Seminar on geometry, symmetry, and physics


(aka mirror symmetry/related topics)

Thursdays (usually) 1:30pm - 2:30 or 3pm in SEC 218.


Organized by Lev Borisov, Diuliu Diaconescu, and Chris Woodward

Thursday, 9/24, Rahul Pandharipande, Princeton in SEC 218.

Title: Quivers, curves, and symplectomorphisms of the algebraic 2-torus.

Abs. I will talk about the relationship between Euler chars of moduli\\ spaces of quivers, curve counts on toric surfaces, and commutators in the symplectomorphism group of the algebraic 2-torus. I will cover results of Reineke, Gross-Siebert, and others. (Joint with String Group Meeting)

Friday, 10/2 Lev Borisov, Rutgers, 11:45 in Hill 425

"In search of families of dg-algebras related to resolutions of Gorenstein toric singularities" Abstract: A Gorenstein toric singularity can be described by simple combinatorial data, namely a convex polytope $P$ in ${bf Z}^n$ with integer vertices. Different triangulations of $P$ with vertices given by integer points of $P$ give rise to different resolutions of the singularity. It has been shown that bounded derived categories of coherent sheaves on these resolutions are equivalent. It is reasonable to expect that there is in fact a continuous family of triangulated categories that includes these categories as its limit points. This is very much work in progress, and the main questions are still wide open. It is my hope that by bringing this problem to your attention I can inspire someone to find such construction. (Joint with quantum math/ Lie theory/algebra)

Thursday, 10/8 Katrin Wehrheim, MIT, 1:30pm in SEC 218.

Calculations of Floer homology by reduction":

I will give some examples of calculating monotone Floer homology from a general strip shrinking isomorphism in quilted Floer homology (for sequences of Lagrangian correspondences). Examples include the Clifford torus in CP^n (previously known by Cho) and nondisplaceable T^{n-k}\times S^{2k-1} in CP^n\times CP^{k-1}. Moreover, the bijection of trajectory moduli spaces can be somewhat generalized to multiply covered compositions of correspondences, yielding e.g. calculations of the Floer homology between Clifford tori and RP^n in CP^n (confirming work by Allston). Finally, "figure eight" bubbling obstructions can be understood explicitly. Work is in progress on overcoming these for the Chekanov/Polterovich torus in S^2\times S^2; using symmetries and twisted coefficients.

Thursday, 10/15 Emanuel Diaconescu, Rutgers

Title: ADHM sheaves, wallcrossing and local stable pair invariants

Abstract: The local stable pair theory of a curve will be constructed in terms of ADHM sheaves satisfying ceratin stability conditions. Wallcrossing formulas for the resulting invariants will be derived, and applications to local Gopakumar-Vafa theory will also be presented,

Thursday, 10/22 Yaron Ostrover, IAS

Title: Algebraic properties of the quantum homology.

Abstract: In this talk we discuss certain algebraic properties of the quantum homology algebra of toric Fano manifolds. In particular, we describe an easily-verified sufficient condition for the semi-simplicity of the quantum homology. (This is a joint work with Ilya Tyomkin.)

Thursday, 10/29 Ron Donagi, U Penn

Approx title: F-theory and its compactifications Abstract: F-theory is a "12 dimensional variant of string theory" whose study has seen great progress in the past year or two. This will be an introduction for mathematicians to F-theory and its global and local compactifications, including some of the recent progress towards F-theory based phenomenology. I will discuss the issue of local versus global in F-theory (and strings), and explore connections to the geometry of del Pezzo surfaces, Higgs bundles, and Noether-Lefschetz loci.

Thursday, 11/5 Bohan Fang, Northwestern

Title: Coherent-constructible correspondence for toric varieties and stacks

Abstract: This is a talk on joint works with Chiu-Chu Liu, David Treumann and Eric Zaslow. I will describe a coherent-constructible correspondence for toric varieties motivated by homological mirror symmetry and T-duality. To each ample line bundle one can assign a polytope-shaped constructible sheaf on a real vector space. This assignment turns out to be a tensorial quasi-equivalence. The correspondence can be extended to toric stacks, using Borisov-Chen-Smith's definition through stacky fans.

Thursday, 11/12 Davesh Maulik, MIT

Title: Gromov-Witten theory of K3 surfaces

Abstract: In this talk, we will survey some recent progress in understanding the Gromov-Witten theory of K3 surfaces. In particular, we will prove and exploit relations with sheaf theory to prove some old conjectures in the subject (joint with R. Pandharipande and R. Thomas)

Thursday, 11/19 no seminar

Thursday, 11/26 no seminar (Thanksgiving)

Thursday, 12/3 no seminar

Thursday 12/10 Chris Woodward, Rutgers

Title: Gauged Gromov-Witten theory and the mirror map

Abstract: I will explain how mirror theorems of Givental etc. are a special case of "quantum non-abelian localization" relating the Gromov-Witten invariants of a symplectic quotient with the gauged Gromov-Witten invariants of the action. In particular, the "mirror map" is generalized to a "morphism of CohFT"s which counts "affine vortices". Applications to abelianization conjecture of Hori-Vafa, Bertam-Ciocan-Fontanine-Kim etc. will be discussed. This is joint in progress with Ziltener, Gonzalez, Ott, and Venugopalan. Thursday 12/10 Lev Borisov, Rutgers

Tuesday, 2/9 Melissa Liu, Columbia

Thursday, 2/??? Mohammed Abouzaid, MIT