Informal Mathematical Physics Seminar
organized by Igor Krichever and Andrei Okounkov
Mondays, 5:30, Room 507
Note a special lecture
course on D-modules by A.
Braverman, on Fridays 12-1:30, Math 507
Schedule of talks for Fall 2014:
A note to
the speakers: this is an
informal
seminar, meaning that the talks are longer than usual (1:30)
and are expected to include a good introduction to the subject as well
as a maximally accessible (i.e. minimally general & minimally
technical) discussion of the main result. The bulk of the audience is
typically formed by beginning graduate students. Blackboard talks are
are particularly encouraged.
Abstracts
Sept 2
The length of the boundary of a J-holomorphic curve with Lagrangian
boundary conditions is dominated by a constant times its area. This
inequality is useful for showing the adic convergence of power series
arising from A-infinity algebras associated with Lagrangian
submanifolds. It gives rise to an invariant of Lagrangian submanifolds
called the optimal isoperimetric constant. And it proves the compactness
of certain moduli of J-holomorphic curves with Lagrangian boundary
conditions in toric Calabi-Yau threefolds. This is joint work with Y.
Groman.
Sept 15
Let $t$ be a complex number, and $V$ a complex vector space. I will
explain how to define the tensor power $V^{\otimes t}$. This can be done
canonically if we fix a nonzero vector in $V$. However, the result is
not a vector space but rather an (ind-)object in the tensor category
${\rm Rep}(S_t)$, defined by P. Deligne as an interpolation of the
representation category of the symmetric group $S_n$ to complex values
of $n$. This category is semisimple abelian for $t\notin \bold Z_+$, but
only Karoubian (=idempotent complete) for $t\in \bold Z_+$, in which
case it projects onto the usual representation category of $S_n$. I will
define the category ${\rm Rep}(S_t)$, and explain how Schur-Weyl duality
works in this category when $t \notin \bold Z_+$. If time permits, I
will explain what happens at integer $t$, which is more subtle and is
due to Inna Entova-Aizenbud.
Oct 6
In this talk, I will build up the equivalence between the deformation
quantization on a symplectic manifold and the perburbative BV
quantization of a Chern-Simons type theory on its loop space. This
allows us to borrow symmetries of deformation quantization to analyze
the correlation functions of quantum observables. As an application, I
will show that the homotopy of a rescaling symmetry within our QFT model
leads to a simple proof of algebraic index theorem. This is joint work
with Qin Li.
Oct 20
The tautological ring of the moduli space of stable curves is a subring
of the Chow ring consisting of the cycles that arise naturally in
geometry through forgetful and gluing morphisms. Relations in this ring
can be pulled back to give recursion relations in Gromov-Witten theory.
I will introduce the notion of a "primitive" tautological relation, one
that cannot be derived from simpler relations using certain basic
operations. We know very little about primitive tautological relations
in genus 2 or greater, but I will describe one interesting infinite
family of them.
Oct 27
We describe a different proof of the theorem of Hausel, Letellier, and
Rodriguez-Villages on counting indecomposable quiver representations
over a finite field, using character sheaves. If time permits, we show
how similar ideas can be used for counting indecomposable vector bundles
on an algebraic curve.
Nov 17
In this talk, I will give the restriction formulas for stable basis in
T^*G/B, and generalize them to T^*G/P case. This allows us to identify
the quantum multiplication of a divisor in equivariant quantum
cohomology of T^*G/P.
Nov 24
This is going to be an introductory talk where I would like to discuss
different properties of
noncommutative schemes such as smoothness, regularity and properness. I
am also going to talk about different examples of quasi-phantoms and
phantoms.