Seminar on geometry, symmetry, and physics, Spring 2013
Mondays (usually) 1:30 pm - 2:30pm in Hill 425
Organized by Lev Borisov, Diuliu Diaconescu, and Chris Woodward
February 11 Benoit Charbonneau, St. Jerome's University in the
University of Waterloo
Title: Fake G_2 instantons, singular monopoles and pairs
SPECIAL PLACE: Serin E372.
In joint work with Jacques Hurtubise (McGill), we introduced some time
ago a relation between singular monopoles on the product of a circle
and a Riemann surface with stable pairs on the Riemann surface. In
current joint work with Spiro Karigiannis and Aaron Smith (both
uWaterloo), we exploit a similar approach to link G_2 instantons (and
more) on the product of a circle and a Calabi--Yau 3-fold and stable
pairs on the Calabi--Yau. This talk is an account of how these
stories intertwine and how we failed to answer big questions about
G_2-instantons and yet provide an interesting sandbox to test certain
ideas in higher dimensional gauge theory.
February 18 Chris Woodward, Rutgers
Title: Quantum Kirwan and quantum Martin
Abstract: I will discuss joint work with E. Gonzalez on
generalizations of the Kirwan map and Martin formula for symplectic
quotients, and applications to abelianization of solutions to quantum
differential equations for e.g. quiver varieties.
February 25 Michael McBreen, Columbia
Title: Quantum cohomology and mirror symmetry for hypertoric varieties
Abstract: Hypertoric varieties are holomorphic symplectic analogues of
toric varieties. I will describe joint work with Daniel Shenfeld
computing their quantum cohomology and producing a mirror formula for
their quantum connection, and describe potential applications to the
representation theory of the Yangian.
March 4 No seminar, conference.
March 11 Howard Nuer, Rutgers
Abstract: We discuss the construction of families of Calabi-Yau 3-folds
containing Enriques surfaces and their birational geometry. By
considering certain 1-parameter families in the boundary of our moduli
space, we find an example of a family of Calabi-Yau 3-folds exhibiting
very surprising properties with regard to mirror symmetry. The first
part of the talk is joint work with Lev Borisov, while the second is
work-in-progress with Pat Devlin.
March 18 No seminar, spring break.
March 25 P. Georgieva, Princeton
Title: Orientability in real Gromov-Witten theory.
Abstract: For a symplectic manifold M, equipped with an
one can consider the moduli space of J-holomorphic maps from a
surface to M commuting with the involutions on the domain and the
moduli spaces play an important role in real enumerative geometry and
theory, as seen in the works of J.-Y. Welschinger and J. Walcher. The
goal of this
talk is to describe what the orientability of the moduli spaces
depends on, which
is an essential ingredient in defining real Gromov-Witten type
invariants. This is
a joint work A. Zinger.
April 1, Alberto Garcia-Raboso, U. Penn.
Title: A nonabelian Hodge theorem for twisted vector bundles
April 8 A. Zinger, IAS/Stonybrook
Mirror Symmetry for Stable Quotients Invariants
We describe a mirror formula for the direct analogue of Givental's
J-function in the SQ theory. The mirror formula in the SQ theory is
remarkably similar to that in the Gromov-Witten theory, but the former
does not involve a change of variables. This suggests that the mirror
map relating the GW-invariants to the B-model of the mirror is more
reflective of the choice of curve counting theory on the A side than
of mirror symmetry. The proof of the mirror formula in the Fano case
is as in the GW-theory. On the other hand, the proof in the Calabi-Yau
case consists of showing that it is a consequence of the Fano
case. This is joint work with Y. Cooper.
April 15 No seminar.
April 22 Yi-Zhi Huang, Rutgers New Brunswick.
Meromorphic open-string vertex algebras,
nonlinear sigma models and Witten's Dirac operators on
Abstract: Mathematicians have been searching for
a construction of nonlinear sigma models for many
years. Such a construction would allow us to turn the
insights of physicists on the geometry and topology
of Riemannian manifolds into a rigorous mathematical
theory. There exist constructions of vertex operator
(super)algebras from suitable manifolds in the literature.
But unfortunately even the resulting algebras for flat
manifolds in these constructions do not agree with the
known sigma models for flat manifolds.
In this talk, I will discuss a construction of modules of meromorphic
open-string vertex algebras (suitable noncommutative generalizations
of vertex algebras) generated from functions on Riemmannian
manifolds. I believe that these modules are the basic ingredients
needed in the future full construction of nonlinear sigma models. I
will also discuss a recent construction of a Dirac-like operator on
the loop space of a Riemmannian manifold proposed first by Witten. I
will present some conjectures on the connections of the representation
theory of these meromorphic open-string vertex algebras with elliptic
genera, Calabi-Yau manifolds and the chiral de Rham complexes of
Malikov, Schechtman and Vantrob.
April 29 Sushmita Venugopalan (Tata Inst.), Introduction to
May 6 No seminar, last day of class