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Sergio Albeverio
(University of Bonn)
Title: Stochastic FitzHugh-Nagumo equations, asymptotic
expansions and quantum graphs
Abstract: Deterministic FitzHugh-Nagumo equations are among
the most used PDE's in neurobiology. Stochastic perturbations of
them. We shall report on recent work dedicated to the study of
their stochastic perturbations and the corresponding asymptotics.
The systems of such equations describe the dynamics of neural
networks. We briefly discuss the relation between the study of
such systems and the area of quantum dynamics on their manifolds
and on their graph limits.
- Fabrice Baudoin (Purdue University)
Title: Heat kernel analysis and generalized Ricci lower
bounds in sub-Riemannian geometry
Abstract: We study a class of rank two
sub-Riemannian manifolds encompassing Riemannian manifolds, CR manifolds
with vanishing Webster-Tanaka torsion, orthonormal bundles over Riemannian
manifolds, and graded nilpotent Lie groups of step two. These manifolds
admit a canonical horizontal connection and a canonical sub-Laplacian. We
construct on these manifolds an analogue of the Riemannian Ricci tensor and
prove Bochner type formulas for the sub-Laplacian. As a consequence, it is
possible to formulate on these spaces a sub-Riemannian analogue of the
so-called curvature dimension inequality. The heat kernel analysis on
sub-Riemannian manifolds for which this inequality is satisfied is shown to
share many properties in common with the heat kernel analysis on Riemannian
manifolds whose Ricci curvature is bounded from below. This is mainly a
joint work with N. Garofalo.
- Rene Carmona (Princeton University)
Title: Mathematical Challenges of the Emissions Markets
Abstract: The first part of the talk will be
devoted to a review of the mathematical models used by
economists and financial engineers for the quantitative analysis
of cap-and-trade schemes. We will concentrate on models inspired
by the European Union Emissions Trading Scheme, and we will
discuss design issues related to the existing allocation
processes and those touted by policy pundits.
The second part of the talk will address some of the puzzling
questions raised by the prices of options on EU Allowances
published by the exchanges: Do they really use Black's formula
to compute these prices? How reasonable are they? How can we
reconcile these prices with the results of the existing
equilibrium theories? How do these models suggest we price these
options?
- Eric Carlen (Rutgers University)
Title: Quantum Entropy and Non-commutative
Brascamp-Lieb Inequalities
Abstract: We formulate and prove, in sharp form,
non-commutative analogs of the Brascamp-Lieb inequality, which is a
generalized form of Young's inequality for convolutions. The proof is based
in part on a duality between such inequalities and certain subadditivity
inequalities for quantum entropy. In proving these, ideas of Len Gross,
first employed in his investigation of fermion hypercontractivity, play an
important role. This talk is based on joint works with Dario Cordero-Erausquin
and Elliott Lieb.
- Brian Hall (University of Notre Dame)
Title: Holomorphic function spaces and heat operators
for symmetric spaces
Abstract: Motivated by Len Gross’s “Fock space” decomposition
for functions on a compact Lie group, the speaker introduced in 1994 a
Segal-Bargmann transform for compact Lie groups, which is a unitary map from
L^2 of the compact group to an L^2 space of holomorphic functions on the
associated complex group. The transform itself consists of applying the heat
operator on the compact group and then analytically continuing to the
complex group. Since then, there has been quite a bit of work, by various
authors, in developing Segal-Bargmann-type transforms for various groups and
symmetric spaces, both compact and noncompact. I will survey this work,
concentrating on the cases where the results can be stated most explicitly.
- Todd Kemp (UCSD (visiting MIT))
Title:
Chaos and the Fourth Moment
Abstract: In
2006, Nualart and collaborators proved a remarkable
central limit theorem in the context of
the Wiener chaos. If $X_k$
is a sequence of $n$th Wiener-Ito integrals (in the
$n$th chaos), then necessary and
sufficient conditions that $X_k$ converge weakly to a normal law are that
its (second and) fourth
moments converge -- all other moments are controlled by these. Their proof
is heavily steeped in Malliavin calculus.
Considering $N\times N$-matrix-valued Brownian motion and letting $N\to\infty$,
there is an analogue of the Wiener chaos in free probability theory. In
this talk, I will discuss this free "Wigner" chaos, and the free analogue of
Nualart's theorem for the semicircle law (the central limit law for the
eigenvalues of random matrices). The qualitative proof uses some beautiful
combinatorics; more quantitative estimates require free versions of tools
from the Malliavin calculus, which I will also discuss.
This is joint work with Roland Speicher.
- H.-H. Kuo (Louisiana State University)
Title: An extension of the Ito stochastic
integration
Abstract: We introduce the class of instantly independent
stochastic processes, which serves as the counterpart of the Ito theory of
stochastic integration. This class provides a new approach to anticipating
stochastic integration. The crucial idea is to evaluate an instantly
independent stochastic process at the right endpoint of a subinterval, while
in the Ito theory the evaluation of an adapted stochastic process is at the
left endpoint. We present some new results on Ito's formula and stochastic
differential equations for the new stochastic integral.
- Michel Ledoux (Université de Toulouse, France)
Title:
Heat kernel measures, logarithmic Sobolev inequalities an measure
concentration
Abstract:
Elementary semigroup calculus
provides simple proofs of various functional inequalities for heat kernel
measures, including Gross's celebrated logarithmic Sobolev inequality, as
well as reversed forms. The approach furthermore reveals dimensional forms
of the logarithmic Sobolev inequality connecting on the one end to Shannon's
entropy power inequality and on the other to the Li-Yau parabolic
inequality. The reversed forms allow besides for a converse to the Herbst
argument from logarithmic Sobolev inequalities to concentration inequalities
in spaces with non-negative curvature, providing thus a new, functional,
proof of E. Milman's recent results.
- Artem Pulemotov (University of Chicago)
Title: The heat equation and the Ricci flow.
Abstract: The first part of the talk is devoted to gradient
estimates for the heat equation on a manifold M with a fixed
Riemannian metric. We will consider the case where M is complete with
∂M=Ř as well
as the case where M is compact with ∂M≠Ř.
Our attention will be centered around Li-Yau-type inequalities. The second
part of the talk focuses on the situation where the Riemannian metric on M evolves under the Ricci flow. We will describe the known options for
imposing the boundary conditions on the flow when ∂M≠Ř.
We will then outline the relevant existence theorems. Our ultimate goal will
be to discuss Li-Yau-type inequalities for the heat equation on M
with respect to the evolving metric. The talk is partially based on my joint
work with Mihai Bailesteanu and Xiaodong Cao.
- Michael Roeckner (University of
Bielefeld, Germany)
Title: Fokker Planck equations on Hilbert spaces.
Abstract: We consider a stochastic differential
equation in Hilbert space with time dependent coefficients for which no
general existence and uniqueness results are known. We prove, under suitable
assumptions, existence and uniqueness of a measure valued solution, for the
corresponding Fokker-Planck equation. In particular, we verify the Chapman-Kolmogorov
equations and get an evolution system of transition probabilities for the
stochastic dynamics informally given by the stochastic differential
equation.
- Ambar Sengupta (Louisiana State University)
Title: Free Yang-Mills on the Plane
Abstract: Quantum Yang-Mills theory on the plane for
the gauge group U(N) has a meaningful large-N limit. We will
describe this limit in terms of notions from free probability
theory.
- Dan Stroock (MIT)
Title:
(Tentative) An interesting but
often ignored theorem of Len Gross
Abstract: Many years ago, Gross gave a definitive
formulation of the fact that a Hilbert space does not determine the
Banach space in the abstract Wiener space for which it is the
Cameron-Martin space. In this talk, I will recall Gross's theorem
and show how it facilitates the proof of a couple of important
results.
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