The SGGTC seminar meets on Fridays in Math 407, at 10:45 am unless noted otherwise (in red).
Abstract: The Milnor fibre of any isolated hypersurface singularity contains exact Lagrangian spheres: the vanishing cycles associated to a Morsification of the singularity. Moreover, for simple singularities, it is known that the only possible exact Lagrangians are spheres. I will explain how to construct exact Lagrangian tori in the Milnor fibres of all non-simple singularities of real dimension four. I will use these to give examples of fibres whose Fukaya categories are not generated by vanishing cycles. Time allowing, I will explain progress towards mirror symmetry for unimodal singularities (one level of complexity up from the simple ones).
Abstract: Orderability of conact manifolds is related in some non-obvious ways to the topology of a contact manifold Σ. We know, for instance, that if Σ admits a 2-subcritical Stein filling, it must be non-orderable. By way of contrast, in this talk I will discuss ways of modifying Liouville structures for high-dimensional Σ so that the result is always orderable. The main technical tool is a Morse-Bott Floer theoretic growth rate, which has some parallels with Givental's nonlinear Maslov index. I will also discuss a generalization to the relative case, and applications to bi-invariant metrics on Cont(Σ).
Abstract: A simple case of the problem of passing from classical to quantum mechanics in a coherent way is to find a functorial quantization of linear canonical relations between symplectic vector spaces. A standard quantization of such relations leads to unbounded operators which cannot always be composed, reflecting some deviation from transversality in the composition of the relations themselves. In this talk, I will describe possible mathematical settings on the classical and quantum sides, give examples, and describe some attempts to resolve the problems associated with "bad compositions".
Abstract: Within the framework of Symplectic Field Theory, there are several conjectural invariants of Legendrian submanifolds. Of these, only the simplest, Legendrian contact homology, has been well-defined. In this talk, I will describe how string topology arises in the formulation of rational SFT as well as recent results on a suitable model of string topology in dimension 2.
Abstract: The classical problem of enumerating rational curves in projective spaces is solved using a recursion formula for Gromov-Witten invariants. In this talk, I will describe a similar relation for real Gromov-Witten invariants with conjugate pairs of constraints. An application of this relation provides a complete recursion for counts of real rational curves with such constraints in odd-dimensional projective spaces. I will outline the proof and discuss some vanishing and non-vanishing results. This is joint work with A. Zinger.
Abstract: Heegaard Floer homology is a powerful 3-manifold invariant, but computing it in general is difficult. I will describe a method for computing HF-hat in the case of graph manifolds. A graph manifold decomposes nicely into pieces which are circle bundles over surfaces. The key machinery for computing HF-hat is bordered Heegaard Floer homology, an extension of the Heegaard Floer package to manifolds with boundary. By computing the bordered invariants of the circle bundles over surfaces and piecing them together in an appropriate way, we can compute HF-hat of any graph manifold.
Abstract: We consider the problem of defining cylindrical contact homology, in the absence of contractible Reeb orbits, using "classical" methods. The main technical difficulty is failure of transversality of multiply covered cylinders. One can fix this difficulty by using S^1-dependent almost complex structures, but at the expense of introducing another difficulty which we will explain. We outline how fixing the latter difficulty ultimately leads to a different theory, an analogue of positive symplectic homology. This talk is intended to be part of a series of expository talks on the foundations of contact homology, but prerequisites should be minimal.
Abstract: In 2004, Khovanov defined a TQFT which categorifies the Kuperberg bracket (this is a flat version of the sl3 polynomial for links). After having reminded this construction, I'll explain how it can be naturally extended to a 0+1+1 TQFT. The Khovanov-Kuperberg algebras are central in this construction. By contrast with the sl2 case, the natural candidates (the so called web-modules) for being a complete family of indecomposable projective modules, may decompose. However I'll explain that that one can characterise the indecomposability with a very simple criteria. The proof is based on some combinatorial objects called red graphs.
Abstract: There are a number of ways to define a Floer homology for three-manifolds using Seiberg-Witten theory. Two such examples are Kronheimer and Mrowka's monopole Floer homology and Manolescu's Seiberg-Witten Floer spectrum, each of which has its own advantages and applications. We will define these and discuss the relationship between these two objects, after describing an analogue in Morse theory. This is work in progress with Ciprian Manolescu
Abstract: Cochran and Gompf defined a notion of positivity for concordance classes of knots that simultaneously generalizes the usual notions of sliceness and positivity of knots. Their positivity essentially amounts to the knot being slice in a positive-definite simply-connected four manifold. We discuss an analogous property for links, and describe relationships with (generalized) Sato-Levine invariants, Milnor's linking invariants, the Conway polynomial, and some modern invariants.
Abstract: In the 1970's, Graeme Segal proved that the space of holomorphic maps from a Riemann surface to a complex projective space is homology equivalent to the corresponding space of continuous maps through a range of dimensions increasing with degree. I will address if a similar result holds when other almost complex structures are put on projective space. For CP^2, I prove that the inclusion map from the space of J-holomorphic maps to the space of continuous maps induces a homology surjection through a range of dimension tending to infinity with degree. The proof involves comparing the scanning map of topological chiral homology with gluing of J-holomorphic curves.
Abstract: To a link L in a thickened annulus, Asaeda-Przytycki-Sikora assigned a Khovanov-type homology theory which categorifies the skein module of the thickened annulus and which is related to a certain knot Floer homology by work of Roberts. In this talk, I will show that this homology theory carries a natural action of sl(2) and, in the case where L is the n-cable of a framed knot K, a commuting action of the symmetric group S_n. In the case where K is the 0-framed unknot, we recover classical Schur-Weyl duality for the nth tensor power of the fundamental representation of sl(2). This is joint work with Eli Grigsby and Tony Licata.
Abstract: I will describe how to use Khovanov homology to detect the trivial braid. The proof is completely combinatorial, relying on an interesting connection between Plamenevskaya's invariant of transverse links and Dehornoy's strict, total order on the braid group. Portions of this talk are joint with John Baldwin, and other portions are joint with Stephan Wehrli.
Abstract: I will present some recent joint work with Richard Siefring on the behavior of finite energy foliations of pseudoholomorphic curves in symplectizations of contact manifolds which undergo either a 0-surgery (i.e. a connect sum) or a 2-surgery. We then discuss applications to the restricted three body problem.
Ana Rita Pires
Abstract: The topology of a toric symplectic manifold can be read directly from its orbit space (a.k.a. moment polytope), and much the same is true of the topological generalizations of toric symplectic manifolds and projective toric varieties: quasitoric manifolds, topological toric manifolds and torus manifolds. An origami manifold is a manifold endowed with a closed 2-form with a very mild degeneracy along a hypersurface, but this degeneracy is enough to allow for non-simply-connected and non-orientable manifolds. In this talk we examine how the topology of a toric origami manifold can be read from the polytope-like object that represents its orbit space and how these results hold for the appropriate topological generalization of the class of toric origami manifolds, which includes quasitoric manifolds, and some torus manifolds. These results are from ongoing joint work with Tara Holm.
Abstract: I will present recent joint work with Vinicius Gripp and Michael Hutchings relating the volume of any closed contact three-manifold to the length of certain finite sets of closed orbits of the Reeb vector field. I will also explain why this result implies that any Reeb vector on a closed three-manifold always has at least two closed orbits.
Other relevant information
- Princeton Topology Seminar