The SGGTC seminar meets on Fridays in Math 520, at 10:45 am unless noted otherwise (in red).

Previous semesters: Fall 2013, Spring 2013, Fall 2012, Spring 2012, Fall 2011, Spring 2011, Fall 2010, Spring 2010, Fall 2009, Spring 2009, Fall 2008, Spring 2008, Fall 2007.

Other area seminars and conferences. Our e-mail list.

# Abstracts

#### January 24, 2014 at 9:30 am

Samuel Lisi

**Abstract:** I will discuss a symplectic embedding problem for polydisks in
dimension 4. Symplectic embeddings are a key phenomenon of symplectic rigidity. In the
case I consider, the obstruction for embedding the polydisk comes from a certain Lagrangian
torus and uses the theory of J-holomorphic foliations in dimension 4. This is joint work
with Richard Hind.

#### January 24, 2014 at 10:45 am

Kai Cieliebak

**Abstract:** Rationally and polynomially convex domains in $\C^n$ are
fundamental objects of study in the theory of functions of several
complex variables. After defining and illustrating these notions, I will
explain joint work with Y.Eliashberg giving a complete characterization
of the possible topologies of such domains in complex dimension at least
three. The proofs are based on recent progress in symplectic topology,
most notably the h-principles for loose Legendrian knots and Lagrangian
caps.

#### January 24, 2014 at 2:00 pm

Sheel Ganatra

**Abstract:** We show that the natural open-closed map from Hochschild
homology of the Fukaya category to symplectic cohomology intertwines relevant
circle actions, at a suitable chain level. As a consequence, we deduce that
there are natural maps from various cyclic homology theories to the
corresponding S^1-equivariant symplectic cohomology theories, intertwining
the usual Gysin exact sequences, and that all of these maps are isomorphisms
whenever the non-equivariant map is. This is work in progress.

#### January 31, 2014 at 10:45 am

Luis Diogo

**Abstract:** Symplectic homology is a very useful tool in symplectic topology,
but it can be hard to compute explicitly. We will review the definition of this invariant
and some of its features. Then, we describe a procedure for computing symplectic homology
in terms of certain Gromov-Witten invariants. This method is applicable to a class of
manifolds that are obtained by removing, from a closed symplectic manifold, a symplectic
hypersurface of codimension 2. This is joint work with Samuel Lisi.

#### February 7, 2014 at 10:45 am

Faramarz Vafaee

**Abstract:** Heegaard Floer theory consists of a set of invariants of three-
and four-dimensional manifolds. Three-manifolds with the simplest Heegaard Floer
invariants are called L-spaces and the name stems from the fact that lens spaces
are L-spaces. The primary focus of this talk will be on the question of which
knots in the three-sphere admit L-space surgeries. We will also discuss about
possible characterizations of L-spaces that do not reference Heegaard Floer homology.

#### Feb 21, 2014 at 10:45 am

Anton Zeitlin

**Abstract:** I will talk about the homotopy Gerstenhaber algebras describing the
symmetries of 2d first order sigma models and their relation to the
structure of Einstein equations.

#### February 28, 2014 at 10:45 am

Michael Abel

**Abstract:** In 2006 Khovanov gave a construction of HOMFLY-PT homology
using Rouquier's braid group action on the homotopy category of Soergel bimodules.
Soergel bimodules have natural filtrations in terms of bimodules representing virtual
crossings. We will describe how to represent Soergel bimodules as mapping cones of
virtual crossings in a derived category of graded bimodules. We will then describe
how this mapping cone presentation fits into the construction of HOMFLY-PT homology.
The mapping cone presentation gives a filtration on HOMFLY-PT homology which is
preserved by Reidemeister moves I and II and conjecturally by Reidemeister III.
We will finish by presenting computations and evidence for the conjectural theory.

#### March 7, 2014 at 10:45 am

John Pardon

**Abstract:** An implicit atlas on a (moduli) space consists of certain
auxiliary (moduli) spaces satisfying a precise set of axioms. We will
summarize the construction of implicit atlases on moduli spaces of
J-holomorphic curves, under the assumption of a precise "strong gluing"
theorem. We will also describe an algebraic "theory of virtual
fundamental cycles" (which does not use perturbation) in the abstract
setting of spaces equipped with implicit atlases. This "VFC package" is
sufficient to define Floer-type homology theories from a collection of
(moduli) spaces equipped with a compatible system of implicit atlases.

#### March 14, 2014 at 10:45 am

Chris Cornwell

**Abstract:** In the knot contact homology of a knot K there are augmentations
that may be associated to a flat connection on the complement of K. We show that all
augmentations arise this way. As a consequence, a polynomial invariant of K called
the augmentation polynomial represents a generalization of the classical A-polynomial.
A recent conjecture, similar to the AJ conjecture concerning the A-polynomial,
relates a 3-variable augmentation polynomial to colored HOMFLY-PT polynomials.
Our results can be seen as motivation for this conjecture having an augmentation
polynomial in place of the A-polynomial.

#### April 4, 2014 at 10:45 am

Yin Tian

**Abstract:** Khovanov gave a graphical calculus for a categorification of
a Heisenberg algebra. In this talk we describe a diagrammatic
categorification of a Clifford algebra and its Fock space
representation via differential graded categories.
The motivation is from the {\em contact categories}
$\mathcal{C}(\Sigma)$ of infinite strips $\Sigma$ introduced by Honda
which describe contact structures on $\Sigma \times [0,1]$.

#### April 11, 2014 at 9:30 am

Denis Auroux

**Abstract:** We illustrate some recent progress on Kontsevich's homological
mirror symmetry conjecture by considering a specific example, the pair of pants
(i.e., the sphere minus three points). This will serve as a pretext to introduce
notions such as wrapped or fiberwise wrapped Fukaya categories, explain the statement
of the homological mirror symmetry conjecture in this setting, and discuss how it can
be verified explicitly. Despite the simplicity of the example we consider, it already
exhibits many of the features common to more complicated affine or general type
examples. (This is based partly on joint work with M. Abouzaid, A. Efimov, L. Katzarkov,
and D. Orlov, and partly on work in progress with M. Abouzaid).

#### April 11, 2014 at 10:45 am

David Krcatovich

**Abstract:** The set of knots up to a four-dimensional equivalence relation
can be given the structure of a group, called the (smooth) knot concordance group.
We will discuss how to compute concordance invariants using Heegaard Floer homology.
We will then introduce the idea of a "reduced" knot Floer complex, see how it can be
used to simplify computations, and give examples of how it can be helpful in
distinguishing knots which are not concordant.

#### April 18, 2014 at 10:45 am

Emmy Murphy

**Abstract:** The Lutz-Martinet theorem states that any 2-plane field on a
3-manifold is homotopic to a contact structure. This construction lead to
Eliashberg's definition of overtwisted contact manifolds, and in this
context the existence theorem of Lutz-Martinet can be extended to a
uniqueness result: any two overtwisted contact structures which are
homotopic as plane fields are in fact isotopic. We discuss a recent
extension of these results to contact manifolds of all dimensions. We will
focus on showing that any almost contact structure is homotopic to a
contact structure, and seeing how this leads to a new definition of
overtwistedness in high dimensions. As time allows we will discuss a proof
that a homotopy class of almost contact structures is realized by a unique
isotopy class of overtwisted contact structure. This project is joint work
with Borman and Eliashberg.

#### April 25, 2014 at 10:45 am

Anton Zeitlin

**Abstract:** I will talk about the construction of an analogue of the
continuous series for affine sl(2,R). The approach is based on the study
of correlation functions of the generators and renormalization
of the emerging divergencies.

#### May 2, 2014 at 10:45 am

Lenny Ng

**Abstract:** Recently Vivek Shende, David Treumann, and Eric Zaslow introduced
a category of constructible sheaves associated to a Legendrian knot. They conjectured
that this category is equivalent to a category studied by Bourgeois and Chantraine and
built from augmentations of Legendrian contact homology. In fact this conjecture is false,
but it can be fixed. I will introduce a new augmentation category for Legendrian knots
that we now know (as of last week) is equivalent to the STZ category, and describe some
of its properties (including comparing and contrasting with the Bourgeois-Chantraine category).
This reports on joint work in progress with some subset of {Shende, Treumann, Zaslow,
Dan Rutherford, Steven Sivek}.

#### May 9, 2014 at 10:45 am

Emma Carberry

**Abstract:** Constant mean curvature (CMC) tori in S ^ 3, R ^ 3 or H ^ 3 are in
bijective correspondence with spectral curve data, consisting of a hyperelliptic curve,
a line bundle on this curve and some additional data, which in particular determines the
relevant space form. This point of view is particularly relevant for considering moduli-space
questions, such as the prevalence of tori amongst CMC planes. I will address these periodicity
questions for the spherical and Euclidean cases, using Whitham deformations, which I will explain.

# Other relevant information

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Announcements for this seminar, as well as for related seminars and events, are sent to the "Floer Homology" e-mail list maintained via Google Groups.