The SGGTC seminar meets on Fridays in Math 417, at 10:45 am unless noted otherwise.

Previous semesters: 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 21, 2011

Organizational Meeting

with Geometric Topology Seminar

#### January 21, 2011

Jen Hom, "Concordance and the knot Floer complex"

**Abstract:** We will use the knot Floer complex, in particular the
invariant epsilon, to define a new smooth concordance homomorphism.
Applications include a formula for tau of iterated cables, better
bounds (in many cases) on the 4-ball genus than tau alone, and a new
infinite family of smoothly independent topologically slice knots. We
will also discuss various algebraic properties of this new
homomorphism.

#### January 28, 2011

Penka Georgieva, "Open Gromov-Witten disk invariants"

**Abstract:** In the presence of an anti-symplectic involution with a non-empty
fixed locus on a symplectic manifold M, open Gromov-Witten disk invariants
were defined by Cho and Solomon when the dimension of M is less than or
equal to 6. I will describe a generalization to higher dimension under
some technical conditions. Time permitting, I will discuss a connection to
real algebraic geometry and an approach to relax the conditions.

#### February 4, 2011

Pierre Schapira, "Sheaf quantization of Hamiltonian isotopies and applications"

**Abstract:** Recently Tamarkin presented a new approach to
symplectic topology based on the microlocal theory of sheaves. For
that purpose he had to adapt this theory which relies on the
homogeneous symplectic structure to the non homogeneous
case. Here, we remain in the homogeneous symplectic setting and
prove various results of non displaceability, including the
conservation of Morse inequalities as well as some results specific to
positive isotopies. The main tool is a theorem which asserts that any
Hamiltonian isotopy admits a unique sheaf quantization.

#### February 11, 2011

Chris Woodward, "Behavior of Gromov-Witten invariants under birational transformations of git type"

**Abstract:** Various authors (Ruan etc.) have noticed that a
functoriality property
for Gromov-Witten invariants under certain birational transformations
seems to hold.
I will describe a formula for the change in Gromov-Witten invariants under a
birational transformation of git type, that is, induced by change of
git polarization/shift of moment map;
the wall-crossing terms are a sum of { — gauged Gromov-Witten invariants}.
In the special case of a Calabi-Yau flop, the graph (or descendent)
potentials are almost everywhere equal in the
quantum parameter. (Many special cases of these results were already
known; this is part of a joint
project with E. Gonzalez.)

#### February 25, 2011

James Pascaleff, "Floer cohomology in the mirror of CP^2 relative to a conic and a line"

**Abstract:** In the Strominger-Yau-Zaslow description of mirror symmetry, singularities of the torus fibration lead to difficulties in the construction of mirror spaces and the computation of algebraic structures associated to these spaces. We will discuss one such algebraic structure, the Floer cohomology of Lagrangian sections of the torus fibration, in a space with a simple type of singularity, the Landau-Ginzburg mirror of CP^2 relative to a conic and a line, along with some natural generalizations. Of particular interest will be the canonical basis for the Floer cohomology groups (and hence of sheaf cohomology groups on CP^2) that our construction gives rise to.

#### March 4, 2011

Bulent Tosun, "Legendrian and transverse knots in cabled knot types"

**Abstract:** In this talk we will exhibit many new phenomena in the structure of Legendrian and transverse knots by giving a complete classification of all cables of the positive torus knots. We will also provide two structural theorems to ensure when cable of a Legendrian simple knot type is also Legendrian simple. Part of the results are joint work with John Etnyre and Douglas LaFountain

#### March 11, 2011

Thomas Kragh, "Fibration properties of symplectic homology and the Nearby Lagrangian conjecture"

**Abstract:** I will start by describing finite reductions of the action
integral and the
geometric intuition behind them. I will then use this on an exact
Lagrangian L in the
cotangent bundle of N, and describe why the symplectic homology associated
to the
cotangent bundle of L can be viewed as fibrant over N. Considering
products this yields
the result that the map L to N is a homology equivalence after passing to
a finite
covering space lift of N.

#### March 18, 2011

Spring break. No seminar.

#### March 25, 2011

Alexandra Popa, "On Mirror Formulas in Open and Closed Gromov-Witten Theory"

**Abstract:** In 2007 J. Walcher defined annulus Gromov-Witten invariants for some Calabi-Yau threefolds endowed with an anti-holomorphic involution as sums over fixed loci graphs of rational expressions in torus weights. He predicted these sums to be rational numbers and conjectured an explicit formula. This formula indeed holds for the quintic and bicubic threefolds (the only projective complete intersection Calabi-Yau threefolds for which these invariants are nonzero). I will explain Walcher’s definition, the key role played by an explicit mirror formula for two point closed Gromov-Witten invariants in the proof of his annulus formula, and ideas behind the proof of the two point formula. This is joint work with Aleksey Zinger.

#### April 1, 2011

Yanki Lekili, "Fukaya categories of the torus and Dehn surgery"

**Abstract:** In joint work with Tim Perutz, we extend the Heegaard Floer theory of
Ozsvath-Szabo to compact 3-manifolds with two boundary components. In
the particular case of 3-manifolds bounding the 2-sphere and the
2-torus, the simplest version of this extension takes the form of an
A-infinity module over the Fukaya category of a once punctured torus.
After giving an overview of this extension, I will show that the
A-inﬁnity structures on the graded algebra A underlying the Fukaya
category of the punctured 2-torus are governed by just two parameters,
extracted from the Hochschild cohomology of A. Finally, I will discuss
that the dg-categories of sheaves on the Weierstrass family of elliptic
curves yield a way to realize all such A-infinity structures. This pins
down a complete description of the Fukaya A-infinity algebra of the
punctured torus, which prove to be non-formal.

#### April 8, 2011, 9:30 am

Mohammed Abouzaid, "On Homological Mirror Symmetry for Toric varieties"

**Abstract:** I will explain a criterion involving Quantum cohomology, used to detect whether a collection of Lagrangians split-generate the Fukaya category of a symplectic manifold. We will then see that this criterion applies to toric manifolds, and shows that there is always a finite collection of fibres of the moment map which generate the Fukaya category in these examples. Time permitting, I will explain how to derive a proof of the Homological Mirror Symmetry conjecture in this case. This is joint work with Fukaya, Oh, Ohta, and Ono.

#### April 8, 2011, 10:45 am

Dan Rutherford, "A combinatorial Legendrian knot DGA from generating families"

**Abstract:** This is joint work with Brad Henry. A generating family for a
Legendrian knot $L$ in standard contact $\mathbb{R}^3$ is a family of
functions $f_x$ whose critical values coincide with the front projection
of $L$. Pushkar introduced combinatorial analogs of generating families
which have become known as Morse complex sequences. In this talk, I will
describe how to
associate a differential graded algebra (DGA) to a Legendrian knot with
chosen Morse complex sequence. In addition, I will discuss the geometric
motivation from generating families and the relationship with the
Chekanov-Eliashberg invariant.

#### April 15, 2011

Frol Zapolsky, "On the Hofer geometry on the space of Lagrangians"

**Abstract:** Consider the set of Lagrangian submanifolds of a symplectic manifold obtained by applying the elements of the Hamiltonian group to a fixed Lagrangian. This space inherits a quotient metric from the Hofer metric on the Hamiltonian group. Very little is known about this metric space in general and I'll present some new results shedding light on it. This is a work in progress, joint with Misha Khanevsky.

#### April 22, 2011, 9:30 am

Chris Schommer-Pries, "The Structure of Fusion Categories via Topological Quantum Field Theories"

**Abstract:** Fusion categories arise in several areas of mathematics and physics -
conformal field theory, operator algebras, representation theory of
quantum groups, and others. They have a rich an fascinating structure.
In this talk we will explain recent work, joint with Christopher
Douglas and Noah Snyder, which ties this structure to the structure of
3-dimensional topological quantum field theories. In particular we
show that fusion categories are fully-dualizable objects in a certain
natural 3-category and identify the induced O(3)-action on the `space'
of fusion categories, as predicted by the cobordism hypothesis. In
light of Hopkins' and Lurie's work on the cobordism hypothesis, this
provides a fully local 3D TQFT for arbitrary fusion categories.
Moreover by understanding various homotopy fixed point spaces, we will
uncover how many familiar structures from the theory of fusion
categories are given a natural explanation from the point of view of
3D TQFTs.

#### April 22, 2011, 10:45 am

Yongbin Ruan, "Gromov-Witten theory of elliptic orbifold P^1 and quasi-modular form"

**Abstract:** Gromov-Witten invariants counts the number
of pseudo-holomorphic curves. One often write them
in terms of generating functions. Occasionally, it
posses some very beautiful properties such as being
a quasi-modular form. In the talk, we will explain
this phenomenon for elliptic orbifold P^1. This is
a joint work with Milanov, Krawitz and Shen.

#### April 29, 2011

Tye Lidman, "Heegaard Floer Homology and Triple Cup Products"

**Abstract:** We use the recent link surgery formula of Manolescu and Ozsv\'ath as well as the theory of surgery equivalence of three-manifolds due to Cochran, Gerges, and Orr to relate Heegaard Floer homology to the cup product structure for a closed, oriented three-manifold. In particular, we give a complete calculation of the infinity flavor of Heegaard Floer homology for torsion Spin$^c$ structures, which is related to the analogous computation for monopole Floer homology with rational coefficients.

#### May 13, 2011

David Treumann, "Constructible sheaves, plumbings, and homological mirror symmetry"

**Abstract:** The talk will be based on joint work with Nicolo Sibilla and Eric
Zaslow. Kontsevich proposed in 2009 that the Fukaya category of an
exact symplectic manifold should be computable locally on a Lagrangian
skeleton. I will discuss the compatibility of this idea with
Kashiwara and Schapira's microlocal sheaf theory and results of Nadler
and Zaslow. Microlocal sheaf theory allows us to associate a category
(the "constructible plumbing model" or CPM) in a combinatorial way to
a "formal skeleton." Conjecturally CPM is equivalent to the Fukaya
category of a suitable symplectic neighborhood. Formal skeleta arise
from a T-duality process at the large complex structure limit X_0 of a
projective hypersurface X, and we can prove that the category of
perfect complexes on X_0 is equivalent to CPM of the associated
skeleton.

# Other relevant information

## Conferences

**The Versatility of Integrability**

A Conference on Integrable systems in algebra, geometry, and physics: celebrating Igor Krichever's 60th Birthday

Columbia University

May 4 – 7, 2011

**Workshop on Sheaf-Theoretic Methods in Symplectic TopologyInstitute for Advanced Study**

May 9 – 12, 2011

**Equivariant Quantum Cohomology, Mirror Symmetry, and Symplectic Geometry WorkshopSimons Center for Geometry and Physics**

May 16 – 20, 2011

**Homological Invariants in Low-Dimensional Topology WorkshopSimons Center for Geometry and Physics**

June 13 – 18, 2011

**Moduli Spaces and Moduli Stacks**

A conference on moduli of curves, stable maps, vector bundles, coherent sheaves, and the like, and on recent developments in the theory of algebraic stacks.

Columbia University

May 23 – 27, 2011

**Geometric and Algebraic Structures in Mathematics**

A conference in honor of Dennis Sullivan

Stony Brook

May 26 – June 4, 2011

**Walterfest!**

Faces of Geometry: 3-Manifolds, Groups, and Singularities

A Conference in Honor of Walter Neumann

Columbia University

June 6–10, 2011

## Other area seminars

- Columbia Geometric Topology Seminar
- Columbia Algebraic Geometry Seminar
- Eilenberg
lecture series at Columbia:

Claude Viterbo on "Symplectic topology from insights to interactions" - Princeton Topology Seminar

## Our e-mail list.

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. You can subscribe directly via Google Groups or by contacting R. Lipshitz.