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

Previous semesters: 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

#### September 14, 2012 at 10:45 am

Adam Knapp, "Cotangent bundles of open 4-manifolds"

**Abstract:** Using results of Eliashberg and Cieliebak, I show that: if
X1 and X2 are two homeomorphic open 4-manifolds, then their cotangent
bundles are symplectomorphic. As a corollary, all exotic R4s smoothly
embed in the standard symplectic R8 as Lagrangian submanifolds.

#### September 21, 2012 at 10:45 am

Olga Plamenevskaya, "Towards flexibility for higher-dimensional contact manifolds"

**Abstract:** By a classical result of Eliashberg, contact manifolds in dimension 3 come in two flavors:
tight (rigid) and overtwisted (flexible). Characterized by presence of an "overtwisted disk", the overtwisted
contact structures form a class where isotopy and homotopy classifications are equivalent.

In higher dimensions, a class of flexible contact structures is yet to be found. However, some attempts to
generalize the notion of an overtwisted disk have been made. One such object is a "plastikstufe" introduced by
Niederkruger following some ideas of Gromov. We show that under certain conditions, non-isotopic contact structures
become isotopic after connect-summing with a contact sphere containing a plastikstufe. This is a small step towards
finding flexibility in higher dimensions. (Joint with E. Murphy, K. Niederkruger, and A. Stipsicz.)

#### September 28, 2012 at 10:45 am

Strom Borman, "The size of a Weinstein neighborhood of a Lagrangian"

**Abstract:** The width of a Lagrangian is the largest capacity of ball that can be symplectically embedded into the
ambient manifold such that the ball intersects the Lagrangian exactly along the real part of the ball. This notion provides
a quantitative answer to the question "What is the maximal size of a Weinstein neighborhood for the Lagrangian?" In this talk
I will present a wrapped Floer cohomology construction that can upper bound a Lagrangian's width in terms of its displacement energy.
This is joint work in progress with Mark McLean.)

#### October 5, 2012 at 10:45 am

Ko Honda, "HF=ECH via open book decompositions"

**Abstract:** The goal of this talk is to sketch a proof of the
equivalence of Heegaard Floer homology (due to Ozsvath-Szabo) and
embedded contact homology (due to Hutchings). This is joint work with
Vincent Colin and Paolo Ghiggini.

#### October 12, 2012 at 10:45 am

Jiuzu Hong, "Polynomial functors and categorifications"

**Abstract:** The theory of symmetric functions is very classical and fundamental in
mathematics. The space B of symmetric functions admits many
symmetries. For example, there exists representation structures of
type A affine Lie algebra and Heisenberg algebra on the space B.
Usually they are called Fock space representation. The theory of polynomial functors is also very fundamental in
mathematics. It is closely related to general linear groups. The
category P of polynomial functors is a natural categorification of the
space B. There exists natural categorifications of Fock space
representations of affine Lie algebra and Heisenberg algebra on the
category P. Schur-Weyl duality relates polynomial representation of general linear
groups and representation of symmetric groups. The Schur-Weyl duality
functor can be enriched to be a morphism of various categorifications
from the cateogry P to the category of representations of all
symmetric groups.

#### October 26, 2012 at 10:45 am

Vasily Dolgushev, "The Grothendieck-Teichmueller group in algebra, geometry, and physics"

**Abstract:** Inspired by Grothendieck's lego-game, Vladimir Drinfeld introduced, in
1990, the Grothendieck-Teichmueller group GRT.
This group has interesting links to the absolute Galois group of
rationals, moduli of algebraic curves, solutions of the
Kashiwara-Vergne problem, and theory of motives. My talk will be
devoted to manifestations of the group GRT in the study of extended
moduli of algebraic varieties and in quantum theory. If time will
permit, I will discuss potential consequences of recent results
related to GRT for physics.

#### November 2, 2012 at 10:45 am

Yu-Shen Lin, "Discs Counting on Elliptic K3 Surfaces and Wall-Crossing"

**Abstract:** Strominger-Yau-Zaslow conjecture suggests that the Ricci-flat metric on Calabi-Yau manifolds might be
related to holomorphic discs. In this talk, I will define a new open Gromov-Witten invariants on elliptic K3 surfaces.
The new invariant satisfies certain wall-crossing formula and multiple cover formula. I will also establish a
tropical-holomorphic correspondence. Moreover, this invariant is expected to be equivalent to the generalized
Donaldson-Thomas invariants in the hyperK\"ahler metric constructed by Gaiotto-Moore-Neitzke.

#### November 9, 2012 at 10:45 am

Inanc Baykur, "Topological complexity of symplectic 4-manifolds and Stein fillings"

**Abstract:** Following the ground-breaking works of Donaldson and Giroux,
Lefschetz pencils and open books have become central tools in the
study of symplectic 4-manifolds and contact 3-manifolds. An open
question at the heart of this relationship is whether or not there
exists an a priori bound on the topological complexity of a symplectic
4-manifold, coming from the genus of a compatible Lefschetz pencil on
it, and a similar question inquires if there is such a bound on any
Stein filling of a fixed contact 3-manifold, coming from the genus of
a compatible open book. We will present our solutions to both
questions, making heroic use of positive factorizations in surface
mapping class groups of various flavors. This is joint work with J.
Van Horn-Morris.

#### November 16, 2012 at 11:00 am

Yanki Lekili, "An arithmetic refinement of homological mirror symmetry for the 2-torus"

**Abstract:** We establish a derived equivalence of the Fukaya category of the 2-torus, relative to a basepoint, with the category of perfect
complexes on the Tate curve over Z[[q]]. It specializes to an equivalence, over Z, of the Fukaya category of the punctured torus with perfect complexes
on the nodal Weierstrass curve y^2+xy=x^3, and, over the punctured disc Z((q)), to an integral refinement of the known statement of homological mirror
symmetry for the 2-torus. This is joint work with Tim Perutz.

#### November 16, 2012 at 1:20 pm

Tye Lidman, "Left-orderability and Heegaard Floer homology"

**Abstract:** We will consider the seemingly arbitrary question of when a three-manifold group can be given a left-invariant order. We will see
that left-orderability is in fact related to other topological features of a three-manifold, especially its Heegaard Floer homology groups. Intuition
from this correspondence will be used to study both of these structures.

#### November 30, 2012 at 10:45 am

Florent Schaffhauser, "The Narasimhan-Seshadri correspondence for vector bundles on a smooth projective curve defined over the field of real numbers"

**Abstract:** The Narasimhan-Seshadri correspondence is a bijection between S-equivalence classes of semistable holomorphic vector bundles on a
compact Riemann surface and conjugacy classes of projective unitary representations of the fundamental group of the surface. In this talk, we
assume that the Riemann surface X under consideration is endowed with an anti-holomorphic involution s admitting fixed points and we study
conjugacy classes of representations of the orbifold fundamental group of X/s into semi-direct products of projective unitary groups by the group
Z/2Z. We show that these are in bijective correspondence with S-equivalence classes of semistable real and quaternionic vector bundles over (X,s).
The heart of the proof is a gauge-theoretic construction of moduli spaces of such bundles.

#### December 7, 2012 at 10:45 am

Nathan Sunukjian, "The relationship between surgery and surface concordance in 4-manifolds"

**Abstract:** Understanding embedded surfaces in 4-manifolds is a starting point to understanding smooth structures on 4-manifolds. One reason
for this is because by performing some sort of surgery on a surface, it is sometimes possible to change the smooth structure of the 4-manifold without
altering its homeomorphism class. A considerable hurdle to this approach is that it can be very difficult to find appropriate surfaces to perform the
surgery on. Surface concordance is one way of trying to organize the surfaces in a 4-manifold. In this talk we will, (1) compute the concordance group
for a class of surfaces, and (2) explain when surgery on concordant surfaces gives the same result. As a further application, we will classify up to
isotopy the surfaces in a given homology class that have simply connected complement.

#### December 14, 2012 at 10:45 am

Mohammad Tehrani, "Counting real curves in symplectic manifolds"

**Abstract:** There are various kinds of $J$-holomorphic curves in a symplectic manifold invariant under an antisymplectic involution (called real
curves). In genus 0, there are two: those that have a fixed point locus and those that do not. The former are described by moduli spaces of $J$-holomorphic
disks, which are well studied in the literature. In this talk, we first study moduli spaces describing the latter and then combine the two types of moduli
spaces to get a well-defined theory of counting real curves of genus 0. We show that these invariants are the same for the two types of involutions on
$\P^{2n-1}$. We then discuss the higher genus case, that how the ideas from genus 0 can help to define real invariants of higher genus.

# Other relevant information

## Conferences

## Other area seminars

- Columbia Geometric Topology Seminar
- Columbia Algebraic Geometry Seminar
- Eilenberg lecture series
- 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.