Seminar on geometry, symmetry, and physics, Spring 2016
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
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
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
March 31. No talk
April 7. No talk
(Joint with Geometric Analysis) Tuesday, April 12, 3pm. Guangbo Xu,
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
study of noncommutative deformation theory. In this talk, we will
how to use contraction algebra to study the three dimensional flopping
contraction. We will show that the derived category of singularities
the subcategory of complexes support on the exceptional curve can be
reconstructed from the contraction algebra. These reconstruction
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
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
Around 2008-2010, Bridgeland and myself studied wall-crossing formulas
of rank one DT invariants in the derived category and showed that
series is a rational function. In this talk, I will update this result
to higher rank DT invariants.