Gauge Theory and Topology Seminar - Harvard University

The seminar meets Fridays 3:30-4:30 in Science Center 507.

All fully vaccinated members of the Boston-area mathematical community are welcome to attend. The fifth floor of the Science Center is accessible by elevator without Harvard ID on Fridays 3:00-6:00.




Spring 2022

Date

Speaker

Title

Feb 11 John Baldwin (Boston College) Fixed points and Khovanov homology
Feb 18 Gage Martin (Boston College) Annular links, double branched covers, and annular Khovanov homology
Feb 25 - no seminar - - talk postponed due to weather -
Mar 4 Melanie Wood (Harvard University) The distribution of the profinite completions of 3-manifold groups
Mar 11 - no seminar - - organizer out of town -
Mar 18 - no seminar - - spring break -
Mar 25 Danny Ruberman (Brandeis University) On the Thom Conjecture in CP^3
Apr 1 - no seminar - - department event -
Apr 8 Lisa Piccirillo (MIT) 4-manifolds with boundary and fundamental group Z
Apr 15 - no seminar - - talk postponed -
Apr 22 - no seminar - - organizer out of town -
Apr 29 - seminar cancelled - - seminar cancelled -
May 6 Matthew Hedden (MSU) Knot theory and complex curves



Abstracts

February 11

Speaker: John Baldwin (Boston College)
Title: Fixed points and Khovanov homology
Abstract: We use a relationship between Heegaard Floer homology and the symplectic Floer homology of surface diffeomorphisms to partially characterize knots with the same knot Floer homology as the torus knot T(2,5). We then combine this with classical results on the dynamics of surface homeomorphisms, and tools from gauge theory, Khovanov homology, and Khovanov homotopy to prove that Khovanov homology detects T(2,5). The ideas introduced in this work have also recently been used to solve problems in Dehn surgery stemming from Kronheimer and Mrowka's resolution of the Property P conjecture, which I will survey if there is time. This is mostly joint work with Ying Hu and Steven Sivek.


February 18

Speaker: Gage Martin (Boston College)
Title: Annular links, double branched covers, and annular Khovanov homology
Abstract: Given a link in the thickened annulus, you can construct an associated link in a closed 3-manifold through a double branched cover construction. In this talk we will see that perspective on annular links can be applied to show annular Khovanov homology detects certain braid closures. Unfortunately, this perspective does not capture all information about annular links. We will see a shortcoming of this perspective inspired by the wrapping conjecture of Hoste-Przytycki. This is partially joint work with Fraser Binns.


March 4

Speaker: Melanie Wood (Harvard University)
Title: The distribution of the profinite completions of 3-manifold groups
Abstract: It is well-known that for any finite group G, there exists a closed 3-manifold M with G as a quotient of pi_1(M). However, we can ask more detailed questions about the possible finite quotients of 3-manifold groups, e.g. if G and H are finite groups, does there exist a 3-manifold group with G as a quotient but not H as a quotient? We give an answer to all such questions. To prove existence of 3-manifolds with certain finite quotients but not others, we use a probabilistic method, by first proving a formula for the distribution of the (profinite completion of) the fundamental group of a random 3-manifold in the Dunfield-Thurston model of random Heegard splittings as the genus goes to infinity. This is joint work with Will Sawin.


March 25

Speaker: Danny Ruberman (Brandeis University)
Title: On the Thom Conjecture in CP^3
Abstract: The most famous version of the Thom conjecture asks for the minimal genus of a surface smoothly embedded in CP^2 carrying the homology class of degree d. Kronheimer-Mrowka proved that the minimal genus is that of a smooth degree d complex curve. One can ask a similar question about the 'simplest' (2k-2) manifold in CP^k of degree d, where simplicity might be measured by the rank of the homology. In particular, Thom’s question (or speculation, or conjecture, depending on whose history you believe) in higher dimension would ask if the degree d complex hypersurface is the simplest. Freedman’s PhD thesis showed that when k is even and bigger than 2, one can find submanifolds simpler than a complex hypersurface.

In joint work with Marko Slapar and Sašo Strle, we showed that for degree d at least 5, there are smooth 4-manifolds embedded in CP^3 with smaller homology than the degree d hypersurface. Hence the Thom conjecture fails in complex dimension 3.


April 8

Speaker: Lisa Piccirillo (MIT)
Title: 4-manifolds with boundary and fundamental group Z
Abstract: In this talk I will discuss work in progress in which we classify topological 4-manifolds with boundary and fundamental group Z, under some mild assumptions on the boundary. We apply this classification to provide an algebraic classification of surfaces in simply-connected 4-manifolds with S^3 boundary, where the fundamental group on the surface complement is Z. We also compare these homeomorphism classifications with the smooth setting, showing for example that every Hermitian form over Z[t^{\pm 1}] arises as the equivariant intersection form of a pair of exotic smooth 4-manifolds with boundary and fundamental group Z. This work is joint with Anthony Conway and Mark Powell.


May 6

Speaker: Matthew Hedden (Michigan State University)
Title: Knot theory and complex curves
Abstract: I'll recall the braid-theoretic characterization of knots in the 3-sphere bounding complex curves in the 4-ball due to Rudolph and Boileau-Orevkov, discuss generalizations to other 3-manifolds and fillings, a conjectural characterization in terms of transverse knot theory, and proof of this conjecture using knot Floer homology in some special cases. Parts of the talk will touch upon joint work in progress with Tovstopyat-Nelip and Baykur-Etnyre-Hayden-Van Horn-Morris.