**Harvard Meeting**

Saturday, December 1, 9AM-12:15PM

Science Center, Room 507

Speakers:

Title: Fukaya categories and mirror symmetry: cylinders, pants and beyond.

Abstract: We will give a gentle introduction to some recent developments in the area of homological mirror symmetry. We will use simple examples to illustrate Kontsevich's conjecture and its extension beyond the Calabi-Yau setting in which it was first formulated. We will mostly focus on a one-dimensional example, the pair of pants, to give a flavor of the geometric concepts involved in a general formulation of homological mirror symmetry. If time permits, we will then describe extensions to higher genus surfaces and to higher-dimensional examples.

Title: Twisted h-invariants

Abstract: In 1999, Froyshov defined an integer invariant of homology 3-spheres obtained from Floer's instanton homology groups. Since then, similar invariants have been defined for a variety of Floer theories, including the d-invariant of Heegaard Floer homology. Levine and Ruberman defined a generalization of the d-invariant for closed 4-manifolds with the homology of S^1 x S^3. Inspired by the isomorphism between Heegaard Floer and monopole Floer, I will explain a construction of an invariant of homology S^1 x S^3's obtained from monopole Floer homology.

Title: The homology cobordism group and involutive Floer homology

Abstract: The homology cobordism group has a largely unknown structure, although it has been successfully studied using various Floer theories. In this talk, we discuss what can be said about the homology cobordism group using involutive Floer homology, a variant of Heegaard Floer homology introduced recently by Hendricks-Manolescu. In particular, we construct infinitely many homomorphisms out of the homology cobordism group using involutive Floer homology. This is joint work with Irving Dai, Jen Hom, and Linh Truong.