Path integral quantization
(pdf)


Adrian P. C. Lim (Cornell)
A typical path integral on a manifold M, is an informal
expression of the form
(1)
1/Z \int_{s \in H(M)} f(s) exp(E(s)) Ds
where H(M) is a space of paths in M with energy E(s) < \infty,
f is a real valued function on H(M), Ds is a
"Lebesgue measure," and Z is a normalization constant. The use of path integrals for
"quantizing" classical mechanical systems (whose classical energy is E) started with
Feynman in [2] with very early beginnings being
traced back to Dirac [1]. In this talk, I will give
several rigorous definitions to Equation (1), by reviewing
work done by Driver and Andersson and recently by me. The idea is
to approximate H(M) by finite dimensional subspaces consisting
of broken geodesics and then to pass to the limit of finer and finer approximations.
[1] P. A. M. Dirac, Physikalische Zeitschrift der Sowjetunion 3 (1933), 64.
[2] R. P. Feynman, Spacetime approach to nonrelativistic quantum mechanics,
Rev. Modern Physics 20 (1948), 367387.
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Taylor map on complex path groups
(pdf)


Matt Cecil (U Conn)
The classical Taylor expansion can be interpreted as describing a unitary map from
holomorphic functions on C^d which are square integrable with respect to a
suitable Gaussian measure to the set of its "derivatives at the origin," interpreted as
the symmetric tensor algebra (Fock space) along with a suitable norm. This map is
related to the Fock space representation of classical quantum mechanics. Analogous
maps have been shown to exist when C^d is replaced by a Hilbert space or
by a noncommutative Lie group G. In this talk, I will review these known
cases and the challenges they present. The goal will be to describe a Taylor map
which takes functions on the space of paths on a Lie group which are square integrable
with respect to a heat kernel measure to subspace of the dual of the universal
enveloping algebra, a setting in which the base group is both noncommutative and
infinite dimensional.
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Voter models with heterozygosity selection
(pdf)


Anja Sturm (U Del)
In this talk we consider variations of the usual voter model which favor
types that are locally less common. This may be understood as a selective
advantage for rare types leading to more variation in the particle
population. The voter models with selection considered here are dual to
certain systems of branching annihilating random walks that are parity
preserving. Coexistence of types in one model is related to survival of
particles in the other. We consider conditions for the existence of
homogeneous invariant laws in which types coexist as well as convergence
to these laws.
This is joint work with Jan Swart (UTIA Prague).
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Random network models: Sampling and estimation
(pdf)


Joe Blitzstein (Harvard)
The usual ErdosRenyi model for random graphs is mathematically
beautiful, but is rarely useful as a model for realworld networks.
Thus, many morecomplicated models are being explored for such
applications as social, biological, and information networks. We will
describe a method for converting combinatorial theorems into efficient
sampling schemes, and compare sequential importance sampling techniques
to MCMC techniques. For the widelystudied case of exponential random
graph models, we discuss challenges and approaches for parameter
estimation. Parts based on joint work with Sourav Chatterjee and Persi
Diaconis.
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Bandwidth sharing networks in overload
(pdf)


Bert Zwart (Georgia Tech)
Processor Sharing (PS) queues were originally introduced to analyze the
performance of timesharing in computer networks. Nowadays, PS queues
are one of the most popular congestion models for TCP traffic on the
Internet. From a methodological perspective, PS is a challenging service
discipline since a measurevalued state descriptor is necessary to analyze the
system.
This talk is focused on such systems in overload. We look at a single
queue and consider the scenario where customers leave impatient as their
waiting time grows too large. We propose a fluid (functional LLN)
approximation of this system and investigate the fixed point of this
approximation. This leads to several qualitative insights into the dynamics of
PS models with impatience. We show that the impact of impatience on the
performance of the system can be quite substantial and propose an admission
control scheme to reduce its effect.
We will also consider the extension of some of these results
to bandwidth sharing networks. The main challenge there is to prove uniqueness
of solutions of certain fixed point equations.
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Characterizations of the invariant measure for a class of reflected
diffusions via the extended Skorokhod map
(pdf)


Weining Kang (Carnegie Mellon)
In this talk we focus on a class of reflected diffusions in polyhedrons defined via the extended
Skorokhod map. We show that the reflected diffusion has certain boundary property, i.e.,
the pushing process in the reflected diffusion charges zero on the amount time the reflected
diffusion spends at the boundary away from a "bad" set V where two or more faces meet.
Further, under suitable conditions, the reflected diffusion has an unique invariant
measure and such an invariant measure satisfies a basic adjoint relationship.
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