I am an organizer for the Gauge Theory, Topology, and Symplectic Geometry seminar at Harvard. The seminar meets on Fridays from 3:30-4:30, in Science Center 507.

Here is the current schedule for this semester [some titles and abstracts still to come]:

Fall 2015:

September 11, Richard Schwartz (Brown)

Title: The plaid model and outer billiards

Abstract: Outer billiards is a dynamical system that is

similar in spirit to ordinary billiards, except that the

ball moves around the outside of the table. Even for

simple tables, such as kites (quadrilaterals with bilateral

symmetry) the orbits produced have a rich combinatorial

structure. I'll describe a combinatorial model, which I call

the plaid model, which predicts the shapes of these orbits

for kites. The model involves mathematics which is similar

in spirit to what one sees for quasi-periodic tilings.

September 18, Jianfeng Lin (UCLA)

Title: The unfolded Seiberg-Witten-Floer spectrum and its applications

Abstract: In 2003, Manolescu defined the Seiberg-Witten-Floer stable homotopy type for rational homology three-spheres. In this talk, I will explain how to construct similar invariants for a general three-manifold and discuss some applications of these new invariants. This is a joint work with Tirasan Khandhawit and Hirofumi Sasahira.

September 25, Peter Kronheimer (Harvard) [Special joint seminar with MIT]

Title: The four-color theorem and spatial graphs

Abstract: Given a trivalent graph embedded in 3-space, we associate to it an instanton homology group, which is a finite-dimensional **Z**/2 vector space. The main result about the instanton homology is a non-vanishing theorem, proved using techniques from 3-dimensional topology: if the graph is bridgeless, its instanton homology is non-zero. It is not unreasonable to conjecture that, if the graph lies in the plane, the dimension of its instanton homology is equal to the number of Tait colorings of the graph (essentially the same as four-colorings of the planar regions that the graph defines). If the conjecture were to hold, then the non-vanishing theorem for instanton homology would imply the four-color theorem and would provide a human-readable proof. This program is joint work with Tom Mrowka.

October 2, Chris Wendl (UCL)

October 9, Shamil Shakirov (Harvard)

October 16, Cheuk Yu Mak (Minnesota)

Title: Divisorial caps, uniruled caps and Calabi-Yau caps

Abstract: We illustrate how a nice symplectic cap captures properties of symplectic fillings of a contact 3-manfiold. Three kinds of symplectic caps are introduced. Divisorial caps are motivated from compactifying divisors in algebraic geometry. Uniruled caps give strong restriction to symplectic fillings for a class of contact 3-manifolds strictly larger than the planar ones. Calabi-Yau caps, in particular, can be used to derive uniform Betti numbers bounds on Stein fillings of the standard unit cotangent bundle of any hyperbolic surface. This is a joint work with Tian-Jun Li and Kouichi Yasui.

October 23, Mohammed Abouzaid (Columbia)

October 30, Richard Hind (Notre Dame)

November 6, Thomas Walpuski (MIT)

November 13, Caitlin Leverson (Duke)

December 4, Chris Woodward (Rutgers)

Spring 2015:

January 30: R. Inanc Baykur (UMass Amherst)

Multisections of Lefschetz fibrations and topology of symplectic

4-manifolds

Abstract: We initiate an extensive study of multisections of Lefschetz

fibrations via positive factorizations in framed mapping class groups.

Using our techniques, we can reformulate and tackle various interesting

conjectures and problems related to the topology of symplectic

4-manifolds. In this talk we will focus on the conjectural smooth

classification of symplectic Calabi-Yaus and their fundamental groups in

real dimension 4. If time permits, we will discuss some other applications

as well; such as Stipsicz's conjectures on minimality and fiber sum

decompositions, constructions of inequivalent Lefschetz fibrations and

exotic pencils. Several parts of this work is joint with K. Hayano and N.

Monden.

February 6: Eli Grigsby (BC)

(Sutured) Khovanov homology and representation theory

Abstract: Khovanov homology associates to a link L in the three-sphere a bigraded vector space arising as the homology groups of an abstract chain complex defined combinatorially from a link diagram. It detects the unknot (Kronheimer-Mrowka) and gives a sharp lower bound (Rasmussen, using a deformation of E.S. Lee) on the 4-ball genus of torus knots.

When L is realized as the closure of a braid--or more generally, of a "balanced" tangle--one can use a variant of Khovanov's construction due to Asaeda-Przytycki-Sikora and L. Roberts to define its *sutured *Khovanov homology, an invariant of the tangle closure in the solid torus. In this talk, I will describe some of the representation theory of the sutured Khovanov homology of a tangle closure. It (perhaps unsurprisingly) carries an action of the Lie algebra sl(2). More surprisingly, this action extends to the action of a slightly larger Lie superalgebra whose structure hints at a unification with the Lee deformation. This is joint work with Tony Licata and Stephan Wehrli.

February 13: No seminar (go see Polterovich at MIT!)

February 20: Alexandra Kjuchukova (Penn)

On the classification of branched covers of four-manifolds

Abstract: Given two simply-connected closed oriented topological four-manifolds X and Y, I ask: *can Y be realized as a branched cover of X?* For B a surface embedded in X with an isolated singularity, I will prove a necessary condition for the existence of an irregular dihedral branched covering map $f: Y \to X$ with branching set B. Conversely, given a simply-connected oriented closed four-manifold X, I will outline a construction realizing as irregular dihedral covers of X infinitely many of the manifolds Y afforded by the necessary condition.

February 27: Dan Halpern-Leistner (IAS)

Abstract: For a Hermitian vector bundle on a Riemann surface, the gradient descent flow of the Yang-Mills functional on the space of Hermitian connections defines a stratification of the moduli of complex structures (up to gauge equivalence) on that vector bundle. In algebraic geometry, this corresponds to the Harder-Narasimhan stratification of the moduli of algebraic vector bundles on the Riemann surface -- the gradient descent flow line for an unstable bundle is analogous to its Harder-Narasimhan filtration. The question of existence and uniqueness of the Harder-Narasimhan filtration can be equivalently phrased as a convex optimization problem on a large topological space associated to the vector bundle. This leads to a framework for constructing stratifications of this kind for moduli problems which are not specifically described as infinite dimensional symplectic reductions.

March 6: Laura Starkston (Texas)

Symplectic fillings of Seifert fibered spaces

Abstract: Although the existence of a symplectic filling is well-understood for many contact 3-manifolds, complete classifications of all symplectic fillings of a particular contact manifold are more rare. Relying on a recognition theorem of McDuff for closed symplectic manifolds, we can understand this classification for certain Seifert fibered spaces with their canonical contact structures. Even in cases without complete classification statements, the techniques used can suggest constructions of symplectic fillings with interesting topology. These fillings can be used in cut-and-paste operations to construct examples of exotic 4-manifolds.

March 13: Jingyu Zhao (Columbia)

Periodic Symplectic Cohomologies

Periodic cyclic homology group associated to a mixed complex was introduced by Goodwillie. In this talk, I will explain how to apply this construction to the symplecticcochain complex of a Liouville domain and obtain two periodic symplectic cohomology theories, which are called periodic symplectic cohomology and finitely supported periodic symplectic cohomology, respectively. The main result is that there is a localization theorem for the finitely supported periodic symplectic cohomology.

March 27: Cagatay Kutluhan (Buffalo)

A Heegaard Floer analogue of algebraic torsion

Abstract: We explain how to construct a refinement of the contact invariant in Heegaard Floer homology that is better at distinguishing overtwisted contact structures from tight ones.

April 3: Tara Holm (Cornell)

The Topology of Toric Origami Manifolds

Abstract: A folded symplectic form on a manifold is a closed 2-form

with the mildest possible degeneracy along a hypersurface. A special

class of folded symplectic manifolds are the origami manifolds. In the

classical case, toric symplectic manifolds can classified by their

moment polytope, and much about their topological invariants can be

read directly from the polytope. In this talk we examine the toric

origami case: we will describe how toric origami manifolds can also be

classified by their combinatorial moment data, and present some

results about the topology of toric origami manifolds. This is joint

work with Ana Rita Pires.

April 10: Daniel Pomerleano (Imperial)

Symplectic cohomology in the topological limit

Abstract: In Sheridan's proof of homological mirror symmetry for Calabi-Yau hypersurfaces in projective space, he highlighted an interesting class of affine varieties known as "Kahler pairs." In this talk, we will explain how to compute their symplectic cohomology. This is joint work with Sheel Ganatra.

April 17: Danny Ruberman (Brandeis)

April 24: Nick Sheridan (Princeton/IAS)

May 1: Yanir Rubinstein (Maryland)

The degenerate special Lagrangian equation

Abstract: In joint work with J. Solomon, we introduce the {\it degenerate special Lagrangian equation} and develop the basic analytic tools to construct and study its solutions. This equation governs geodesics in the space of positive Lagrangians. It has intriguing relations to uniqueness, existence and stability notions related to special Lagrangians, Lagrangian mean curvature flow, and the topology of Hamiltonian isotopy classes.

Here are the speakers from previous semesters:

Fall 2014:

September 19: Emmy Murphy (MIT)

Existence of overtwisted contact structures on high dimensional manifolds

Abstract: The Lutz-Martinet theorem states that any 2-plane field on a 3-manifold is homotopic to a contact structure. This construction led to Eliashberg's definition of over twisted contact manifolds, and in this context the existence theorem of Lutz-Martinet can be extended to a uniqueness rest: any two over twisted 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 buy a unique isotopy class of overtwisted contact structure. This project is joint work with Borman and Eliashberg.

September 26: Jo Nelson (Columbia/IAS)

Cylindrical contact homology in dimension 3 via intersection theory and more.

Abstract: After reviewing the difficulties in proving d²=0 for dynamically convex contact manifolds, we explain how intersection theory and automatic transversality come to the rescue, as implemented by Hutchings-Nelson. So that these methods do not remain shrouded in mystery, we provide an overview of the relevant ideas from intersection theory for pseudoholomorphic curves in 4-dimensional symplectizations a la Siefring and Hutchings. Time permitting we end with a brief sketch of how to obtain invariance of cylindrical contact homology over Q via nonequivariant formulations, obstruction bundle gluing, domain dependent almost complex structures, and S¹-equivariant constructions. The discussion of invariance is based on work in progress by Hutchings-Nelson.

October 3: Sobhan Seyfaddini (MIT)

The displaced disks problem via symplectic topology

Abstract: We will show that a C^0-small area preserving homeomorphism of S^2 can not displace a disk of large area. This resolves the displaced disks problem posed by F. Béguin, S. Crovisier, and F. Le Roux.

October 10: Tristan Collins (Harvard)

A tour of Sasakian-Einstein geometry

Abstract: I will discuss the problem of finding Sasakian-Einstein metrics, which are a special type of contact metric structures. The primary focus will be connections to algebraic geometry and Kahler geometry, as well as algebro-geometric conditions which imply existence or obstruction of Sasakian-Einstein metrics. I hope to discuss how these conditions lead to the construction of an abundance of Einstein metrics on spheres.

October 17: Dan Freed (Texas)

Chern-Weil forms and abstract homotopy theory

Abstract: I will discuss the theorem (joint w/Hopkins) that Chern-Weil forms are the only natural differential forms associated to a connection on a principal G-bundle. We use abstract homotopy theory to eventually reduce the problem to classical invariant theory. In the process we define a "generalized manifold" which plays the role of EG and whose de Rham complex is precisely the Weil algebra. There is a corresponding geometric interpretation of the Weil model in equivariant de Rham theory.

October 24: Sheng-Fu Chiu (Northwestern)

Contact non-squeezability and Microlocal category

Abstract: This talk focuses on the relation between microlocal categories and symplectic/contact topology. We will briefly describe how to assign a triangulated category to a given geometric object and how this assignment varies under Hamiltonian flows in the ambient manifolds. This allows us to retrieve Hamiltonian invariants from homological data in a systematic way. Finally, we will discuss its application to a contact non-squeezability problem posed by Eliashberg, Kim and Polterovich. This is a joint work with Dmitry Tamarkin.

October 31: Nate Bottman (MIT)

Singular quilts and a proposed A-infinity 2-category

Abstract: I will describe work-in-progress with Katrin Wehrheim to construct the "symplectic A-infinity 2-category `Symp' ", whose objects are symplectic manifolds and where hom(M,N) is the immersed Fukaya category Fuk(M- x N). The structure maps will be defined by counting moduli spaces of pseudoholomorphic quilts with a figure eight singularity. A formal consequence of Symp is a symplectic analogue of Fourier-Mukai functors. After describing the blueprint for Symp, I will present two analytic results about singular quilts: a removal of singularity for the figure eight singularity, and a Gromov compactness theorem for strip-shrinking. This talk will be based partly on the preprint arXiv:1410.3834 and partly on an upcoming preprint with Wehrheim.

November 7: Mark McClean (Stony Brook)

Minimal Log Discrepancy of Isolated Singularities and Reeb Orbits

Abstract: Let A be an affine variety inside a complex N dimensional vector space which either has an isolated singularity at the origin or is smooth at the origin. The intersection of A with a very small sphere turns out to be a contact manifold called the link of A. Any contact manifold contactomorphic to the link of A is said to be Milnor fillable by A. If the first Chern class of our link is 0 then we can assign an invariant of our singularity called the minimal discrepancy. We relate the minimal discrepancy with indices of certain Reeb orbits on our link. As a result we show that the standard contact 5 dimensional sphere has a unique Milnor filling up to normalization. This generalizes a Theorem by Mumford.

November 14: Tom Mark (Virginia)

Tight contact structures on positive surgeries

Abstract: Motivated by the question of existence of tight contact structures on 3-manifolds, we study the question of whether a contact structure obtained by positive contact surgery along a Legendrian knot in the 3-sphere is tight. If the classical invariants of the knot satisfy a maximality condition, and the surgery coefficient is larger than 2g_4(K) - 1, we prove that the Heegaard Floer contact invariant of a resulting contact structure is nonvanishing; in particular the structure is tight. Our proof applies to integral surgery coefficients, with an extension in progress to rational surgeries. The method uses a generalization of the “capping off” procedure for open book decompositions, and may be of further use in studying open books with reducible monodromies. This is joint work with Bulent Tosun.

November 20: Michael Hutchings (Berkeley)

Beyond ECH capacities

Abstract: ECH (embedded contact homology) capacities give obstructions to symplectically embedding one four-dimensional symplectic manifold with boundary into another. Cristofaro-Gardiner showed that these obstructions are sharp when the domain is a “concave toric domain” and the target is a “convex toric domain”. However ECH capacities often do not give sharp obstructions, for example in many cases when the domain is a polydisk. We use refined information from ECH to give stronger symplectic embedding obstructions when the domain is a convex toric domain. In particular we obtain new and sometimes sharp symplectic embedding obstructions when the domain is a polydisk and the target is a polydisk or an ellipsoid.