The SGGT seminar meets on Fridays in Math 417, at 10:45 AM unless noted otherwise. There is also an informal symplectic geometry seminar, which meets at a different time.

Previous semesters: spring 2009, fall 2008, spring 2008, fall 2007, spring 2007, fall 2006.

Other area seminars. Our e-mail list.

SGGT seminar schedule.
Date Speaker Title
Sept. 3 2:30-3:30 p.m.: Egor Shelukhin (Tel Aviv) Invariants of Hamiltonian loops
4-5 p.m.: Felix Schlenk (Lausanne) Lagrangian product tori
Sept. 18

Organizational meeting.
Oct. 2 10:45 am Michael Brandenbursky (Technion) Knot theory and quasi-morphisms
NY Joint Symplectic Geometry Seminar
Meets in Math 520
  
1:15 p.m.: Mu-Tao Wang (Columbia) Mean curvature flows of Lagrangian submanifolds and isotopy problems in symplectic geometry.
3:45 p.m.: Dusa McDuff (Columbia/Barnard) Symplectic embeddings and continued fractions
Oct. 9 András Stipsicz (Rényi Institute) Contact structures on 5-manifolds
Oct. 16 Weiyi Zhang (U of Minnesota) A new cohomology decomposition of almost complex 4-manifolds and comparing symplectic cones.
Oct. 23 Bohan Fang (Northwestern) Coherent-constructible correspondence for toric varieties and stacks.
Oct. 26 10:30am: Denis Auroux (MIT/Berkeley)
Meets in Math 622		 Fukaya categories of symmetric products and bordered Heegaard-Floer homology
Oct. 30 Adam Levine (Columbia) Sliceness of Whitehead and Bing Doubles
Nov. 6
10:45 a.m.: Yaron Ostrover (IAS) Symplectic measurements and convex geometry
2:30 p.m. Joel Kamnitzer (U of Toronto)
(Joint with the Columbia Algebraic Geometry Seminar)		 Categorical Lie algebra actions, quiver varieties, and braid group actions
3:45 p.m.: Ely Kerman (UIUC)
Meets in Math 520		 New obstructions to symplectic embeddings in dimensions greater than four.
Nov. 13 Joan Licata (Stanford) Differential algebras for Legendrian knots in contact lens spaces.
Nov. 20 Mark McLean (MIT) Exotic non-finite type Stein manifolds.
Nov. 27
No seminar this week.
Dec. 4 9:30 a.m.: Hendryk Pfeiffer (UBC) Fusion categories in terms of graphs and relations
Rumen Zarev (Columbia) Bordered Floer homology and sutured Floer homology
3:45 p.m.: Florent Schaffhauser
Meets in Math 507		 Moduli spaces of vector bundles over a Klein surface
Dec. 9 5 p.m.: Mark Branson (Columbia) The Action-Maslov Homomorphism for Monotone Symplectic Manifolds
Dec 11. 9:30 a.m.: Matt Hedden (MSU) Recent results in concordance
10:45 a.m.: Leonid Polterovich (Tel Aviv) Poisson brackets and symplectic invariants
Dec. 18

 

Abstracts.

September 3, 2009.

Felix Schlenk, "Lagrangian product tori"

Abstract: A Lagrangian product torus in C^n is a torus of the form T(a_1)x...xT(a_n), where T(a) denotes the circle enclosing area a. Given a symplectic manifold M and a Darboux chart B \to M from a ball to M, we obtain Lagrangian product tori in M. I'll try to explain how such tori can be classified up to Hamiltonian isotopy if the symplectic form and the first Chern class vanish on \pi_2(M). But already in products of spheres, the situation is different, and the classification of product tori is not comlpete. If time permits, I'll also give the construction of many Lagrangian tori that are not Hamiltonian isotopic to product tori. This is joint work with Yuri Chekanov.

 

October 2, 2009.

Michael Brandenbursky (Technion), "Knot theory and quasi-morphisms"

Abstract (pdf): Quasi-morphisms on a group are real-valued functions which satisfy the homomorphism equation "up to a bounded error". They are known to be a helpful tool in the study of the algebraic structure of non-Abelian groups. I will discuss a construction relating:
a) certain knot and link invariants-in particular, the ones that come from the knot Floer homology and a Khovanov-type homology,
b) braid groups,
c) the dynamics of area-preserving diffeomorphisms of a two dimensional disc,
d) quasi-morphisms on the group of all such compactly supported diffeomorphisms of the disk.

 

October 2, 2009.

Mu-Tao Wang, "Mean curvature flows of Lagrangian submanifolds and isotopy problems in symplectic geometry"

Abstract: When a symplectic manifold is equipped with an appropriate Riemannian metric, there is a natural way to deform a Lagrangian submanifold called the mean curvature flow. This can be considered as the heat equation for submanifolds, just like the Ricci flow is the heat equation for metrics. In this talk, I shall discuss some recent applications of this flow to the study of symplectomorphism groups of complex projective spaces and Lagranian isotopy problems in cotangent bundles.

 

October 9, 2009.

András Stipsicz, "Contact structure on 5-manifolds"

Abstract: We show a simple way to produce contact structures on 5-manifolds which are products of lower-dimensional manifolds. The case of S^1xCP^2 will be analyzed in detail.

 

October 16, 2009.

Weiyi Zhang, "A new cohomology decomposition of almost complex 4-manifolds and comparing symplectic cones"

Abstract: In this talk, I will introduce a new cohomology decomposition of second cohomology group of almost complex 4-manifolds. This can be viewed as a generalization of (real) Dolbeault decomposition for complex structures. I will calculate and estimate the rank of the subgroups in some cases. I will explain the relations with the comparison of J-tamed and J-compatible symplectic cones. This talk is based on joint work with Tedi Draghici and Tian-Jun Li.

 

October 23, 2009.

Bohan Fang, "Coherent-constructible correspondence for toric varieties and stacks."

Abstract: This is a talk on joint work with Chiu-Chu Liu, David Treumann and Eric Zaslow. I will describe a coherent-constructible correspondnece for toric varieties and stacks. Motivated by SYZ conjecture for mirror symmetry, from a line bundle on a toric variety one constructs a Lagrangian brane on its mirror, and further a polytope-shaped constructible sheaf on a real vector space. This correspondence, which is also a version of homological mirror symmetry of Kontsevich, can be extended to toric stacks.

 

October 26, 2009.

Denis Auroux, "Fukaya categories of symmetric products and bordered Heegaard-Floer homology"

Abstract: We will present an interpretation of Lipshitz-Ozsvath-Thurston's "bordered Heegaard-Floer homology" in terms of the symplectic topology of symmetric products. More specifically, we will explain how to understand the algebra A(F) associated to a surface in terms of a (relative) Fukaya category of the symmetric product, and the corresponding interpretation of the module CFA(Y) associated to a bordered 3-manifold.

 

October 30, 2009.

Adam Levine, "Sliceness of Whitehead and Bing Doubles"

Abstract: Links obtained using the operations of Whitehead and Bing doubling (and combinations thereof) are of great interest in the study of concordance, since they play a fundamental role in the work of Freedman on topological 4-manifolds. I will discuss recent work on this topic and prove some new results on the smooth sliceness of such links. For example, we can prove that the positive Whitehead double of the Borromean rings is not smoothly slice; whether or not it is topologically slice remains a major unsolved question.

 

Nov 6, 2009.

Yaron Ostrover, "Symplectic measurements and convex geometry"

Abstract: In this talk we discuss some interrelations between symplectic and convex geometry. In particular, we will show how tools from one field can be used to tackle questions in the other field.

 

Nov 6, 2009.

Joel Kamnitzer, "Categorical Lie algebra actions, quiver varieties, and braid group actions"

I will discuss the notion of categorical Lie algebra actions, as introduced by Rouquier and Khovanov-Lauda. In particular, I will give examples of categorical Lie algebra actions on derived categories of coherent sheaves on quiver varieties. These categorical Lie algebra actions lead to actions of braid groups, generalizing known actions by spherical twists. I will explain applications to constructions of knot homology theories and connections to symplectic geometry.

 

November 6, 2009.

Ely Kerman, "New obstructions to symplectic embeddings in dimensions greater than four"

In this talk I will describe new restrictions on symplectic embeddings of certain convex domains into symplectic vector spaces of dimension greater than four. More precisely, these restrictions hold for symplectic embeddings of the product of a symplectic unit disc with a symplectic ball of arbitrarily large radius. The restrictions obtained are stronger than those implied by the Ekeland-Hofer capacities. By refining an embedding technique due to Guth, they can also be shown to be sharp. This is joint work with Richard Hind.

 

Nov 13, 2009.

Joan Licata, "Differential algebras for Legendrian knots in contact lens spaces"

Abstract: Relative contact homology is a Floer-theoretic invariant for Legendrian knots in contact manifolds. In special cases, it can be computed combinatorially from the Lagrangian projection of hte knot. In this talk, I'll introduce the family of combinatorial invariants, focusing on the case of primitive knots in lens spaces.

 

Nov 20, 2009.

Mark McLean, "Exotic non-finite type Stein manifolds"

Abstract: A Stein manifold is a complex manifold that can be properly holomorphically embedded in C^n. Let M be an even dimensional manifold admitting an exhausting Morse function with finitely many critical points of index at most half the dimension. Then there are uncountably many non-finite type Stein manifolds diffeomorphic to M which are pairwise distinct as symplectic manifolds. I use an invariant called symplectic homology to distinguish these Stein manifolds.

 

December 4, 2009.

Hendryk Pfeiffer, "Fusion categories in terms of graphs and relations"

Abstract: Every fusion category C that is linear over a suitable field, is the category of finite-dimensional comodules of a Weak Hopf Algebra H. I show how to obtain a presentation of H that makes the truncated tensor product of C fully transparent: H is a quotient H=H[G]/I of a Weak Bialgebra H[G] which has a combinatorial description in terms of a finite directed graph G that depends on the choice of a generator M of C and on the fusion coefficients of C. The algebra underlying H[G] is the path algebra of the quiver GxG, and so the composability of paths in G parameterizes the truncation of the tensor product of C. The ideal I is generated by two types of relations. The first type enforces that the tensor powers of the generator M have the appropriate endomorphism algebras (Schur-Weyl dual description). If C is braided, this includes relations of the form "RTT=TTR" where R contains the coefficients of the braiding on M(x)M, a generalization of the construction of Faddeev-Reshetikhin-Takhadjan to truncated tensor products. The second type of relations removes a suitable set of group-like elements in order to make the category of finite-dimensional comodules equivalent to C over all tensor powers of the generator M.

 

December 4, 2009.

Rumen Zarev, "Bordered Floer homology and sutured Floer homology"

Abstract: Bordered Floer homology (CFD,CFA), and sutured floer homology (SFH) are two different invariants of three manifolds with boundary, derived from Heegaard Floer homology. A natural question is whether these different invariants are equivalent to each other. We show that both are special cases of a more general construction call Bordered Sutured Floer homology (BSD, BSA). Using this more general invariant, we show that CFD and CFA contain at least as much information as SFH, and give evidence supporting the conjecture that the converse is also true.

 

December 4, 2009.

Florent Schaffhauser, "Moduli spaces of vector bundles over a Klein surface"

Abstract: A compact topological surface S, possibly non-orientable and with non-empty boundary, always admits a Klein surface structure (an atlas whose transition maps are dianalytic). Its complex cover is, by definition, a compact Riemann surface X endowed with an antiholomorphic involution which determines topologyically the original surface S. In this talk, we relate dianalytic vector bundles over S and holomorphic vector bundles over X, devoting special attention to the implications this has for moduli spaces of semistable bundles over X. We construct, starting from S, Lagrangian submanifolds of moduli spaces of semistable bundles of fixed rank and degree over X. This relates the present work to the constructions of Ho and Liu over non-orientable compact surfaces with empty boundary. arXiv: http://arxiv.org/abs/0912.0659

 

December 9, 2009.

Mark Branson, "The Action-Maslov Homomorphism for Monotone Sympectic Manifolds"

Abstract: The action-Maslov homomorphism, introduced by Polterovich, is a map from the fundamental group of the Hamiltonian group of a symplectic manifold to the real numbers. I will explore conditions under which this homomorphism vanishes on monotone symplectic manifolds. My strategy involves showing that the units in the quantum homology, and thus the Seidel element, have a very specific form. I prove that these conditions hold for monotone products of projective spaces and for the Grassmannian of 2-planes in C^4. I will also discuss the consequences of this result for the topology of the Hamiltonian group.

 

Dec 11, 2009.

Matt Hedden, "Recent results in concordance"

Abstract: Under an equivalence relation called concordance, knots form a group with operation provided by the conneted sum. This group depends on the category (smooth or topological) in which it is defined, and highlights iteresting aspects of four-dimensional topology. My talk will focus on attempts to understand the group through its subgroups; that is, given a collection of knots one can consider the subgoup which they generate within the larger concordance group. I will discuss various geometrically relevant subgroups and some recent progress. Depending on which results become the focus, this is joint work with subsets of {Paul Kirk, Chuck Livingston, Danny Ruberman}.

 

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