Harvard Gauge Theory and Topology Seminar Fall 2023
The seminar meets Fridays 3:30-4:30 pm in Science Center 507. The organizers are Peter Kronheimer, Fan Ye, and Gage Martin. If you'd like to get future announcements, you could send email to gagemartin@math.harvard.edu.
For those without Harvard IDs, the 5th floor of the Science Center can be accessed by elevator during 3:00-6:00 pm.
September 15th: Fan Ye
Title: Towards isomorphisms among Floer homologies
Abstract: Since Floer's work in 1988, various Floer homologies have been constructed for closed 3-manifolds, knots, and sutured manifolds. In 2008, Kronheimer-Mrowka proposed a conjecture about isomorphisms among Floer homologies. In this talk, I will introduce an approach to proving the isomorphisms and mention some partial results, based on combinatorial version of Floer homology. This work is joint with Baldwin, Li, and Sivek.
September 22nd: No Seminar
No seminar on account of the CMSA welcome event
September 29th: Ian Montague
Title: Equivariant Eta, Rokhlin, and Kappa Invariants of Seifert Fibered Homology Spheres
Abstract: In previous work I used Seiberg-Witten Floer K-theory to define a suite of invariants for homology spheres equipped with cyclic group actions. Involved in the construction is a so-called equivariant correction term, whose mod 2 reduction constitutes an equivariant refinement of the Rokhlin invariant with values in the representation ring of the corresponding cyclic group. In this talk I will explain how to calculate the equivariant correction terms with respect to cyclic group actions contained in the canonical S^1-action on Seifert-fibered homology spheres. As an application we show that for any manifold X homeomorphic to a Gompf nucleus N(2k) and any odd prime p, the Z/p-action given by rotation in the fibers of the Seifert-fibered boundary cannot extend smoothly to a Z/p-action over X, whereas some of these actions do extend topologically.
October 6th: Constantin Teleman
Title: Quantization commutes with Reduction again
Abstract: Frances Kirwan described the cohomology of smooth symplectic quotients $X//G$ in terms of the equivariant cohomology of X and the Morse stratification by the square-norm of the moment map. A quantum version of that result was later established by Woodward. In this talk, I will review a recent result giving the right 2-dimensional version of that result for monotone symplectic manifolds: the quantum cohomology of X//G is the gauged quantum cohomology of X, so that A-model quantization commutes with symplectic reduction This is joint work with Dan Pomerleano.
October 13th: Elia Portnoy
Title: On Freedman's Link Packings
Abstract: Freedman recently posed a new question in quantitative topology about link packings. Given a link L, define the $\epsilon$-diagonal packing number $n_{L(\epsilon)}$ to be the number of copies of L that can be simultaneously embedded in $[0,1]^3$ so that (1) Each copy of $L$ is contained in a ball which is disjoint from the other copies. (2) Within each copy, the components are separated by a distance of at least $\epsilon$. We'll discuss a new construction for obtaining a lower bound on $n_{L(\epsilon)}$ and expand on Freedman's ideas to obtain an upper bound on $n_{L(\epsilon)}$ when $L$ has a non-trivial Milnor Invariant. At the end we'll mention several related open problems about link packings. This is joint work with Fedya Manin.
October 20th: Deeparaj Bhat
Title: Surgery Exact Triangles in Instanton Theory
Abstract: We prove an exact triangle relating the knot instanton homology to the instanton homology of surgeries along the knot. As the knot instanton homology is computable in many instances, this sheds some light on the instanton homology of closed 3-manifolds. We illustrate this with computations in the case of some surgeries on the trefoil which are different from the analogous groups in other Floer theories such as Heegaard Floer and monopole Floer. Finally, we sketch the proof of the triangle.
October 27th: Sunghyuk Park
Title: Quantum UV-IR map and curve counts in skeins
Abstract:
Quantum UV-IR map (a.k.a. q-nonabelianization map), introduced by Neitzke and Yan, is a map from UV line defects in a 4d N=2 theory of class S to those of the IR. Mathematically, it can be described as a map between skein modules and is a close cousin of quantum trace map of Bonahon and Wong.
In this talk, I will discuss how quantum UV-IR map can be generalized to a map between HOMFLYPT skein modules, using skein-valued curve counts of Ekholm and Shend
November 10th: Dan Freed
Title: Complex Chern-Simons invariants of 3-manifolds via abelianization
Abstract: A hyperbolic 3-manifold M carries a flat PSL(2;C)-connection whose Chern-Simons invariant has been much studied since the early 1980's. For example, its real part is the volume of M. Explicit formulas in terms of a triangulation involve the dilogarithm. In joint work with Andy Neitzke we use 3-dimensional spectral networks to abelianize the computation of complex Chern-Simons invariants. The locality of the Chern-Simons invariant, expressed in the language of topological field theory, plays an important role. The dilogarithm arises from a novel construction involving Chern-Simons invariants of flat C*-connections over a 2-torus.
November 17th: Ali Daemi
Title: Rank three instantons, representations and sutures
Abstract: Yang-Mills gauge theory with gauge group SU(2) has played a significant role in the study of the topology of 3- and 4-manifolds. It is natural to ask whether we obtain more topological information by working with other choices of gauge groups such as SU(n) for higher values of n. Mariño and Moore formulated a conjecture essentially stating that there is no new information in Donaldson invariants of smooth 4-manifolds defined using SU(n) Yang-Mills gauge theory. Despite this "negative" prediction, one might still hope that there is still novel information about 3-manifolds in higher rank gauge theory. In this talk, I will discuss a result about the topology of 3-manifolds obtained using gauge theory with respect to the Lie group SU(3): for any knot K in the 3-dimensionl sphere (or more generally an integer homology sphere) there is a non-abelian representation of the knot group of K into SU(3) such that the homotopy class of the meridian of K is mapped to a matrix with eigenvalues 1, w, w^2 with w being a primitive third root of unity. As a byproduct of the proof, we obtain a structure theorem for SU(3) Donaldson invariants of 4-manifolds, analogous to Kronheimer and Mrowka's structure theorem for SU(2) Donaldson invariants. This can be regarded as a piece of evidence supporting Mariño and Moore's conjecture. This talk is based on a recent joint work with Nobuo Iida and Chris Scaduto.
December 1st: Hannah Turner
Title: The (fractional) Dehn twist coefficient and infinite-type surfaces
Abstract: The fractional Dehn twist coefficient (FDTC) is an invariant of a self-map of a surface which is some measure of how the map twists near a boundary component of the surface. It has been studied for compact (or finite-type) surfaces; in this setting the invariant is always a fraction. I will discuss work to extend this invariant to infinite-type surfaces and show that it has surprising properties in this setting. In particular, the invariant no longer needs to be a fraction - any real number amount of twisting can be achieved! I will also discuss a new set of examples of (tame) big mapping classes called wagon wheel maps which exhibit irrational twisting behavior. This is joint work with Diana Hubbard and Peter Feller.