The SGGT seminar meets on Fridays in Math 407 from 11:00 am to 12:00 pm, unless noted otherwise (in red).

Previous semesters: Fall 2023, Spring 2023, Fall 2022, Spring 2020, Fall 2019, Spring 2019, Fall 2018, Spring 2018, Fall 2017, Spring 2017, Fall 2016, 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.

Schedule


Date Speaker Title
Jan 19, 11:00 am
José Simental (UNAM)
Cluster structure on braid varieties
Jan 26, 11:00 am
Alex Xu (Columbia)
The Seiberg-Witten Equations and Einstein Metrics on Finite Volume 4-Manifolds with Asymptotically Hyperbolic Ends
Feb 02, 11:00 am
Chris Woodward (Rutgers)
Tropical disk counting in almost toric manifolds
Feb 09, 11:00 am
Yi Wang (Purdue)
A simple chain model of loop spaces and application to the study of Lagrangian submanifolds
Feb 16, 11:00 am
Spencer Cattalani (Stony Brook)
Complex Cycles and Symplectic Topology
Feb 23, 11:00 am
Deeparaj Bhat (MIT)
Surgery exact triangles in instanton theory
Mar 01, 11:00 am
Joseph Breen (U Iowa)
The Giroux correspondence in arbitrary dimensions
Mar 08, 11:00 am
Daniel Pomerleano (UMass Boston)
TBA
Mar 22, 11:00 am
Oliver Edtmair (UC Berkeley)
TBA
Mar 29, 11:00 am
Thomas Guidoni (Sorbonne)
TBA
Apr 05, 11:00 am
Roman Krutowski (UCLA)
TBA
Apr 12, 11:00 am
Eric Zaslow (Northwestern)
TBA
Apr 19, 11:00 am
Mohan Swaminathan (Stanford)
TBA
Apr 26, 11:00 am
Jiakai Li (Harvard)
TBA

 

Abstracts

Jan 19: José Simental (UNAM) "Cluster structure on braid varieties"

Abstract: The braid varieties of the title are smooth affine algebraic varieties that naturally generalize important Lie-theoretic varieties such as double Bruhat cells, positroid varieties and, more generally, open Richardson varieties on the flag variety. Thanks to works of Kálmán and Casals-Ng, they also appear as the augmentation variety of a class of (-1)-closures of positive braids. In this talk, based on joint work with Roger Casals, Eugene Gorsky, Mikhail Gorsky, Ian Le and Linhui Shen, I will explain how to give the coordinate algebra of a braid variety the structure of a Fomin-Zelevinsky cluster algebra. The main combinatorial input is that of a weave, a colored graph that encodes positions of flags, and defines an open torus inside the braid variety. No prior knowledge of cluster algebras will be assumed.

Jan 26: Alex Xu (Columbia) "The Seiberg-Witten Equations and Einstein Metrics on Finite Volume 4-Manifolds with Asymptotically Hyperbolic Ends"

Abstract: One of the features of the Seiberg-Witten equations is that existence of irreducible solutions give rise to apriori estimates for the total scalar curvature. This was used by LeBrun in the late 90s when he constructed the first examples of closed 4-manifolds that satisfy the strict Hitchin-Thorpe inequality yet do not admit any Einstein metrics. In this talk, I'll describe a method of constructing irreducible solutions to the Seiberg-Witten equations on certain finite volume 4-manifolds with asymptotically hyperbolic $T^3$ ends. These 4-manifolds arise as complements of smoothly embedded tori with zero self-intersection in an ambient closed 4-manifold. As an application, this allows us to construct infinitely many examples of finite volume 4-manifolds with $T^3$ ends that do not admit any asymptotically hyperbolic Einstein metrics yet satisfy a strict logarithmic version of the Hitchin-Thorpe inequality due to Dai-Wei. By extending the construction to the Pin^-(2) monopoles due to Nakamura, we are able to construct examples with signature 0.

Feb 02: Chris Woodward (Rutgers) "Tropical disk counting in almost toric manifolds"

Abstract: In joint work with Sushmita Venugopalan (Chennai), we generalize Mikhalkin's tropical curve counting formula to the case of disks bounding tropical Lagrangians in almost toric manifolds, such as moment fibers or tropical Lagrangians in the ADE Lagrangian configurations in del Pezzo surfaces. This allows, for example, the direct computation of disk potentials of these Lagrangian tori and spheres and allows one to identify split-generators for the Fukaya category of del Pezzo's. (For tori in del Pezzo surfaces, similar results appear after an algebro-geometric detour in Sam Bardwell-Evans, Man-Wai Mandy Cheung, Hansol Hong, Yu-Shen Lin. For cubic surfaces, split generators appear in Sheridan's thesis. A formula of this type was widely expected, as in, for example, Vianna's thesis.)

Feb 09: Yi Wang (Purdue) "A simple chain model of loop spaces and application to the study of Lagrangian submanifolds"

Abstract: It was a proposal of Fukaya that one can use chain level string topology to study a Lagrangian submanifold L, as pseudoholomorphic disks induce loops in L and bubbling of disks is governed by string topology operations. A rigorous construction of chain level loop bracket was worked out by Irie, whose applications include a proof of Audin's conjecture. In this talk, I will describe a simplification of Irie's chain model of loop spaces, which greatly simplifies some technicalities that arise in related work when applying the chain model to symplectic topology, and which is also interesting from a purely algebraic topological point of view. Then I will discuss a variant of the chain model in the S^1-equivariant context, which allows one to lift the Fukaya A_infinity of L to a Maurer-Cartan element in the dg Lie algebra of cyclic invariant chains on the free loop space of L. Applications include the nonexistence of Lagrangian submanifolds in C^{2n} that are homotopy equivalent to a manifold with negative sectional curvature (this slightly generalizes a theorem attributed to Viterbo, under additional assumptions). Part of the work is joint with Irie and Rivera.

Feb 16: Spencer Cattalani (Stony Brook) "Complex Cycles and Symplectic Topology"

Abstract: Among all almost complex manifolds, those which are tamed by symplectic forms are particularly well studied. What geometric properties characterize this class of manifolds? That is, given an almost complex manifold, how can one tell whether it is tamed by a symplectic form? By a 1976 result of D. Sullivan, this question can be answered by studying complex cycles. I will explain what complex cycles are and their role in two recent results, which confirm speculations posed by M. Gromov in 2000 and 1985, respectively. The first is that an almost complex manifold admits a taming symplectic structure if and only if it satisfies a certain bound on the areas of coarsely holomorphic curves. The second is that an almost complex 4-manifold which has many pseudoholomorphic curves admits a taming symplectic structure. This leads to an almost complex analogue of D. McDuff’s classification of rational symplectic 4-manifolds.

 

Feb 23: Deeparaj Bhat (MIT) "Surgery Exact Triangles in Instanton Theory"

We prove an exact triangle relating the knot instanton homology to the instanton homology of surgeries along the knot. As the knot instanton homology is computable in many instances, this sheds some light on the instanton homology of closed 3-manifolds. We illustrate this with computations in the case of some surgeries on the trefoil. In particular, we show the Poincaré homology sphere is not an instanton L-space (with Z/2 coefficients), in contrast with Heegaard Floer and monopole Floer theories. Finally, we sketch the proof of the triangle inspired by the Atiyah-Floer conjecture and results from symplectic geometry.

 

Mar 1: Joseph Breen (Iowa) " The Giroux correspondence in arbitrary dimensions"

The Giroux correspondence between contact structures and open book decompositions is a cornerstone of 3-dimensional contact topology. While a partial correspondence was previously known in higher dimensions, the underlying technology available at the time was completely different from that of the 3-dimensional theory. In this talk, I will discuss recent joint work with Ko Honda and Yang Huang on extending the statement and technology of the 3-dimensional correspondence to all dimensions.