Seminar on geometry, symmetry, and physics, Spring 2016

(aka mirror symmetry/related topics)

Thursdays (usually) 12:30-1:30 pm in Hill 525

Organized by Lev Borisov, Emanuel Diaconescu, and Chris Woodward

Thursday Feb 25 at 12:30 in Hill 525.

Speaker: Artan Sheshmani, Ohio State and MIT

Title: On proof of S-duality modularity conjecture over compact Calabi-Yau threefolds

I will talk about an algebraic-geometric proof of the S-duality conjecture in superstring theory, made formerly by physicists Gaiotto, Strominger, Yin, regarding the modularity of DT invariants of sheaves supported on hyperplane sections of the quintic Calabi-Yau threefold. Together with Amin Gholampour we use degeneration and localization techniques to reduce the threefold theory to a certain intersection theory over the relative Hilbert scheme of points on surfaces and then prove modularity; More precisely, we have proven that the generating series, associated to the top intersection numbers of the Hilbert scheme of points, relative to an effective divisor, on a smooth quasi-projective surface is a modular form. This is a generalization of the result of Okounkov-Carlsson, where they used representation theory and the machinery of vertex operators to prove this statement for absolute Hilbert schemes. These intersection numbers eventually, together with the generating series of Noether-Lefschetz numbers as I will explain, will provide the ingredients to achieve an algebraic-geometric proof of S-duality modularity conjecture. Our work is based on our earlier results with Richard Thomas and Yukinobu Toda, which I will also discuss as further ingredients, needed for the final proof.

Thursday Mar 3 at 12:30 in Hill 525.

Speaker: Lev Borisov, Rutgers

Title: Combinatorics of Clifford double mirrors

Abstract: In a joint paper with Zhan Li we discuss the toric geometry that underlies the construction of Kuznetsov of Clifford noncommutative varieties which are derived equivalent to complete intersections of quadrics in projective spaces. Better understanding of the toric underpinnings allows us to expand the construction to new examples.

Thursday Mar 10 at 12:30 in Hill 525.

Speaker: Pavel Putrov (IAS)

Title: Fivebranes and 3-manifold homology

Abstract: In my talk I will describe how string/M- theory provides a unified view on various homological invariants of 3-manifolds. A well known example of such an invariant is monopole Floer homology. I will also discuss a possibility to define/compute a 3-manifold analog of Khovanov-Rozansky link homology which categorifies Chern-Simons partition function (a.k.a. WRT invariant).

March 17. Spring Break.

March 24. Pablo Solis, Caltech.

Voronoi Tilings and Loop Groups

I would like to describe a partial compactification of the loop group LT of a torus. All the ingredients are infinite dimensional but the final result is essentially described by a finite dimensional toric variety. In the case of T= C^* the compactification recovers the Tate curve which has a central fiber which is an infinite chain of projective lines which is closely related to the moduli of line bundles on a genus 0 nodal curve. A similar modular interpretation is available for higher rank tori. It seems likely that there is also a connection with Aleexev and Nakamura's work on degenerations of Abelian varieties.

March 31. No talk

April 7. No talk

(Joint with Geometric Analysis) Tuesday, April 12, 3pm. Guangbo Xu, Princeton.

Thursday, April 14, 12:30pm, David Duncan, McMaster.

April 21. Zheng Hua (Hong Kong).

Title: Contraction algebra and invariants associated to three dimensional flopping contraction

Abstract: The contraction algebra is defined by Donovan and Wemyss in the study of noncommutative deformation theory. In this talk, we will explain how to use contraction algebra to study the three dimensional flopping contraction. We will show that the derived category of singularities and the subcategory of complexes support on the exceptional curve can be reconstructed from the contraction algebra. These reconstruction theorems suggest that the contraction algebra can be viewed as a noncommutative analogue of the Milnor ring of hyper surface singularity. We will also explain how to recover the genus 0 Gopakumar-Vafa invariants from the contraction algebra. This talk is based on a joint work with Yukinobu Toda:arxiv 1601.04881.

May 12. Y. Toda, 12:30 room 525 (backup Serin 372)

Title: Rationality in higher rank Donaldson-Thomas theory

Abstract: The Donaldson-Thomas invariants count stable coherent sheaves on Calabi-Yau 3-folds, and their rank one theory is related to Gromov-Witten invariants. Around 2008-2010, Bridgeland and myself studied wall-crossing formulas of rank one DT invariants in the derived category and showed that their generating series is a rational function. In this talk, I will update this result to higher rank DT invariants.