Symplectic Geometry, Gauge Theory, and Low-Dimensional Topology Seminar: Spring 2023

This semester, Artem Kotelskiy, Donghao Wang, Langte Ma, Olga Plamenevskaya, and myself are organizing a seminar where we invite people working in low-dimensional topology, gauge theory, and symplectic geometry to talk about their work. We meet Thursdays, 1-2:30 pm in Math Tower, 4-130.


Date Speaker Title Abstract
Jan 26 Ian Montague (Brandeis) Seiberg-Witten Floer K-Theory and Cyclic Group Actions
Given a spin rational homology sphere equipped with a cyclic group action preserving the spin structure, I will introduce equivariant refinements of Manolescu's kappa invariant, derived from the equivariant K-theory of the Seiberg--Witten Floer spectrum. These invariants give rise to equivariant relative 10/8-ths type inequalities for equivariant spin cobordisms between rational homology spheres. I will explain how these inequalities provide applications to knot concordance, obstruct cyclic group actions on spin fillings, and give genus bounds for knots in punctured 4-manifolds. If time permits I will explain how these invariants are related to equivariant eta-invariants of the Dirac operator, and describe work-in-progress which provides explicit formulas for the S^1-equivariant eta-invariants on Seifert-fibered spaces.
Feb 2 Shaoyun Bai (SCGP)
Arnold conjecture over integers
We show that for any closed symplectic manifold, the number of 1-periodic orbits of any non-degenerate Hamiltonian is bounded from below by a version of total Betti number over Z, which takes account of torsions of all characteristics. The proof relies on an abstract perturbation scheme (FOP perturbations) which allows us to produce integral pseudo-cycles from moduli space of J-holomorphic curves, and a geometric regularization scheme for moduli space of Hamiltonian Floer trajectories generalizing the recent work of Abouzaid-McLean-Smith. I will survey these ideas and indicate potential future developments. This is joint work with Guangbo Xu.
Feb 9 Soham Chanda (Rutgers) Invariance of Floer cohomology under higher mutation via neck-stretching
Pascaleff-Tonkonog defined higher mutations for monotone toric fibers and proved an invariance of disc potential under a change of local system.  In this talk, I will define a local version of higher mutations for locally mutable Lagrangians. I will then use neck-stretching to show the invariance of Lagrangian intersection cohomology under a change of local system which agrees with the mutation formula in Pascaleff-Tonkonog.
Feb 16 Kyle Hayden (Rutgers-Newark) A handle-holding approach to Wall-type stabilization problems
In dimension four, the differences between continuous and differential topology are significant but fundamentally unstable. A longstanding question due to Wall aims to quantify this instability. In the first part of the talk, I will introduce Wall's stabilization problem and some its friends, highlight recent breakthroughs proven via Floer homology, and sketch a new "atomic" approach to these stabilization problems. As a proof of concept, in the second part of the talk, I will use this approach to produce new counterexamples to an analog of Wall's problem for knotted surfaces. The key obstruction comes from the "universal" version of Khovanov homology, and I will close with some speculative connections to Floer homology.
Feb 23 Daniel Pomerleano (UMass-Boston)
Singularities of the quantum connection on a Fano variety
The small quantum connection on a Fano variety is one of the simplest objects in enumerative geometry. Nevertheless, it is the subject of far-reaching conjectures known as the Dubrovin/Gamma conjectures. Traditionally, these conjectures are made for manifolds with semi-simple quantum cohomology or more generally for Fano manifolds whose quantum connection is of unramified exponential type at q=∞.

I will explain a program, joint with Paul Seidel, to show that this unramified exponential type property holds for all Fano manifolds M carrying a smooth anticanonical divisor D. The basic idea of our argument is to view these structures through the lens of a noncommutative Landau-Ginzburg model intrinsically attached to (M,D).
Mar 2 Jae Hee Lee (MIT) Quantum Steenrod Operations for the Local Rational Curve
The quantum cohomology of a symplectic manifold with coefficients in positive characteristic admit additional equivariant operations known as the quantum Steenrod operations, due to Fukaya and Wilkins. In all previously known examples, these operations were determined by the (small) quantum cup product and classical Steenrod operations. I will describe the first nontrivial computation where this is no longer the case, in the geometry of a local rational curve in a Calabi-Yau 3-fold. If time permits, I will explain some ongoing work on relating this computation with the equivariant quantum connection of Maulik-Okounkov and mirror symmetry.
Mar 9 Ipsita Datta (IAS) Lagrangian cobordisms between enriched knot diagrams
In this talk, we discuss Lagrangian cobordisms in R^4 between smooth knots. We define enriched knot diagrams of knots, which turns out to be a good framework to develop obstructions to the existence of Lagrangian cobordisms. We present some such obstructions arising from studying moduli spaces of holomorphic disks with boundaries on a Lagrangian tangle, which is an immersed Lagrangian surface we define. Time permitting we discuss difficulties in developing algebraic invariants or capacities that capture these obstructions.
Mar 16 (spring recess) No meeting

Mar 23 Zhenkun Li (Stanford) Instanton Floer homology and Heegaard diagrams
Instanton Floer homology was introduced by Floer in the 1980s and has become a power invariant for three manifolds and knots since then. It has led to many milestone results, such as the approval of Property P conjecture. Heegaard diagrams, on the other hand, is a combinatorial method to describe 3-manifolds. In principle, Heegaard diagrams determine 3-manifolds and hence determine their instanton Floer homology as well. However, no explicit relations between these two objects were known before. In this talk, for a 3-manifold Y, I will talk about how to extract some information about the instanton Floer homology of Y from the Heegaard diagrams of Y. Additionally, I will explain some of the applications and future directions of this work. This is a joint work with Baldwin and Ye.
Mar 24 Kevin Sackel (UMass-Amherst) Obstructions for exact submanifolds with symplectic applications
Suppose a closed oriented manifold comes with a fixed real cohomology class. This cohomology class singles out those submanifolds for which the restriction of the given cohomology class is trivial. We may ask in what integral homology classes we find such "exact submanifolds." We build an infinite sequence of obstructions which are readily computable by (finite-dimensional) linear algebra performed on the de Rham complex. As the impetus for this work arose out of (locally conformal) symplectic geometry, which we shall partially describe, our main applications are symplectic in nature. For example, the following symplectic manifolds admit no non-separating (a fortiori contact-type) hypersurfaces: Kähler manifolds, symplectically uniruled manifolds, and the Kodaira-Thurston manifold.
Mar 30 Dusa McDuff (Barnard)
Cuspidal Curves and Symplectic Embeddings
I will first explain the relation between resolution of cuspidal singularities of rational curves in CP^2 (the complex projective plane) and existence of exceptional divisors in blowups of CP^2; then explain their relation to the question of when you can symplectically embed an ellipsoid into CP^2.  I will then discuss the relation between Orevkov's twist construction of cuspidal curves and the Fibonacci staircase for symplectic embeddings into CP^2, and (if there is time) into other Fano 4-folds.  If there is time I will discuss the construction of cuspidal curves  using methods from tropical geometry.  This is joint work with Kyler Siegel.
Apr 6 Tim Large (Columbia) Localization, Morava K-theory Gromov-Witten Invariants, and Multiple Covers
Gromov-Witten (GW) invariants as originally defined are rational numbers: for instance, Aspinwall-Morrison's multiple cover formula computes the contribution to the genus zero GW invariant of degree d covers of a rigid holomorphic sphere in a Calabi-Yau threefold to be 1/d^3. Abouzaid-McLean-Smith recently defined analogues of GW invariants with values in certain complex oriented cohomology theories, such as Morava K-theory; the advantage is that this can be done in finite characteristic. In this talk, I will explain how to compute GW invariants in complex oriented cohomology in the presence of a circle action, by using an extension of the Atiyah-Bott formula. As an application, I will revisit the multiple cover formula in characteristic p.
Apr 13 Beibei Liu (MIT) Heegaard Floer Homology and Plane Curves with Singularities
It is an interesting question to ask about the possible configuration of singular points occurring on general algebraic curves. In this talk, we use Heegaard Floer's homology to study the singularity links and give some obstructions for the configuration. The method works for any type of singularity link, though the difficult part is to compute the corresponding Heegaard Floer package. In the second part of the talk, we use link lattice homology to give an algorithm to compute the Heegaard Floer homology for all algebraic links. This is joint work with Borodzik and Zemke.
Apr 20 Hokuto Konno (University of Tokyo) Homological Instability for Moduli Spaces of 4-Manifolds
We prove that homological stability with respect to connected sums of S^2×S^2 fails for moduli spaces BDiff(X) of simply-connected closed 4-manifolds X. This makes a striking contrast with other dimensions: in all even dimensions except for 4, analogous homological stability for moduli spaces has been established by work of Harer and of Galatius and Randal-Williams. The proof of the above result is based on a characteristic class constructed using the Seiberg-Witten equations. This is joint work with Jianfeng Lin.
Apr 27 Vardan Oganesyan (UC-Santa Cruz) Symplectic Homotopy Theory
Inspired by motivic homotopy theory we define a new homotopy theory of symplectic manifolds. In particular, we define a new cohomology theory. This cohomology theory is functorial, homotopy invariant and has other interesting properties. As an application, using only our new theory we prove that C^n is not symplectomorphic to DxC^{n-1}, where D is the unit disk. This shows that our homotopy theory detects some nontrivial symplectic invariants.
May 4 Vinicius Ramos (IMPA) The Toda Lattice, Billiards and the Viterbo Conjecture
The Toda lattice is one of the earliest examples of non-linear completely integrable systems. Under a large deformation, the Hamiltonian flow can be seen to converge to a billiard flow in a simplex. In the 1970s, action-angle coordinates were computed for the standard system using a non-canonical transformation and some spectral theory. In this talk, I will explain how to adapt these coordinates to the situation to a large deformation and how this leads to new examples of symplectomorphisms of Lagrangian products with toric domains. In particular, we find a sequence of Lagrangian products whose symplectic systolic ratio is one and we prove that they are symplectic balls. This is joint work with Y. Ostrover and D. Sepe.

Symplectic Geometry, Gauge Theory, and Low-Dimensional Topology Seminar: Fall 2022


Date Speaker Title Abstract
Sept 8 Johan Asplund (Columbia) Simplicial descent for Chekanov-Eliashberg dg-algebras
The Chekanov-Eliashberg dg-algebra offers a way to compute wrapped Fukaya categories of Weinstein manifolds. In this talk we introduce a type of surgery decomposition of Weinstein manifolds we call simplicial decompositions. We will discuss the result that the Chekanov-Eliashberg dg-algebra of the top attaching spheres of a Weinstein manifold satisfies a descent (cosheaf) property with respect to a simplicial decomposition. We will also touch on the subject of defining the Chekanov-Eliashberg dg-algebra for singular Legendrians and applications to knot contact homology for tangles. This talk is partially based on joint work with Tobias Ekholm.
Sept 15 Sebastian Haney (Columbia) Hyperbolic Lagrangians in Calabi-Yau threefolds
The existence of Lagrangian submanifolds of the quintic Calabi-Yau threefold which admit a hyperbolic metric was predicted by Jockers, Morrison, and Walcher using mirror symmetry. They conjectured that the limiting value of a certain normal function on the mirror quintic should correspond to the hyperbolic volume of a Lagrangian in the quintic. I will discuss some work in progress constructing (singular) hyperbolic Lagrangians using techniques from tropical geometry. As part of the construction, I will explain how to construct Lagrangian lifts of certain nonsmooth tropical varieties.
Sept 22 Yash Deshmukh (Columbia) A Homotopical description of Deligne-Mumford compactifications
I will present a description of the Deligne-Mumford compactification of moduli spaces of curves (of all genera) as arising from moduli spaces of framed curves by homotopy trivializing certain circle actions in an appropriate sense. I will sketch how such a description is relevant to the problem of relating GW invariants (in all genera) with the Fukaya categories. Finally, I will indicate how our result relates to other statements available in the literature.
Sept 29 Boyu Zhang (UMaryland) The existence of irreducible SU(2) representations of link groups
Representations of 3-manifold groups into groups such as SU(2) and SL(2,C) have been an active field of study for decades. Many topological invariants are defined by considering these representations, such as the Casson invariant, the Casson-Lin invariant, and the A polynomial. In 2010, Kronheimer-Mrowka showed that the fundamental group of every non-trivial knot in S^3 admits an irreducible representation in SU(2), which answered a conjecture of Cooper. In this talk, I will present a result that generalizes Kronheimer-Mrowka's theorem to the case of links. We show that for every link L that is not the unknot, the Hopf link, or a connected sum of Hopf links, its fundamental group admits an irreducible SU(2) representation such that the image of every meridian is traceless. This is joint work with Yi Xie.
Oct 6 Gage Martin (MIT) Annular links, double branched covers, and annular Khovanov homology
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.
Oct 13 Marco Marengon (Alfréd Rényi) Sliceness and the rank of some knot homologies
A popular notion in knot theory is that of sliceness: a knot in S^3 is called slice if it bounds a smooth disc in B^4. There are various reasons why this concept is so fundamental: for example, sliceness is at the core of a trendy strategy proposed to disprove the smooth 4-dimensional Poincaré conjecture, and it has recently been shown that a generalisation of this concept to 4-manifolds other than B^4 can detect exotic pairs, i.e. 4-manifolds that are homeomorphic but not diffeomorphic to each other.
We study whether sliceness poses any restrictions on the rank of certain homology theories associated to knots. We prove some results and formulate some curious conjectures. This is joint work with Hockenhull and Willis, and partly also with Dunfield and Gong.
Oct 20 Fan Ye (Harvard) Reinterpretation of knot surgery formula via Fukaya category of punctured torus
Ozsvath-Szabo and Rasmussen introduced knot Floer chain complex CFK of a knot K inside a closed 3-manifold Y and provided formulae computing Heegaard Floer homology of the surgered manifold Y_r(K) with any surgery coefficient r in Q. In this talk, I will reinterpret the formulae and many useful exact triangles in Heegaard Floer theory via the immersed curve invariant introduced by Hanselman-Rasmussen-Watson. The immersed curve invariant can be regarded as an object in the Fukaya category of the punctured torus (the boundary of Y\N(K) with a basepoint) and I make use of the mapping cone construction and the octahedral lemma in the Fukaya category. As a result, we prove the knot surgery formulae for instanton Floer homology, which has implied many applications on the SU(2) representations of the fundamental group, e.g., the 3-surgered manifold on any nontrivial knot in S^3 has an irreducible representation to SU(2). This is a joint work with Zhenkun Li.
Oct 27 Nick Wilkins (MIT) S^1 localisation by pseudocycles and applications to equivariant Borman-Sheridan classes
In this talk, we discuss a new way to view Atiyah-Bott S^1-localisation, comparing an S^1-manifold and its fixed-point-set by constructing a specific pseudocycle bordism. Applying this to an S^1-equivariant moduli space of holomorphic curves is not immediate, as genericity prevents such spaces from being a homotopy quotient, but we demonstrate how one lifts to moduli spaces in the example of Borman-Sheridan classes.
Nov 3 Peter Feller (ETH Zurich)
On the length of knots on a Heegaard surface of a 3-manifold
3-manifold theory has expanded its tool box in recent decades: topological, (Floer and quantum) homological, and geometrical methods all have been employed with success. However, often the relation between these different approaches remains mysterious.

In this talk we explore connections between the topology and the geometry of 3-manifolds by using Heegaard-splittings (topology) of a 3-manifold to describe hyperbolic structures (geometry) on it. More concretely, for a knot K that lies on a Heegaard surface of a closed oriented connected 3-manifold M, we describe a sufficient condition for M to carry a hyperbolic structure. Furthermore, whenever our criterion applies, we determine the length of K up to a multiplicative constant.

Upshot of our approach: there is NO Ricci-flow machine running in the background. Instead, the motor behind what we do is an effective version of Thurston's hyperbolic Dehn surgery (pioneered by Hodgson-Kerckhoff). Applications include a Ricci-flow free proof of Mather's result that random 3-manifolds (in the sense of Dunfield-Thurston) are hyperbolic, and bounds on the diameter and injectivity radius of a random 3-manifold.
Nov 10 Lea Kenigsberg (Columbia) Coproduct Structures: a Tale of Two Outputs
I will tell the elusive story of coproduct structures in Floer theory and string topology, and explain why we care about them. I will then define a new coproduct structure on the symplectic cohomology of Liouville manifolds and compute it in an example to show that this structure is not trivial. This is based on my thesis work, in progress.
Nov 17 William Ballinger (Princeton) The SO(8) invariant of colored trivalent graphs and its categorification
After discussing some skein theory for the Reshetikhin-Turaev invariant of knotted trivalent graphs coming from the vector and spinor representations of SO(8), I will describe a homology theory that appears to categorify it. When the knotted trivalent graph is just a link with a single color, this Reshetikhin-Turaev invariant is a specialization of the Kauffman polynomial and the homology theory reduces to a special case of one constructed and conjectured to be a link invariant by Khovanov and Rozansky, showing that in this case their theory is in fact an invariant.
Nov 24 (Thanksgiving) No meeting

Dec 1 Mike Miller (Columbia) Instantons and Dehn surgery
There exist many integer homology spheres which are not Dehn surgery on any knot, the first examples being due to Gordon--Luecke. A longstanding question asks whether there exist integer homology spheres Y_n which are not Dehn surgery on any link of fewer than n components; no examples for n > 2 have appeared in the literature.

I will discuss forthcoming work with Ali Daemi which establishes many examples of such Y_n, including hyperbolic examples. The argument relies on formal properties of Froyshov's homology cobordism invariant q_3, in particular that q_3(Y) gives information about the intersection form of any 4-manifold W (whose homology has no 2-torsion) with boundary Y, even indefinite W. The key technical point is the definition of an algebraic object designed to support mod-2 relative invariants for manifolds with b^+(W) > 0.