Harvard Gauge Theory and Topology Seminar 

The seminar meets Fridays 3:30-4:30 pm in Science Center 507. The organizers are Peter Kronheimer, Gage Martin, Dylan Galt, William Ballinger, and Clair Xinle Dai. If you'd like to get future announcements, you could send email to xdai@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.

Fall 2025

October 24: Yingying Wu (University of Houston）

Title: Singularity models for Z/2 harmonic 1-forms and spinors

Abstract: We introduce local models for a Z/2 harmonic 1-form or spinor on R^{3} and R^{4} near a singular point in their zero loci based on symmetry considerations. The local models are homogeneous with respect to rescaling, with their zero loci given by rays from the origin in R^{3} and cones on the 1‑skeleton of regular 4‑polytopes invariant under subgroups of SO(4).

November 14: Elise LePage (Columbia）

Title: Aganagic’s invariant is Khovanov homology

Abstract: Recently, Aganagic proposed a categorification of quantum link invariants (corresponding to U_q(g) where g is an ADE Lie algebra) using Lagrangian Floer theory in multiplicative Coulomb branches equipped with a potential. Her original proposal was based on insights from string theory, but the resulting definition of categorified link invariants can be made mathematically rigorous. In this talk, I will review her proposal for link invariants and explain my recent proof (joint with Vivek Shende) that Aganagic’s invariant recovers Khovanov homology in the case g=sl(2).

November 21: Isabella Khan (MIT）

Title: Bordered algebras and the wrapped Fukaya category

Abstract: This talk will discuss an isomorphism between endomorphism algebras from the wrapped Fukaya category of a type of punctured surface, and a class of A-infinity algebras related to bordered knot Floer homology. After a brief outline of the constructions and algebraic methods used in the proof, this talk will focus on the model calculations with Fukaya categories that both motivate and help to verify this result.

December 5: Thibault Decoppet (Harvard)

Title: Mixed Verlinde Categories and Invariants of 4-Dimensional 2-Handlebodies  

Abstract: I will review how a skein-theoretic construction of Costantino--Geer--Haïoun--Patureau-Mirand produces an invariant of 4-dimensional 2-handlebodies from a finite unimodular ribbon tensor category. I will then discuss more specifically the invariants associated to the mixed Verlinde categories, which are derived from quotients of the Temperley-Lieb category in positive characteristic. This is based on joint work with Benjamin Haïoun.

Spring 2026

Febuary 13: Seraphina Lee (Harvard)

Title: Nielsen realization problem for del Pezzo surfaces

Abstract: The (topological) Nielsen realization problem for a closed, oriented manifold $M$ asks which finite subgroup of $\pi_0(\Homeo^+(M))$ admits a lift to $\Homeo^+(M)$. I will discuss an assortment of results about the Nielsen realization problem for del Pezzo surfaces $M_n := \mathbb{CP}^2 \# n \overline{\mathbb{CP}^2}$, $n \leq 8$, including a classification of finite subgroups $G \leq \pi_0(\Homeo^+(M_2))$ admitting lifts to $\Diff^+(M_2)$ and a comparison of the smooth, complex, and metric versions of the Nielsen realization problem for certain ``irreducible'' finite cyclic subgroups of $\pi_0(\Homeo^+(M_n))$. This talk is based in part on joint work with Tudur Lewis and Sidhanth Raman.

Febuary 27: Shunyu Wan (Georgia Tech)

Title: Surgeries on knots and tight contact structures

Abstract: The existence and nonexistence of tight contact structures on 3-manifolds are interesting and important topics studied over the past thirty years. Etnyre-Honda found the first example of a 3-manifold that does not admit tight contact structures, and later Lisca-Stipsicz extended their result and showed that a Seifert fiber space admits a tight contact structure if and only if it is not smooth (2n-1)-surgery along the T(2,2n+1) torus knot for any positive integer n. Surprisingly, since then no other example of a 3-manifold without tight contact structures has been found. Hence, it is interesting to study if all such manifolds, except those mentioned above, admit a tight contact structure. Towards this goal, I will discuss the joint work with Zhenkun Li and Hugo Zhou about showing any negative surgeries on any knot in S^3 admit a tight contact structure.

March 6: Shuhao Li (Stony Brook)

Title: Topology of Lagrangian submanifolds via open-closed string topology

Abstract: We study Lagrangian submanifolds in standard symplectic vector spaces R^{2n} using ideas from open-closed string topology. Specifically, for a closed, oriented, spin Lagrangian L, we construct a (possibly curved) deformation of the dg associative algebra of chains on the based loop space of L ("open string"), using moduli spaces of pseudo-holomorphic discs with boundaries on L viewed as chains in the free loop space ("closed string"). As an application, we prove that if the second homotopy group of L is zero, then L has non-vanishing Maslov class, generalizing previous results due to Viterbo, Cieliebak-Mohnke, Fukaya and Irie. If time permits, we will speculate on the possible classification of the topology of such Lagrangians in dimension 3.

March 13: Ollie Thakar (Harvard)

Title: A homotopy moduli space of G2 structures with infinite fundamental group

Abstract: An irreducible G2-manifold is a Riemannian 7-manifold M with holonomy group equal to the exceptional Lie group G2. When M is closed, the Teichmüller space T(M) of G2 metrics on M divided by diffeomorphisms isotopic to the identity is a smooth, finite-dimensional manifold by a result of Joyce. Yet its topology, and that of its quotient by the smooth mapping class group, remains elusive. Using ideas of Crowley, Goette, and Hertl, we exhibit the first known example of a G2-manifold M together with infinitely many diffeomorphisms that both act freely on T(M) and preserve a connected component. The diffeomorphisms are 7-dimensional analogs of diffeomorphisms of K3 surfaces constructed recently by Farb and Looijenga.

March 27: Yash Deshmukh (IAS)

Title: Deligne–Mumford field theories from categories

Abstract: Open-closed Deligne–Mumford field theories (oc-DMFTs) are a variant of 2D topological field theories (TFTs). These are based on moduli spaces of stable nodal curves and encode the algebraic structure of Gromov–Witten invariants. Obtaining TFTs and related structures from suitable categories has been a subject of significant interest. I will explain how to obtain an oc-DMFT from a smooth, proper Calabi–Yau category and a splitting of the non-commutative Hodge–de Rham spectral sequence. This provides an enhancement of the categorical enumerative invariants of Costello and Caldararu–Tu. The construction has a precise universal property that can be used to compare it with the geometrically defined Gromov–Witten oc-DMFT, provided the latter can be constructed.

April 3 (2:15 - 3:15 pm): Ciprian Manolescu (Stanford) 

Title: Canonical orientations in Heegaard Floer theory

Abstract: Heegaard Floer homology was originally defined over the integers by Ozsvath and Szabo using choices of coherent orientations on the moduli spaces. In this talk I will explain how to construct orientations in a more canonical way, by using a coupled Spin structure on the Lagrangian tori. This allows us to prove naturality of Heegaard Floer homology over the integers. The talk is based on joint work with Mohammed Abouzaid.

April 3 (3:30 - 4:30 pm): Michael Hutchings (Berkeley)

Title: Quantitative closing lemmas

Abstract: We consider the dynamics of Reeb vector fields on three-manifolds. Irie proved that for a C^\infty generic contact form, the periodic orbits of the Reeb vector field ("Reeb orbits") are dense. The key step is a closing lemma asserting that one can modify a contact form by a C^\infty-small perturbation to create a Reeb orbit passing through a given neighborhood. In this talk we discuss a quantitative refinement of Irie's closing lemma. Roughly speaking this asserts that one can create a Reeb orbit of period at most L via a perturbation of size O(L^{-1}). The proof uses spectral invariants related to embedded contact homology, and a key ingredient is a Weyl law for these invariants arising from Seiberg-Witten theory.

April 10: Zhenghao (William) Zhong (SIMIS)

Title: Landscape of mirrors 

Abstract: Mirror symmetry is a cornerstone of 3d N=4 supersymmetric theories. It is a duality that relates the Coulomb branch and Higgs branch moduli spaces of vacua. Finding such mirror pairs had always been constrained by the existence of type IIB Hanany-Witten brane configurations. In our recent paper, we developed a general way of deriving 3d mirror pairs directly from quiver diagrams without relying on string theory techniques. 

 April 17: Qianhe Qin (Stanford)

Title: How to Pick Out the Slicing Degree of Knots Using a Spork

Abstract: The slicing degree of a knot K is the smallest integer k such that K is k-slice (i.e., bounds a disk with self-intersection number –k) in #n(-CP)^2 for some n. In this talk, we establish bounds on the slicing degrees of knots using Rasmussen’s s-invariant, knot Floer homology, and singular instanton homology.

We also introduce sporks, defined as pairs (W, f) consisting of a contractible 4-manifold W and a boundary diffeomorphism f that extends smoothly inside. Sporks appear naturally in certain k-RBG links and produce knots with the same k-trace; although too blunt to produce exotic smooth structures, they are effective in detecting slicing degree in the examples we consider.

 April 24: Joshua Wang (Princeton)

Title: Some progress towards the stable Khovanov homology of torus knots

Abstract: The stable Khovanov homology of T(n, infinity) is a limit of the Khovanov homology groups of the torus links T(n,m) as m goes to infinity. A conjecture of Gorsky-Oblomkov-Rasmussen '12 states that the stable limit is the homology of a certain explicit Koszul complex. We explain some progress towards this result: there exists a spectral sequence converging to this stable group whose E_2 page is this explicit Koszul complex. Joint with William Ballinger, Eugene Gorsky, and Matthew Hogancamp. 

May 1: Richard Wentworth (Maryland)

Title: Higgs bundles, isomonodromic leaves, and minimal surfaces

Abstract: I will discuss various aspects of the geometry of the joint moduli space and nonabelian Hodge correspondence for Higgs bundles on Riemann surfaces with varying complex structures.  Specifically, there are four objects that are related in a surprising way: the isomonodromic distribution, the degeneracy of the hermitian pairing arising from the Atiyah-Bott-Goldman form, the "Kodaira-Spencer" form, and the energy functional for equivariant harmonic maps. I will show how this leads to the existence of pseudo-Kaehler metrics for certain moduli spaces of minimal surfaces, recovering and extending several recent constructions of various authors. This work is part of a collaboration with Brian Collier and Jeremy Toulisse.