Thursday February 08, 2018 1:00 PM  2:15 PM Math Tower 5127
 Kristen Hendricks, Michigan State University
Connected Heegaard Floer homology and homology cobordismWe study applications of Heegaard Floer homology to homology cobordism. In particular, to a homology sphere Y, we define a module HF_conn(Y), called the connected Heegaard Floer homology of Y, and show that this module is invariant under homology cobordism and isomorphic to a summand of HF_red(Y). The definition of this invariant relies on involutive Heegaard Floer homology. We use this to define a new filtration on the homology cobordism group, and to give a reproof of Furuta's theorem. This is joint work with Jen Hom and Tye Lidman.

Thursday March 01, 2018 1:00 PM  2:15 PM Math Tower 5127
 Aliakbar Daemi, SCGP
ChernSimons functional and the Homology Cobordism GroupThe set of 3manifolds with the same homology as the 3dimensional sphere, modulo an equivalence relation called homology cobordance, forms a group. The additive structure of this group is given by taking connected sum. This group is called the homology cobordism group and plays a special role in low dimensional topology and knot theory. In this talk, I will explain how one can construct a family of invariants of the homology cobordism group by applying ideas from minmax theory in Floer theory. The relationship between these invariants and the Froyshov’s invariant will be discussed. I will also talk about some topological applications.

Thursday March 08, 2018 1:00 PM  2:15 PM Math Tower 5127
 Ian Zemke, Princeton
Heegaard Floer mixed invariants of mapping toriThe Seiberg Witten invariant is a powerful tool for studying 4dimensional topology. The Heegaard Floer mixed invariants conjecturally agree with the SeibergWitten invariants. In this talk we describe how to compute the Heegaard Floer mixed invariant of a mapping torus in terms of Lefschetz numbers on HF^+ using cobordism maps for 4manifolds with embedded graphs. The computation is in terms of traces, cotraces, and a "broken path cobordism" map.

Thursday March 22, 2018 1:00 PM  2:15 PM Math Tower 5127
 Akram Alishani, Columbia
Trivial tangles, compressible surfaces, and Floer homologyHeegaard Floer homology has different extensions for 3manifolds with boundary. In this talk, we will explain how they can be used to give a combinatorially effective way for detecting boundary parallel components of tangles and existence of homologically essential compressing disks. The fact that these are checkable by computer is based on the factoring algorithm of LipshitzOzsvathThurston for computing bordered Floer homology and our extension of it to compute borderedsutured homology. This is a joint work with Robert Lipshitz.

Thursday April 05, 2018 1:00 PM  2:15 PM Math Tower 5127
 Jonathan Hanselman, Princeton
Bordered Heegaard Floer homology with torus boundary via immersed curvesI will describe a geometric interpretation of bordered Heegaard Floer invariants in the case of a manifold M with torus boundary. In particular these invariants, originally defined as homotopy classes of modules over a particular algebra, can be described as collections of decorated immersed curves in the boundary of M. Pairing two bordered Floer invariants corresponds to taking the Floer homology of immersed curves; in most cases this simply counts the minimal intersection number. This framework leads to elegant proofs of several interesting results about closed 3manifolds. As one example,
I will prove a lower bound for the complexity of Heegaard Floer homology of a manifold containing an incompressible torus, reproving and strengthening a recent result of
Eftekhary.

Thursday April 19, 2018 1:00 PM  2:15 PM Math Tower 5127
 Andrew Lobb, Durham University
Quantum invariants and concordanceWe shall give an overview of the information about smooth knot concordance so far extracted from quantum knot cohomologies. Joint work with Lukas Lewark. No prior knowledge assumed.

Thursday May 03, 2018 1:00 PM  2:15 PM Math Tower 5127
 Michael Miller, UCLA
Equivariant instanton homology and group cohomologyFloer's celebrated instanton homology groups are defined for integer homology spheres, but analagous groups in Heegaard Floer and Monopole Floer homology theories are defined for all 3manifolds; these latter groups furthermore come in four flavors, and carry extra algebraic structure. Any attempt to extend instanton homology to a larger class of 3manifolds must be
somehow equivariant  respecting a certain SO(3)action. We explain how ideas from group cohomology and algebraic topology allow us to define four flavors of instanton homology for rational homology spheres, and how these invariants relate to existing instanton homology theories.

