The SGGTC seminar meets on Fridays in Math 407 at 1pm, unless noted otherwise (in red).
Previous semesters: Spring 2016, Fall 2015, Spring 2015, Fall 2014, Spring 2014, Fall 2013, Spring 2013, Fall 2012, Spring 2012, Fall 2011, Spring 2011, Fall 2010, Spring 2010, Fall 2009, Spring 2009, Fall 2008, Spring 2008, Fall 2007.
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Abstract: The Jucys-Murphy elements are known to generate a maximal commutative subalgebra in the Hecke algebra. They can be categorified to a family of commuting complexes of Soergel bimodules. I will describe a relation between a category generated by these complexes and the category of sheaves on the flag Hilbert scheme of points on the plane, using the recent work of Elias and Hogancamp on categorical diagonalization. As an application, I will give an explicit conjectural description of the Khovanov-Rozansky homology of generalized torus links. The talk is based on a joint work with Andrei Negut and Jacob Rasmussen.
September 16, 2016: Heather Lee "Homological mirror symmetry for open Riemann surfaces from pair-of-pants decompositions"
Abstract: We will demonstrate one direction of HMS for punctured Riemann surfaces - the wrapped Fukaya category of a punctured Riemann surface is equivalent to the matrix factorization category MF(X,W) of the toric Landau-Ginzburg mirror (X, W). The category MF(X,W) can be constructed from a Cech cover of (X,W) by local affine pieces that are mirrors of pairs of pants. We supply a suitable model for the wrapped Fukaya category for a punctured Rimemann surface so that it can also be explicitly computed in a sheaf-theoretic way, from the wrapped Fukaya categories of various pairs of pants in a decomposition. The pieces are glued together in the sense that the restrictions of the wrapped Floer complexes from two adjacent pairs of pants to their adjoining cylindrical piece agree.
Abstract: Through 3 general points and 6 general lines in $\mathbb P^3$, there are exactly 190 twisted cubics; 190 is a Gromov-Witten invariant of $\mathbb P^3$. Mirror symmetry is a beautiful conjecture about the structure of all Gromov-Witten invariants of a smooth variety (or orbifold) X. It is also an important step towards solving many interesting and difficult problems, such as Ruan's Crepant Resolution Conjecture, relating the Gromov-Witten invariants of an orbifold to those of a crepant resolution. One of the prime examples of a crepant resolution is the resolution of $Sym^d(\mathbb P^2)$ by the Hilbert scheme of points — this was one of Ruan’s motivating examples. However, mirror symmetry has proved difficult in this case; it is essentially known only for toric orbifolds and some of their complete intersections (after much difficult work). We prove the mirror theorem for $Sym^d(\mathbb P^r)$, and on the way we develop techniques for dealing with nonabelian orbifolds.
Abstract: I will use cablings with infinite torus braids to define a Khovanov homotopy type for $sl_2(C)$ colored links and discuss some of its basic properties. In the case of n-colored B-adequate links, I will describe a stabilization of the homotopy types as the coloring n goes to infinity, generalizing the tail behavior of the colored Jones polynomial. In the case of the unknot, I will also describe a simpler argument for stabilization. If time permits, I will also discuss a more recent result regarding stabilization involving more general infinite braids.
Abstract: In this talk, we introduce a symplectic analogue of log Calabi Yau surfaces and classify them up to symplectic deformation equivalence.
October 14, 2016: Peter Koroteev "Non-commutative instantons from large-n limit of Seiberg-Witten theories"
Abstract: We study moduli space of U(n) instantons with ramification (rank-n torsion free sheaves on P^2 with framing at infinity). It can be argued that the instanton partition function (holomorphic equivariant Euler characteristic) satisfies a system of differential or difference equations which coincide with energy equation for some integrable n-body system of Calogero-Moser or Ruijsenaars type. We then take a limit, when the number of particles becomes large, and recover a different effective integrable model — intermediate long wave hydrodynamics. Spectrum of this model describes quantum multiplication in the small cohomology ring of the moduli space of non-commutative instantons on C^2 and ALE spaces. We shall formulate some results and conjectures on related topics.
Abstract: In this talk we use a construction of a moduli space of stable pairs over a Riemann surface to produce a Floer-theoretic invariant of a surface bundle over the circle (together with a choice of line bundle on it). We then outline a calculation of this invariant for a certain class of torus bundles. This is joint work with Tim Perutz.
Abstract: In this talk we'll describe an approach to defining cobordism maps in link Floer homology for decorated link cobordisms. We'll describe the maps which feature in the construction, and describe some of the steps to proving invariance. We'll describe some basic properties of the maps, such as a grading change formula. We'll describe several applications of the theory, such as a proof of the formula for the "Sarkar map", and quick proofs of several well known bounds on the concordance invariant tau.
Abstract: Over a decade ago, Welschinger defined real enumerative invariants in dimensions 2 and 3. It has remained an open problem to extend these invariants to higher dimensions. I will discuss a solution to this problem in the language of open Gromov-Witten theory. The key idea is that boundary point constraints should be replaced with canonical gauge equivalence classes of Maurer-Cartan elements (bounding chains) in the relevant Fukaya A-infinity algebra. The resulting invariants satisfy an open WDVV equation. All invariants for projective spaces have been calculated. In connection with open WDVV, a relative version of the quantum product appears. Real structures do not play an essential role in our arguments. This is joint work with S. Tukachinsky.
November 4, 2016: C.-M. Michael Wong "Unoriented skein relations for grid homology and tangle Floer homology"
Abstract: Although the Alexander polynomial does not satisfy an unoriented skein relation, Manolescu (2007) showed that there exists a skein exact triangle for knot Floer homology over Z/2Z. In this talk, we will give a combinatorial proof of this result using grid homology. The proof readily generalizes to Z coefficients because of its combinatorial nature. As corollaries, we obtain a cube of resolutions with untwisted coefficients, and an application to quasi-alternating links. If time permits, we will outline a similar skein relation for the Petkova-Vertesi tangle Floer homology (joint work with Ina Petkova).
Abstract: In this talk, I will explain that the open Gromov-Witten invariants on K3 surfaces coincide with the counting of tropical discs, which generalizes the correspondence theorem on toric manifolds of Nishinou-Siebert. In particular, this implies some existence of holomorphic discs which do not arise from standard gluing easily.
Abstract: In this talk, I will prove that all flexible Weinstein fillings of a given contact manifold have isomorphic integral cohomology. As an application, I will show that in dimension at least 5 any almost contact class that has an almost Weinstein filling has infinitely many different contact structures. Using similar methods, I will show how to construct the first known infinite family of almost symplectomorphic Weinstein domains whose contact boundaries are not contactomorphic. These results are proven by studying Reeb chords of loose Legendrians and using positive symplectic homology.
December 9, 2016: Mark McLean " Computing Symplectic Cohomology of Affine Varieties Using Spectral Sequences"
Abstract: In this talk I will construct a spectral sequence converging to symplectic cohomology of an affine variety A whose E1 page is built from strata of a smooth normal crossing compactification of A. We will use this to compute symplectic cohomology of some simple examples and then give a modest application.
Abstract: I will explain how to compute the symplectic homology differential for the complement of a Donaldson-type divisor D on a symplectic manifold X, at least when D and X are both monotone. The answer is in terms of Gromov-Witten invariants of D and of the pair (X,D). This can be thought of as a refinement of the spectral sequence in McLean's talk, in the case when the divisor is smooth (first considered by Seidel). This is joint work with Sam Lisi.
December 9, 2016: Hülya Argüz "Log-geometric invariants of degenerations with a view towards symplectic cohomology: the Tate curve"
Abstract: The talk will start with a quick overview of the use of log Gromov-Witten theory in the Gross-Siebert program in mirror symmetry. We then will discuss an algebraic geometric approach to the symplectic Fukaya category via certain stable logarithmic curves called log corals. For this, our main object of study will be the degeneration of elliptic curves, namely the Tate curve. We show the correspondence between the Lagrangian Floer theory of the general fiber of the Tate curve and the log Gromov-Witten theory of its central fiber. This works via tropical geometry on a half-space. We furthermore propose that a suitable degeneration of log corals can be used to describe the symplectic cohomology of the total space of the Tate curve minus its central fiber. The results are expected to generalise to higher dimensional Calabi-Yau manifolds. Most of this talk is based on joint work with Bernd Siebert, with general ideas based on discussions of Bernd Siebert and Mohammed Abouzaid.