SCHEDULED
TALKS FOR WINTER, 2003
April 4, 2003
Mike Roth
Queen's University
Title: "Rational Connectedness"
Abstract:
Rational Connectedness is a relatively new concept in algebraic
geometry introduced by Kollar, Miyaoka and Mori, which is proving
to be a much better (and much more geometric) substitute for the
venerable notions of rational and unirational varieties. The definition
is amazingly simple, and its strength in applications come from
this simplicity. The talk will survey some of the basic facts about
rational connectedness, as well as some new results.
Friday, March 28
Paul Feehan
Rutgers University
Title: "Seiberg-Witten invariants,
Donaldson invariants, and blow-up formulae"
Abstract:
Compact smooth 4-manifolds have two kinds of gauge-theoretic invariants,
due to Seiberg-Witten and to Donaldson, which can be used to distinguish
between different smooth structures on the same topological 4-manifold.
The invariants are conjecturally related by a formula of Edward
Witten (1994). In this lecture we shall explain how input from an
overarching gauge theory (SO(3) monopoles), the blow-up formulae
for Donaldson and Seiberg-Witten invariants, and properties of the
Donaldson and Seiberg-Witten theories can be used to prove Witten's
formula.
Friday, March 21
Eckhard Meinrenken
University of Toronto
Title: "Chern-Weil constructions in Lie theory"
Abstract:
The classical Chern-Weil theory of characteristic classescassociates
to every principal bundle over a space X a homomorphism from the ring
of invariant polynomials into the cohomology ring of X. It was observed
by Cartan that the Chern-Weil construction admits a very nice algebraic
reformulation, entering his model for the equivariant cohomology of
a space. In this talk I will review the Chern-Weil-Cartan theory,
and show how to apply these ideas in the context of Lie algebras.
Friday, March 14
Jingyi Chen
UBC
Title: " Some recent progress
on mean curvature flow"
Abstract:
The classical Chern-Weil theory of characteristic classes associates
to every principal bundle over a space X a homomorphism from the ring
of invariant polynomials into the cohomology ring of X. It was observed
by Cartan that the Chern-Weil construction admits a very nice algebraic
reformulation, entering his model for the equivariant cohomology of
a space. In this talk I will review the Chern-Weil-Cartan theory,
and show how to apply these ideas in the context of Lie algebras.
Friday, March 7
Jeff Viaclovsky
MIT
Title: "A fully nonlinear
equation on 4-manifolds with positive scalar curvature"
Abstract:
We present a conformal deformation involving a fully nonlinear equation
in dimension 4, starting with positive scalar curvature. Assuming
a certain conformal invariant is positive, one may deform from positive
scalar curvature to a stronger condition involving the Ricci tensor.
A special case of this deformation gives a more direct proof of the
result of Chang, Gursky and Yang. We also give a new conformally invariant
condition for positivity of the Paneitz operator, which allows us
to give many new examples of manifolds admitting metrics with constant
Q-curvature. This is joint work with Matt Gursky.
Friday,
February 28
Speaker: Katia
Consani
University
of Toronto
Title: "Archimedean
fibers and non-commutative geometry"
Abstract:
In Arakelov geometry a completion of an arithmetic surface is
achieved by enlarging the group of divisors by formal linear combinations
of the "closed fibers at infinity" If one enriches Arakelov's
metric structure on a compact Riemann surface of genus at least 2
by choosing a Schottky uniformization, then this extra datum may be
combined with the archimedean cohomology theory on the surface to
determine the structure of a non-commutative manifold.
Friday,
February 14
Speaker: Matheus
Grasselli
McMaster
University
Title: "A
Monte Carlo method for exponential hedging in semimartingale markets"
Abstract:
Utility based methods provide a general economically sound framework
for hedging and pricing contingent claims in markets where the traditional
paradigm of replicating portfolios cannot be applied. Mathematically,
they are a variant of the optimal investment problem in semimartigale
markets which has been solved in great generality in recent years.
We review the uniqueness and existence results for optimal hedging
portfolios for different classes of utility functions. Despite these
theoretical achievements, little has been pursued concerning the way
to construct such trading strategies when one is faced with particular
(albeit complicated) market models. For the case of an exponential
utility, we propose a Monte Carlo method which implements the choice
of the hedging portfolio on arbitrarily specified market conditions
through a learning algorithm reminiscent of the Longstaff-Schwartz
method for pricing American options.
Friday,
February 7
Alexander Tovbis
University of Central Florida
Title: "Semiclassical (zero
dispersion) limit for the focusing Nonlinear Schroedinger Equation
and
method of Riemann Hilbert Problem"
Abstract:
In this talk we discuss the semi-classical limit of the focusing Nonlinear
(cubic) Schroedinger Equation (NLS) for a certain one-parameter family
of initial conditions, which contains both soliton and pure radiation
cases. Starting with the explicit scattering data, we apply the method
of Riemann-Hilbert Problem (RHP) to find the leading order term of
the solution through the technique of inverse scattering. Error estimates
are also provided.
Friday,
January 31
Gordon Sinnamon
University of Western Ontario
Title: "The Level Function
Grows Up"
Abstract:
The level function is a tool for working with monotonicity. From its
introduction as a construction applied to bounded functions in weighted
Lebesgue spaces it has developed to become an important (non-linear)
operator defined on general rearrangement-invariant function spaces.
The level function has been used to improve Holder's inequality, represent
the dual norms of Lorentz spaces, prove Hardy inequalities, and recently
to give necessary and sufficient conditions for the boundedness of
the Fourier transform between weighted Lorentz spaces. We show that
it also plays a natural role in a certain lattice of functions, appearing
as a monotone envelope in two ways. This property enables the transfer
of monotonicity from the kernel to the weight in norm inequalities.
Friday,
January 24
M. Ram Murty
Queen's University
Title: "Ramanujan
Graphs"
Abstract:
Ramanujan graphs are k-regular graphs with certain extremal properties.
These graphs have become important in communication theory. For fixed
k, explicit construction of infinite families is only known when k-1
is a prime power and this construction uses algebraic geometry and
number theory. The talk will be a survey lecture of a very fertile
area of research that fuses number theory, graph theory, representation
theory and algebraic geometry.
Friday,
January 17
Hui
June Zhu
McMaster University
Title: "p-adic exponential
sums with a beautiful geometry"
Abstract:
In his proof of rationality of zeta function of curves (Weil Conjecture),
Dwork developed some great tools in p-adic analysis. Our work is inspired
by Dwork's study of p-adic variation of Lagrange family of elliptic
curves and also by Katz-Mazur theorem of Newton over Hodge. I will
explain the pictures and my recent work in p-adic variation of L functions
of p-adic
exponential sums.