Analysis / Math Physics Seminar, Fall 2007
Friday's from 3:30 to 4:30 PM, meeting alternately at MIT (room 2-136) and at Harvard (Science Center room 232).
Full schedule:
September 21: Benjamin Schlein (Munich and Univ Cambridge)
Hartree approximation for the dynamics of mean field systems
Abstract:
In the first part of the talk I am going to review recent results
concerning the rigorous derivation of the nonlinear Hartree equations (and
the related Gross-Pitaevskii equation) from many body quantum dynamics. In
the second part of the talk I am going to discuss some new bounds on the
rate of convergence towards the limiting Hartree evolution.
September 28: Enno Lenzmann (at Harvard)
Dynamical evolution of self-gravitating relativistic matter in Hartree theory
October 5: Juerg Frohlich 4:00 pm at Harvard -- room 453
Jefferson (physics dept)
(note special time and location)
Recent progress on
mathematical theory of QED
October 12: Todd Kemp (at Harvard)
Semigroup
contraction properties for q-Gaussian processes
Abstract:
The construction of Brownian motion led Wiener to what is now called the
Wiener chaos decomposition; in the world of mathematical physics, this
later became known as the (Bosonic) Fock space. In this language, it was
discovered (by Parthasarathy) that Brownian motion itself can be viewed as
a (free) quantum field -- that is, simply as the real part of a creation
operator.
This same construction can be carried out for the Fermionic Fock space, and
for an interpolating class of q-Fock spaces (-1<q<1) introduced by Bozejko
and Speicher in the 1990s. There is also a (somewhat rudimentary) theory
of stochastic calculus associated with these q-Brownian motions, allowing
the construction of (non-commutative) Markov processes.
In this talk, I will focus on the q-Ornstein-Uhlenbeck process, studied
through its infinitesimal generator. The associated Markov semigroup has
two strong contraction properties: hypercontractivity and
ultracontractivity (as shown by Biane and Bozejko, respectively). In the
q=1 (classical) case, it is known that restricting to the complex wave
representation actually improves the nature of the hypercontractive bound;
I will show the comparable results hold for all q, as well as an improved
ultracontractive bound in the q=0 case.
October 19: Peter Shor (at MIT)
Matrix product states and simulating spin chains
October 26: Hans Christianson (at Harvard)
Phase space analysis near
periodic orbits
November 2: Jinho Baik (University of Michigan) (at MIT)
Asymptotics of Tracy-Widom distribution functions in random matrix theory
Abstract:
The Tracy-Widom distribution functions are certain probability
distribution functions expressed in terms of a solution of second order
nonlinear ODE (Painleve II equation). These distribution functions arise
as universal limits in random matrix theory, statistics, combinatorics and
probability.
After reviewing some of the above models, we discuss the asymptotic tail
behaviors
of these functions. Especially we discuss the question of evaluating the
constant terms
in the asymptotics at negative infinity and related constant problems for
other distribution functions.
November 9: Wei-Min Wang (Universite Paris-Sud) (at
Harvard Science center 232)
Bounds on Sobolev Norms for Time Dependent Schr\"odinger Equations
Abstract:
I prove that for the time dependent 1-D linear Schr\"odinger
equation with (arbitrary) bounded analytic potentials, periodic in $x$,
the growth of Sobolev norms $H^s$ (all $s>0$) of the solutions is at most
logarithmic in time: ${\log t}^C$.
This is obtained by using periodic (in time) approximations.
The above logarithmic bound is consistent with the result when the time
dependence is periodic with fixed integer periods, where I proved that the
Sobolev norms remain bounded for all time. It is also in line with a
previous result for a nonlinear lattice Schr\"odinger equation, where
Bourgain and I proved that the growth in Sobolev norms is at most
polynomial: $t^\delta$
($\delta>0$ arbitrarily small).
November 16: No seminar (CDM conference)
November 23: No seminar (Thanksgiving)
November 30: Alessandro Pizzo (Theoretische Physik ETH Zurich)
(at MIT)
December 7: Andrej Zlatos (University of Chicago)
(at
Harvard)
Dissipation Enhancement by Flows in 2D
Abstract:
We consider a passive scalar advection-diffusion equation in 2D with a
periodic incompressible advecting vector field (flow). We provide a sharp
characterization of all flow profiles that optimally enhance dissipation
in the sense that solutions with compactly supported initial data become
arbitrarily small arbitrarily quickly, provided the flow amplitude is
large enough. Our characterization is expressed in terms of simple
geometric and spectral conditions on the flow. Extensions to higher
dimensions and applications to reaction-advection-diffusion equations will
also be mentioned.
December 14: Gunther Uhlmann (University of Washington)
(at MIT)
Invisibility and Singular Metrics