.

Earlier versions of Heegaard-Floer TQFT are constructed as functors from the category of (pointed) knots and links, along with (decorated) cobordisms between them, to the category of modules along with homomorphisms between them. However, the dependence of these modules and homomorphisms on the decorations introduces certain complexities in computations and applications. In this talk, we will review the applications of Heegaard-Floer TQFTs and introduce a modified Heegaard-Floer functor from the category of oriented links in closed three-manifolds and oriented surface cobordisms in four-manifolds connecting them to the category of F[v]-modules and F[v]-homomorphisms between them, where F is the field with two elements. Notably, this modified functor is independent of the decoration. We will discuss some of its basic properties, provide examples, and explore potential applications.

Bio-Sketch: 

Eaman Eftekhary is currently the head of School of Mathematics at the Institute for Research in Fundamental Sciences, IPM, in Tehran, Iran. He obtained his PhD from Princeton University in 2004, and was a BP Assistant Professor of Mathematics at Harvard before joining IPM as a faculty member in 2007. He is a member of Iran's Academy of Science and has previously served as the president of Iran National Science Foundation.