| Date, location |
Speaker |
Title |
| Fri, Jan 16 |
Ben Davis (Saint Mary's College) |
Linearization of Poisson manifolds
on neighborhoods of symplectic leaves |
| 2:30-3:30pm |
| 104 Osmond |
abstract |
We will explain Vorobjev's model for the linearization of
a Poisson manifold on a neighborhood of a symplectic leaf of arbitrary dimension.
We will explore the possiblity of using Vorobjev linearization as a means
of generalizing the Poisson linearization results of Weinstein, Conn, and
Crainic & Fernandes to neighborhoods of arbitrary symplectic leaves. |
|
| Fri, Jan 23 |
Emma Previato
(Boston University) |
Zooming in on the Hitchin system
in genus 2 |
| 2:30-3:30pm |
| 104 Osmond |
| abstract |
The Hitchin system of the title is an algebraically completely
integrable system whose
integral manifolds are Prymians of dimension $3g-3$, defined on the holomorphic-symplectic
manifold ${\cal T}^*{\cal SU}_X (2,\xi)$, the cotangent bundle to the moduli
space of vector bundles of rank 2 and fixed odd determinant $\xi$, over
a Riemann surface $X$ of genus $g>1$. Further work by Hitchin (1990)
implemented geometric quantization and provided a link of the system with
the KZ (Knizhnik-Zamolodchikov) equations when the Riemann surface varies,
by showing that an analog of the rank-1 heat equation holds over these moduli
spaces.
These results have not been rendered in terms of explicit functions except
in genus 2, and in fact only in an extended sense (even-determinant case),
nor have I been able to identify a source where the interpretation of all
the mentioned aspects is given in a unified way.
In this talk, which is intended to be introductory and expository, I will
detail the mentioned features and their interconnection; more precisely:
(1) explicit Hamiltonians of the Hitchin system in genus 2, even-determinant
case (joint work with B. van Geemen, and work by K. Gawedzky and P. Trang-Ngoc-Bich)
including a `strange duality' of projective-geometric nature;
(2) interpretation of the integrals for the genus 2, odd-determinant case
(W.M. Oxbury, unpublished D.Phil. thesis, Oxford, 1987);
(3) geometric quantization in genus 2, even-determinant case (B. van Geemen
and A. de Jong), as well as a(nother?) heat equation for hyperelliptic Riemann
surfaces of any genus, even-determinant case;
(4) KZ equations in genus 2 (K. Gaw\c{e}dzky and P. Trang-Ngoc-Bich); (5)
ad hoc reduction
of Hitchin to Neumann (rational-parameter Lax equations) in genus 2, even
determinant
(K. Gawedzky and P. Trang-Ngoc-Bich);
(6) Lax representation for the
Hitchin system in terms of Tyurin parameters (I.M. Krichever);
(7) singular cases if there is time (N. Nekrasov; B. Enriques and V. Rubtsov).
This background will serve to formulate a current research program,
articulated as follows:
(1) description of ${\cal SU}_X(2,L)$ for $X$ hyperelliptic, $L$ even/odd
(S. Ramanan, A. Beauville) and of ${\cal SU}_X(2,L)$ for $X$ non-hyperelliptic
of genus 3,
$L$ even (Coble);
(2) construction of a geometric-quantization coordinate space in the cases
given in (1), and of the class of functions for which the generalized heat
equations are to be found.
|
| Fri, Jan 30 |
Aissa Wade (Penn State) |
Complex Dirac structures and
pure spinors |
| 2:30-3:30pm |
| 104 Osmond |
abstract |
In this talk, I will present a survey of new developments
in the theory of Dirac structures. The emphasis will be on results recently
obtained by N. Hitchin and M. Gualtieri. |
| |
| Thu, Feb 5 |
Gregory Bell (Penn State) |
TBA |
| 5:05-6:05pm |
| TBA |
| abstract |
|
| Fri, Feb 13 |
Danny Stevenson (University of
Adelaide) |
Bundle Gerbes and String Structures |
| 2:30-3:30pm |
| 104 Osmond |
abstract |
Suppose that P --> M is a principal bundle with strcture
group the loop group LG (LG is the space of smooth maps from the circle
to G a Lie group). The string class of P is the class in H^3(M;Z) obstructing
the existence of a lift of the structure group of P to LG^, the Kac-Moody
group. We shall show how, using the geometry of bundle gerbes, one can write
a closed 3-form on M representing the image of the string class of P in
real cohomology. We shall also describe a construction of the group LG^. |
| |
| Fri, Feb 20 |
Hassene Siby
(UNC-Chapel Hill & Univ. Montpellier 2)
|
Symplectic double extension |
| 2:30-3:30pm |
| 104 Osmond |
| abstract |
A symplectic Lie group is a Lie group G together with a left
invariant symplectic 2-form $Omega$. Every symplectic Lie group admits a
natural affine structure determined by its symplectic 2-form. The symplectic
double extension gives a procedure for constructing new symplectic Lie groups
starting from two ones. In this talk, we will mainly focus on certain symplectic
Lie groups having an exact symplectic 2-form that can be derived by symplectic
reduction or double extension. We will show that these Lie groups carry
two transversal invariant Lagrangian foliations with affine and closed leaves.
|
| Fri, Feb 27 |
Hanno Sahlmann (PSU gravity)
|
Introduction to loop quantum
gravity |
| 2:30-3:30pm |
| 104 Osmond |
abstract |
Loop quantum gravity is an approach to the quantization of
general relativity starting from a formulation in terms of connections instead
of metrics. It rests on a solid mathematical framework featuring a diffeomorphism
invariant functional measure on connections and a quantization of 3-geometries.
In this talk, I will give an introduction to both physical and
mathematical aspects this formalism. I will also try to hint at interesting
mathematical questions regarding extensions of this formalism that are largely
unexplored. |
| |
| Fri, March 5 |
Jonathan Engle (PSU gravity) |
Black hole entropy calculation
in loop quantum gravity |
| 2:30-3:30pm |
| 104 Osmond |
| abstract |
In the early 70's, classical and semiclassical considerations
of Bekenstein, Hawking, et al. led to the belief that the entropy of a black
hole should be proportional to its horizon area.
I will present a derivation of this result for 3+1 dimensional black holes
in the framework of loop quantum gravity. The derivation features a quantization
of general relativity on a
manifold with an inner boundary, with Chern-Simons "edge states"
living on this inner boundary. |
| Fri, Mar 12 |
Spring Break |
|
| 2:30-3:30pm |
| 104 Osmond |
| abstract |
|
| Fri, Mar 19 |
Murat Gunyadin (PSU gravity) |
Conformal and Quasi-conformal
Realizations of Exceptional Groups and
their minimal unitary representations |
| 2:30-3:30pm |
| 104 Osmond |
abstract |
I review some novel realizations of noncompact exceptional
groups as quasiconformal groups, which reduce to the known conformal realizations
by truncation. The "quantization" of the geometric quasiconformal
actions lead naturally to the minimal unitary representations of the corresponding
noncompact groups. Emphasis will be put on different noncompact real forms
of the largest exceptional group E_8. |
| |
| Fri, Mar 26 |
Alejandro Perez
(PSU gravity)
|
Invariants of three manifolds
and loop quantum gravity in three dimensions |
| 2:30-3:30pm |
| 104 Osmond |
| abstract |
I will show how the state sum invariants of three manifolds
(with boundaries) are recovered in canonical loop quantum gravity in three
dimensions in the definition of the physical scalar product of the theory.
|
| Fri, Apr 9 |
Jean Paul Dufour (Universite
de Montpellier) |
Semi-local normal forms for Poisson
structures |
| 2:30-3:30pm |
| 104 Osmond |
| abstract |
We will present Vorobjev method for describing the neighborhood
of a symplectic leaf in a Poisson manifold. We will focus on his "homotopy"
theorem which gives conditions under which two Poisson structures are isomorphic
in a neighborhood of a common symplectic leaf. We will give some applications.
In particular, we will explain how this gives a generalization of the Darboux-Weinstein
splitting theorem for Poisson structures. |
| Fri, Apr 9 |
Valentin Ovsienko (Universite
de Montpellier) |
Multi-parameter deformations
of Lie algebras and their homomorphisms |
| 3:40-4:30pm |
| 104 Osmond |
| abstract |
The deformation theory of Lie/associative algebras and their
homomorphisms was developed in 60'-70' by Gersternhaber and Nijenhuis-Richardson.
Traditionally, this theory studies one-parameter deformations. A more recent
viewpoint (Fialowski, Fuchs in the end of 90') deals with multi-parameter
deformations. The space of parameters generates a commutative algebra intrinsically
related with the initial Lie algebra. I will present some recent developments
of this approach with some examples and application. |
| Wed, Apr 14 |
Robert Coquereaux (CPT, Marseille) |
Twisted partition functions for
Wn minimal models in conformal field
theory and quantum symmetries of higher Coxeter-Dynkin diagrams |
| 2:30-3:30pm |
| 104 Osmond |
| abstract |
We first review the relations between quantum symmetries of
graphs and partition functions of affine conformal models in quantum field
theory. Usual minimal models can be defined in terms of representation theory
for the Virasoro algebra and are associated with pairs of ADE diagrams.
We then consider more general minimal models defined in terms of representation
theory for the so called Wn algebras (actually Casimir algebras) generalizing
the Virasoro algebra, and show that the corresponding partition functions
-- twisted or not -- are related to quantum symmetries of pairs of graphs
belonging to generalized
Coxeter-Dynkin systems. Our discussion is strongly motivated by the work
of A. Ocneanu on quantum subgroups of Lie groups. |
| Fri, Apr 23 |
Eric R Shape (Urbana-Champaign)
|
Some mathematical aspects of
D-branes
|
| 2:30-3:30pm |
| 104 Osmond |
| abstract |
D-branes are physical objects in string theory that can often
be mathematically modelled by coherent sheaves and, in certain circumstances,
derived categories. Such models are a part of Kontsevich's ``homological
mirror symmetry'' program, an attempt to understand certain physical dualities
between complex Kahler manifolds with trivial canonical bundle.
In this talk we shall review D-branes, and discuss how they can be modelled
with sheaves and derived categories. In particular, such models make predictions
for physical properties of D-branes, which we will discuss and check.
|
| Fri, Apr 30 |
Geoff Mason (UC Santa Cruz) |
The bosonic string and arithmetic
at genus 2 |
| 2:30-3:30pm |
| 104 Osmond |
| abstract |
A requirement of string theory/conformal field theory
is that it should 'work' at all genera g, though little is known (with
mathematical rigor) beyond g = 1. We discuss recent ideas concerned with
the elucidation of this issue in the case of the chiral bosonic string
and lattice compactifications at g = 2. In particular, we show how these
theories are related to Riemann surfaces and Siegel modular forms at g
= 2. |