| Feb 1
|
R. Inanc Baykur (Max Planck Institute): 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
symplectic4-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.
|
| Mar 29
|
Chung-Jun Tsai: Dirac spectral flow on contact 3-manifolds
|
| Apr 5
|
Chris Wendl (3 PM to 4 PM): Tight but nonfillable contact manifolds in all
dimensions
|
|
|
Matt Hedden (4 PM to 5 PM)
|
| Apr 19
|
Eli Grigsby (BC): CANCELLED
|
| Apr 26
|
Tye Lidman (UT Austin): Monopole Floer homology and covering spaces
|
| Sep 21
|
Ke Zhu (Harvard): Thin instantons in G2
manifolds and Seiberg-Witten invariants
For two nearby disjoint coassociative submanifolds C and C' in a G2
manifold, we construct thin instantons (associative submanifolds) with
boundaries lying on C and C' from regular J-holomorphic curves in C. We
explain their relationship with the Seiberg-Witten invariants for C. This
is a joint work with Naichung Conan Leung and Xiaowei Wang.
|
| Oct 5
|
Andy Wand (Harvard): Tightness and open book
decompositions
Abstract: A well known result of Giroux tells us that isotopy classes
of contact structures on a closed three manifold are in one to one
correspondence with stabilization classes of open book decompositions
of the manifold. We will introduce a stabilization-invariant property
of open books which corresponds to tightness of the corresponding
contact structure. We will mention applications to the classification
of contact 3-folds, and also to the question of whether tightness is
preserved under Legendrian surgery.
|
| Oct 12
|
Jae Choon Cha (POSTECH): Amenable L2-signatures and
cobordism of knots and 3-manifolds
Abstract: Recently, new amenable Cheeger-Gromov rho-invariants have been
introduced to study homology cobordism of 3-manifolds and concordance of
knots and links. This reveals new structures beyond those visible via
Cochran-Orr-Teichner type signatures. We introduce the key ideas of
amenable L2 methods and some applications. Also, if time permits, we
discuss the notion of symmetric Whitney tower cobordism which provides a
generalized framework for bordered 3-manifolds and link concordance.
Parts of the results in this talk are joint with Kent Orr.
|
| Oct 19
|
Michael Hutchings (UC Berkeley): Existence of two Reeb
orbits
Abstract: We show that every (possibly degenerate) contact form on a
closed three-manifold has at least two geometrically distinct Reeb orbits.
The proof uses a relation between contact volume and the asymptotics of
the amount of symplectic action needed to represent certain classes in
embedded contact homology. This is joint work with Dan Cristofaro-Gardiner
and Vinicius Gripp.
|
| Nov 2
|
Joshua Batson (MIT): A link splitting spectral
sequence in Khovanov Homology
Abstract: The Khovanov homology of a link L is a bigraded abelian group
Kh(L) with Poincaré polynomial P_L(q,t), such that P(q,-1) recovers
the Jones polynomial. The meaning of the group Kh(L) is topologically
obscure, since its definition is resolutely combinatorial, but a suite of
spectral sequences abutting to more manifestly three-dimensional homology
theories provide some traction. The spectral sequence to singular
instanton homology shows that P_L detects the unknot, and the one to
Heegaard Floer of the branched double cover shows that P_L detects the
two-component unlink. We construct a new spectral sequence from Kh(L) to
the tensor product $\otimes_i Kh(K_i)$ of the Khovanov homologies of the
components K_1,...,K_n of L. From this we show that P_L detects the
n-component unlink for any n, and construct a novel bound on the number of
between-component crossing changes required to split a link. This work is
inspired by a conjectured spectral sequence in the symplectic theory of
Seidel-Smith. Project joint with Cotton Seed.
|
| Nov 9
|
Rosa Sena-Dias (Instituto Superior Técnico): Non-compact scalar-flat Kähler toric metrics and their
energy
Abstract: I will talk about a concrete family of scalar-flat Kähler
toric metrics on non-compact toric surfaces. These generalize some well
know Ricci-flat metrics with toric symmetry: the so-called gravitational
instantons of Gibbons-Hawking and Kronheimer as well as the Taub-NUT
metric. After giving some background and introducing these metrics, I hope
to discuss the relation with some recent developments in Kähler
geometry.
|
| Dec 7
|
John Baldwin (Boston College):
A bordered monopole Floer theory
Abstract: I'll discuss work-in-progress toward constructing monopole Floer
theoretic invariants of bordered 3-manifolds. Roughly, our construction
associates an A-infinity algebra to a surface, an A-infinity module to a
bordered 3-manifold, and a map of A-infinity modules to a 4-dimensional
cobordism of bordered 3-manifolds. I'll focus on the topological and
algebraic aspects of our work and, in particular, will indicate how we
prove a pairing theorem relating the invariants of two bordered
3-manifolds with that of the manifold obtained by gluing the former
together along homeomorphic components of their boundaries. This is joint
work with Jon Bloom.
|
| Feb 10
|
Adam Levine (Brandeis):
Bordered Floer homology and splicing knot complements
Abstract: We use bordered Floer homology to study 3-manifolds obtained by
gluing together two knot complements (gluing meridian to longitude). If
the knots are non-trivial knots in S^3, we show that the Heegaard Floer
homology of the resulting manifold has rank greater than one. By extending
this approach to knots in arbitrary three manifolds, we hope to obtain a
new proof of Eftekhary's claimed result that a manifold whose Heegaard
Floer homology has rank one cannot contain an essential torus. This is
joint work in progress with Matt Hedden.
|
| Feb 17
|
John Pardon (Stanford):
Totally disconnected groups (not) acting on three-manifolds
Abstract: Hilbert's Fifth Problem asks whether every topological group
which is a manifold is in fact a (smooth!) Lie group; this was solved in
the affirmative by Gleason and Montgomery--Zippin. A stronger conjecture
is that a locally compact topological group which acts faithfully on a
manifold must be a Lie group. This is the Hilbert--Smith Conjecture,
which in full generality is still wide open. It is known, however (as a
corollary to the work of Gleason and Montgomery--Zippin) that it suffices
to rule out the case of the additive group of $p$-adic integers acting
faithfully on a manifold. I will present a solution in dimension three.
The proof uses tools from low-dimensional topology, for example
incompressible surfaces, minimal surfaces, and a property of the mapping
class group.
|
| Mar 2
|
Sushmita Venugopalan (Rutgers):
Yang-Mills heat flow on gauged holomorphic maps
Abstract: We study the gradient flow lines of a Yang-Mills-type
functional on the space of gauged holomorphic maps $\mathcal{H}(P,X)$,
where $P$ is a principal bundle on a Riemann surface $\Sigma$ and $X$
is a Kahler Hamiltonian $G$-manifold. For compact $\Sigma$, possibly
with boundary, we prove long time existence of the gradient flow. The
flow lines converge to critical points of the functional. So, there is
a stratification on $\mathcal{H}(P,X)$ that is invariant under the
action of the complexified gauge group.
Symplectic vortices are the zeros of the functional we study. When
$\Sigma$ has boundary, similar to Donaldson's result for the Hermitian
Yang-Mills equations, we show that there is only a single stratum -
any element of $\mathcal{H}(P,X)$ can be complex gauge transformed to
a symplectic vortex. This is a version of Mundet's Hitchin-Kobayashi
result on a surface with boundary.
|
| Mar 9
|
Cliff Taubes (Harvard): SL(2;C) connections with L^2 bounds on curvature
|
| Mar 30
|
Max Lipyanskiy (Columbia):
Gromov-Uhlenbeck Compactness
Abstract: We introduce an analytic framework that, in special circumstances, unites Yang-Mills
theory and the theory of pseudoholomorphic curves. As an application of these ideas, we discuss the
relation between instanton Floer homology and Lagrangian Floer homology of representation varieties.
|
| Apr 6
|
Nathan Dunfield (UIUC): Twisted Alexander polynomials of hyperbolic knots
Abstract: I will discuss a twisted Alexander polynomial naturally
associated to a hyperbolic knot in the 3-sphere via a lift of its
holonomy representation to SL(2, C). It is an unambiguous symmetric
Laurent polynomial whose coefficients lie in a number field coming
from the hyperbolic geometry. The polynomial can be defined as the
Reidmeister torsion of a certain acyclic chain complex, namely the
first homology of the knot exterior with coefficients twisted by the
holonomy representation tensored with the abelianization map. This
polynomial contains much topological information, for instance about
the simplest surface bounded by the knot. I will present
computations showing that for all 313,209 hyperbolic knots in S^3 with
at most 15 crossings it in fact gives perfect such information, in
contrast with a related polynomial coming from the adjoint
representation of SL(2, C) on it's Lie algebra. This is joint work
with Stefan Friedl and Nicholas Jackson
|
| Apr 13
|
Thomas Koberda (Harvard): Mapping class groups, homology and finite covers
of surfaces
|
| Apr 20
|
Ethan Street (Harvard): Towards an
Instanton Floer Homology for Tangles
Abstract: In this thesis defense, I will motivate and study the problem of
computing the cohomology rings of a class of moduli spaces of flat
connections on orbifold Riemann surfaces. I will then discuss the parallel
problem of computing the instanton Floer homology ring of a product link
in product 3-manifold. Complete results in the case of genus 0 will be
given. I will briefly discuss the application of these ideas to defining
of instanton Floer homology for a tangle in a sutured 3-manifold. Emphasis
will be on elementary ideas related to the moduli spaces rather than Floer
homology.
|
| Apr 27
|
Tatyana Kobylyatskaya (Harvard): Symplectic embeddings of
rational homology balls
Abstract: We will begin with a description of the rational homology balls
appearing in Fintushel and Stern's rational blow-down operation for smooth
4-manifolds, a generalization of the standard blow-down. We will then
discuss various smooth and symplectic embedding results of these rational
homology balls. Time permitting, we will also give some explicit
constructions of such symplectic embeddings.
|
| May 4
|
Cagatay Kutluhan (Harvard): Holonomy filtration and
knots
Abstract: Motivated by the construction of the isomorphism between
Heegaard Floer and Seiberg--Witten Floer homologies (joint with Yi-Jen Lee
and Clifford H. Taubes), I will describe a doubly filtered Floer homology
for null-homologous knots in closed oriented 3-manifolds in the context of
Seiberg-Witten theory. I will also explain how the aforementioned
construction leads to an isomorphism between the latter and
Ozsvath-Szabo's knot Floer homology.
|
| May 11
|
Yi-Jen Lee (Purdue)
|
| Sep 16
|
Jon Bloom (MIT):
Floer homology with boundary
Abstract: We extend the TQFT structure of monopole Floer
homology to cobordisms with multiple ends, equipped with
(degenerating) families of metrics. The story is complicated
by the fact that the configuration space has boundary,
consisting of reducible monopoles. We show how to package
the cobordism relations among the resulting moduli spaces
into algebraic structure, using a notion of path DGA on a
directed hypergraph. Our approach is motivated by, and
applies to, the finite-dimensional model: Morse homology on a
manifold with boundary. We discuss some applications.
|
| Sep 23
|
Steven Sivek (Harvard):
Monopole Floer homology and Legendrian knots
Abstract: We will define invariants of Legendrian knots using Kronheimer and Mrowka's construction of monopole
Floer homology for sutured manifolds. These invariants have several interesting properties: their behavior under
stabilization and contact surgery suggests that they are closely related to the Lisca-Ozsvath-Stipsicz-Szabo
invariant in knot Floer homology, and they are functorial with respect to Lagrangian concordance. As an
application, we will construct many examples of non-loose knots in overtwisted contact manifolds.
|
| Sep 30
|
Thomas Kragh (MIT):
Fibration of Symplectic Homology in Cotangent Bundles
Abstract: Let M be a Liouville domain exact embedded in a cotangent bundle
T*N. In this talk I will describe a fiber-wise version of symplectic
homology of M defined for each q in N. I will then sketch a proof of why
this does not depend on q and defines a local coefficient system, and why
there is a Serre type spectral sequence converging to the symplectic
homology of M - Page 2 of which is isomorphic to the homology of N with
coefficients in this local system. Finally I will discuss applications, and
if time permits - products.
|
| Oct 7
|
Paul Seidel (MIT): Symplectic invariants
beyond
Gromov-Witten theory
Abstract: One can define invariants of closed symplectic manifolds
by looking at families of Lagrangian submanifolds (similar in principle
to the classical notion of flux group, but more abstract). These
invariants can be used to distinguish simply-connected symplectic
manifolds which have the same Gromov-Witten invariants.
|
| Oct 14
|
Nikolai Saveliev (Miami): An
index theorem for end-periodic operators
Abstract: We extend the Atiyah, Patodi, and Singer index theorem for Dirac type operators from the context of
manifolds with cylindrical ends to that of manifolds with periodic ends. Our theorem provides a natural complement
to Taubes' Fredholm theory for general end-periodic operators. It expresses the index in terms of a new
end-periodic eta-invariant which equals the Atiyah-Patodi-Singer eta-invariant in the cylindrical setting. This is
a joint project with Tom Mrowka and Daniel Ruberman.
|
| Oct 21
|
Vivek Shende (MIT): Hilbert schemes of
plane curve singularities and knot invariants
Abstract: To a singular point on a complex plane curve in a surface,
one associates a link by intersecting the curve with a small sphere
around the singularity. It has long been understood that there is a
close relationship between the topology of the link and the geometry
of the singularity. I will discuss here a conjectural relationship
between the Khovanov-Rozansky invariants of the link and the
cohomology of Hilbert schemes of points on the singular curve.
|
| Oct 28
|
Hirofumi Sasahira (Nagoya):
Instanton Floer homology for lens space
Abstract: We will define instanton Floer homology for lens space. The difficulty
in the construction comes from the fact that flat connections on lens
space are reducible. Using Floer homology we prove a gluing formula for
a variant of Donaldson invariant. Lastly we will discuss an application
of the gluing formula.
|
| Nov 4
|
Barney Bramham (IAS):
Approximating Hamiltonian systems by integrable systems using pseudo-holomorphic curves
Abstract: I will talk about an approach using finite energy foliations to a
question raised by Anatole Katok: "In low dimensions is every conservative dynamical
system with zero topological entropy a limit of integrable systems?"
|
| Dec 2
|
David Shea Vela-Vick (Columbia): Contact structures and knot Floer homology
Abstract: I plan to discuss a general method for building the minus version of knot Floer homology
using sutured Floer homology and contact structures. I will also show how contact geometric
constructions can be used to recover many of the naturally defined maps connecting the various
version of Heegaard Floer theory. Time permitting, I will also discuss generalizations of these
techniques to the study of tight contact structures on non-compact 3-manifolds with cylindrical
ends.
|
| Dec 9
|
Liam Watson (UCLA):
L-spaces and left-orderability
Abstract: A group is left-oderable if it admits a strict total order of its elements that is
invariant under multiplication on the left; an L-space is a rational homology sphere with simplest
possible Heegaard Floer homology. It has been conjectured that an irreducible rational homology
sphere is an L-space if and only if it has non-left-orderable fundamental group. While this
conjecture seems very optimistic, I will discuss some of the evidence for it. This will centre
around joint projects with S. Boyer and C. Gordon and with A. Clay.
|
| Feb 4
|
Sergey Cherkis (Trinity College Dublin): Yang-Mills Instantons on Asymptotically Locally Flat Spaces and their Moduli Spaces
|
| Feb 11
|
James Pascaleff (MIT): Floer cohomology in the mirror of CP^2 relative to
a conic and a line
|
| Feb 18
|
Steven Sivek (MIT): A bordered Chekanov-Eliashberg algebra
|
| Feb 25
|
Edward Witten (IAS): Khovanov Homology and Gauge Theory
|
| Mar 4
|
Vaibhav Gadre (Harvard): Bounds for minimal pseudo-Anosov translation
lengths in the complex of curves
|
| Mar 25
|
Peter Albers (Purdue): A variational approach to Givental's nonlinear Maslov index
|
| Apr 1
|
Michael Usher (University of Georgia): Boundary depth and the Hofer norm
|
| Apr 8
|
Başak Gürel (Vanderbilt): Conley conjecture for negative monotone symplectic
manifolds
|
| Apr 22
|
Chris Woodward (Rutgers): Quilted Floer theory and fibered Dehn twists
|
| Apr 29 |
Jacob Rasmussen (Cambridge/Simons Center): Khovanov homology of torus knots (at 11 AM)
|
| May 6
|
Vera Vertesi (MIT): Transverse invariants in Heegaard Floer homology
|
| May 13
|
Daniel Mathews (BC): Sutured Floer homology and TQFT
|
| Feb 5
|
Jacob Lurie (Harvard): Nonabelian Poincare Duality
|
| Feb 12
|
Alex Subotic (Harvard): A monoidal structure for Fukaya categories
|
| Feb 19
|
Zhongtao Wu (Princeton): Cosmetic Surgery Conjecture on S^3
|
| Feb 26
|
Jeremy Kahn (SUNY Stony Brook): Essential Immersed Surfaces in Closed Hyperbolic 3-Manifolds
|
| Mar 5
|
Eduardo Gonzalez (U Mass Boston): Area dependence for gauge
Gromov-Witten invariants
|
| Mar 12
|
Bernd Ammann (Universität Regensburg): A surgery formula for the smooth Yamabe invariant
|
| Apr 2
|
John Baldwin (Princeton): Contact monoids and Stein cobordisms
|
| Apr 9
|
Ciprian Manolescu (UCLA): A combinatorial approach to four-manifold invariants
|
| Apr 16
|
Dan Freed (UT Austin): A differential index theorem
|
| Apr 23
|
Yasha Eliashberg (Stanford): Symplectic geometry of Stein manifolds
|