Seminar in Topology and Symplectic Geometry

from Monday
January 01, 2018 to Thursday
May 31, 2018
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Thursday
February 08, 2018

1:00 PM - 2:15 PM
Math Tower 5-127
Kristen Hendricks, Michigan State University
Connected Heegaard Floer homology and homology cobordism

We 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 5-127
Aliakbar Daemi, SCGP
Chern-Simons functional and the Homology Cobordism Group

The set of 3-manifolds with the same homology as the 3-dimensional 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 min-max 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 5-127
Ian Zemke, Princeton
Heegaard Floer mixed invariants of mapping tori

The Seiberg Witten invariant is a powerful tool for studying 4-dimensional topology. The Heegaard Floer mixed invariants conjecturally agree with the Seiberg-Witten 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 4-manifolds 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 5-127
Akram Alishani, Columbia
Trivial tangles, compressible surfaces, and Floer homology

Heegaard Floer homology has different extensions for 3-manifolds 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 Lipshitz-Ozsvath-Thurston for computing bordered Floer homology and our extension of it to compute bordered-sutured homology. This is a joint work with Robert Lipshitz.


Thursday
April 05, 2018

1:00 PM - 2:15 PM
Math Tower 5-127
Jonathan Hanselman, Princeton
Bordered Heegaard Floer homology with torus boundary via immersed curves

I 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 3-manifolds. 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 5-127
Andrew Lobb, Durham University
Quantum invariants and concordance

We 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 5-127
Michael Miller, UCLA
Equivariant instanton homology and group cohomology

Floer'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 3-manifolds; these latter groups furthermore come in four flavors, and carry extra algebraic structure. Any attempt to extend instanton homology to a larger class of 3-manifolds 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.


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