- STEVE AWODEY, Carnegie Mellon University
Algebraic Set Theory
Since its introduction by A. Joyal and I. Moerdijk in 1995, algebraic
set theory has proven to be a flexible and powerful framework for
constructing models of various classical and constructive set theories
of a new and fascinating kind. This introductory survey outlines the
basic theory and indicates some current research advances.
- JOHN CONWAY, Princeton
- NICOLE EL KAROUI, École Polytechnique, Paris, France
Derivatives Market: Recent developments in pricing, hedging
and measuring market risk exposure
In 1973, Black, Scholes and Merton introduced a revolution in the
market risk industry, by stating that the price (for the seller) of a
derivatives product is the cost of the hedging strategy. Based on a
simple framework they deduced the famous Black-Scholes formula and
the associated Delta-hedging strategy.
More sophisticated framework are now used, and a part of the market
risk has been transformed into a model risk, given there exists a
large family of tractable models allowing to recover observable market
data (financial products prices). Risk managers are daily faced with
this model risk, in particular in the pricing of exotic products. How
to measure it is a challenge of the daily risk management.
Moreover, in more integrated point of view, market authorities now
require financial institutions to compute their daily global exposure
(Value at Risk) via their own "internal" models. Motivated by this
challenge, academic and risk-managers are debating the "best
concept" of risk measure (Delbaen et al. ,
Foellmer and Schied ). The dual representation of these
convex functionals yields to a nice interpretation in terms of market
Best adapted than utility maximization criterium, this new tool allows
us to develop an unified point of view about pricing and hedging in
incomplete markets, including model risk. Classical problems as
optimal risk transfer or optimal hedging are studied in this new
context, using inf-convolution technics in static or dynamic
P. Artzner, F. Delbaen, J. M. Eber and D. Heath,
Coherent Measures of Risk.
Math. Finance 9(1999), 203-228.
P. Barrieu and N. El Karoui,
Pricing via minimization of risk measures.
In: Paris-Princeton Lectures, to appear, 2006.
F. Bellini and M. Frittelli,
On the Existence of Minimax Martingale Measures.
Math. Finance 12(2002), 1-21.
H. Föllmer and A. Schied,
Stochastic finance: an introduction in discrete time.
de Gruyter Studies in Mathematics 27, Walter de Gruyter &
Co., Berlin, 2002.
M. Musiela and T. Zariphopoulou,
An Example of Indifference Prices under Exponential
Finance Stoch., to appear, 2004.
- NIGEL KALTON, University of Missouri
Extensions of Banach spaces and their applications
An extension of a Banach space X by a Banach space Y is a short
exact sequence 0® Y® Z® X® 0. We discuss two basic problems
concerning extensions of Banach spaces which were solved in the
1970's. Our main goal is to show how the solution of these problems
has led over the last 30 years to the development of a general theory
which has found links with harmonic analysis, operator theory and
- ALEXANDER S. KECHRIS, California Institute of Technology
Logic, Ramsey theory and topological dynamics
In this talk, I will discuss some recently discovered interactions
between topological dynamics, concerning the computation of universal
minimal flows and extreme amenability, the Fraïssé theory of
amalgamation classes and homogeneous structures, and finite Ramsey
- LÁSZLÓ LOVÁSZ, Microsoft
Very large graphs
There are many huge graphs whose structure we want to understand, from the
internet to the human brain. What kind of questions are meaningful about
these graphs? When should we say that two such graphs are similar? How can
we äpproximate" such graphs, either by a much smaller graphs or by a
continuous object, so that important properties are not lost?
These questions have a rather complete answer in the case of dense graphs,
and partial answers for graphs with bounded degrees.
This is a summary of joint work with Jennifer Chayes, Christian Borgs,
Vera Sos, Balazs Szegedy and Katalin Vesztergombi.
- DAVE MARKER, University of Illinois at Chicago
Model Theory and Exponentiation
In the 90s model theoretic methods were used by Wilkie to show that
sets defined in the real field with exponentiation have many of the
good geometric and topological properties of real algebraic varieties.
For example, any such set has only finitely many connected components.
Complex exponentiation has a very different flavor. The definablility
of the integers leads to pathologies, but there is still some hope for
a reasonable theory of definable sets. In this lecture I will review
some of the older work on the real field and discuss Zilber's program
for understanding complex exponentiation.