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. |
| 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. |