UNIVERSITY OF WESTERN ONTARIO
Department of Mathematics
London, Ontario N6A 5B7
Distinguished Lecture Series
The department
of mathematics is pleased to announce the 1998-1999 distinguished lecture
series . The
lecture will be presented by James
Stasheff ( North Carolina, Feb.
18, 19 ) ,
Steve Gersten (Salt
Lake City, March 15,16,17) , and Lisa Jeffrey
(Toronto,
March 29,30,31) .
...
J. Stasheff
Thursday Feb. 18, 11-12 am, Middlesex College 108
This talk will be a survey of `cohomological
physics` , a phrase that first
appeared in the
context of anomalies in gauge theory. The cohomology referred
to there
was that of differential forms ( the de Rham complex ) . Differential
forms
were
implicit in physics at least as far
back as Gauss (1833) (cf. his electro-
magnetic definition
of the linking number, and more visibly in Dirac`s magnetic
monopole
(1931), which lived in a circle bundle over the complement of the origin
in
3-space ) . Through the magnetic charge ,
characteristic classes ( and by
implication
the cohomology of Lie algebras and of Lie groups) were introduced
into physics. ...
...
.
Cohomological physics had
a major break through with
the `ghosts`
introduced by Fadeev and popov. These were incorporated into what came
to be
known as BRST cohomology , which was applied to a
variety of problems in
mathematical physics .There the ghosts were reinterpreted as generators
of the
Chevalley-Eilenberg complex for Lie algebra cohomology. ......
.
Cohomological physics also makes use of group
theoretic cohomology ,
algebraic deformation theory and especially a novel
extension of homological
algebra , combining Lie algebra cohomology with
the Koszul-Tate resolution.
Combination
of both kinds of cohomology
appeared in the Batalin-Fradkin-
vilkovisky
approach to the cohomological reduction
of constrained Poisson
algebras. ...
...
.
An analogous `odd` version
was developed in the Batalin - Vilkovisky
approach to quantizing particle lagrangians of string
field theory . A revisionist
view of
the Batalin - Vilkovisky machinery
recognizes parts of it as a
reconstruction of homological
algebra with some
powerful new ideas
undreamt of in that
discipline . An essential feature
of this and other
recent developments in cohomological physics
is the rich algbraic structure
of various generalizations
of Poisson brackets ,
including Schouten ,
Schouten-Nijenhuis , Gerstenhaber and Nijenhuis-Richardson
brackets and the
physicists` `anti-bracket` . A
particularly prominent role is
played by the
Master E'quation`.
.
.
Thursday Feb.
18, 3:30-4:30 pm ,Middlesex College 108
String theory regards particles as (tiny) one-dimentional objects in space-time
path or loops.
String Field theory deals with functions defined on the space of all
such strings
. String Field theory is multi-layered ,
often presented as involving
topology, geometry,
algebra and analysis, especially analysis in
the sence of
Riemann
surfaces . The bottom layer is `convolution algebras
of fields. The talk
will provide
interpretation of these algebraic structures from the point of view
of
operads.
This framework in algebraic topology developed for studying spaces of
the homotopy
type of (iterated)loop spaces. The topological operads involved are
moduli
spaces for Riemann surfaces with marked
points and decorations or
compactifications
as well as the more traditional ones for complex structures.
... ..
.
.
Friday Feb. 19, 11-12 am ,Middlesex College 107
Barannikov and Kontsevich construct
a formal solution of the
master
equation or Maurer-Cartan equation in a certain differential
Gerstenhaber algebra
arising from a Calabi-Yau manifold
which also admits a K\"ahler structure. The
master equation may be viewed as defining a twisting cochain . Under appropriate
circumstances , twisting cochains
may be constructed
by means of
homological perturbation theory . A standard
homological perturbation theory
construction yields the Barannikov- Kontsevich
formal solution of the master
equation, even if the underlying manifold
does NOT admit a K\"ahler structure.
Relations with
deformation quantization will be explored.
.. ....
.
Monday March 15, 3.30 pm , Middlesex College 108
This lecture is intended for a broad audience
and uses only the notions of
group and metric space to introduce the ideas of geometric group theory.
Cayley
graph and Cayley 2-complex will be discussed and illustrated with Escher
prints .
Geometric properties are introduced and Gromov`s programme for classification
of finitely generated groups by geometric (quasi-isometry
invariant ) properties
is explained. .
.
Tuesday March 16, 3.30 pm , Middlesex College 108
Dehn`s solution of the word problem
for surface groups when properly
interpreted leads to the general notion of hyperbolic groups.The Dehn function
of a finite presentation is introduced and related to the word problem.
Examples
of isoperimetric functions are given in "Dehn`s zoo."..
....
.
.
.
Wednesday March 17, 3.30 pm , Middlesex College 108
Fillings of circuits in the Cayley graph
are studied , first by surfaces of
genus 0 ("Van Kampen diagrams")and then by orientable surfaces of
arbitrary
genus .The integral and real filling norms on 1-cycles of the
Cayley 2-complex
are introduced and related . The l _\{infty}-cohomology
is introduced in order
to characterise hyperbolic groups by a cohomological
vanishing .Mineyev`s
vanishing theorem is discussed in addition to some open problems. .
.
.
March 29,30,31; 3.30 pm , Middlesex College 108
The idea of Hamiltonian
flow on a symplectic manifold
has its roots in
Hamilton`s equations , which govern trajectory
of a particle in phase space ( the ..
space parametrizing
coordinates and momenta of a classical particle). .
.
A fundamental idea in theoretical physics
is that to every symmetry in a physical
system ( such as a group action ) , there
is an associated conserved quantity:
invariance under translation corresponds to
conservation of linear momentum ,
rotational symmetry corresponds to conservation of angular
momentum and so on
, and these momenta are
functions on the phase
space . The mathematical
formulation of this idea is the idea of the moment map
associated to a group action
on a symplectic manifold the group action is obtained from
the Hamiltonian flow of
the moment map .These lectures will describe some basic features
of moment maps
associated to Hamiltonian group
actions , and some recent results
about the
geometry and topology of symplectic manifolds which have such group actions.
.
.