Seminar Information
Thursdays, 10am AP&M 6402
University of California, San Diego
La Jolla, CA 920930112
Map
Organizer: Todd Kemp
Email: tkemp@math.ucsd.edu
Previous years' webpages:
20102011
20092010
20082009
20072008
20062007
20052006
20042005
20032004
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19981999


Fall Quarter Schedule
Sep 8   Lucian Beznea (Simion Stoilow Institute of Mathematics of the Romanian Academy)
Special Time: 2pm
Title:   The semigroup approach for measurevalued branching processes and a nonlinear Dirichlet problem 
Abstract:   We use a branching Markov process on the space of finite configurations to solve a Dirichlet problem associated with the operator Laplacian(u) + u^2. We follow the pioneering works of M. Nagasawa, N. Ikeda, S. Watanabe, M.L. Silverstein, and the approach of E.B. Dynkin. 


Oct 20   Brian Hall (Notre Dame)
Title:   Analysis on Lie groups from a probabilistic perspective 
Abstract:   I will discuss results in analysis on a compact Lie group that can be obtained using Brownian motion. These include a "Hermite expansion" and an analog of the SegalBargmann transform. Both results can be understood by lifting Brownian motion in the group to Brownian motion in the Lie algebra. I will also briefly discuss an open problem concerning the infinitedimensional limit of these results. 


Oct 27   Thomas Laetsch (UCSD)
Title:   An L^2 metric limit theorem for Wiener measure on manifolds with nonpositive sectional curvature. 
Abstract:   An interpretation for an intuitive expression for the path integral prescription of the Hamiltonian on a manifold is given. Following the direction of the informal derivation of the path integral, a limit theorem is proved using finite dimensional approximations to Wiener measure. 


Nov 3   William Massey (Princeton ORFE)
Title:   SkewnessVariance Approximations for Dynamic Rate, MultiServer Queues with Abandonment. 
Abstract:  
The multiserver queue with nonhomogeneous Poisson arrivals and customer abandonment is a fundamental, dynamic rate, queueing model for large scale
service systems such as call centers and hospitals. Scaling the arrival rates and number of servers arises naturally as designing a staffing schedule in
response to predictable increasing demand. Mathematically, this gives us the fluid and diffusion limits found in Mandelbaum, Massey, and Reiman. The general
rate asymptotics used here for Markovian service networks reduce to the HalfinWhitt scaling for multiserver queues.
The diffusion limit suggests a Gaussian approximation to the stochastic behavior of this queueing process. The mean and variance are easily computed
from a twodimensional dynamical system for the fluid and diffusion limiting processes. Recent work by Gautam and Ko found that a modified version of these
differential equations can be used to obtain better Gaussian estimates of the original queueing system distribution. In this paper, we introduce a new
threedimensional dynamical system that improves on both of these approaches by constructing a nonGaussian estimation of the mean, variance, and third
cumulative moment. This is joint work with Jamol Pender of Princeton University. 


Nov 10   Lionel Levine (Cornell University & University of Michigan)
Title:   Logarithmic fluctuations from circularity. 
Abstract:   Starting with n particles at the origin in Z^d, let each
particle in turn perform simple random walk until reaching an
unoccupied site. Lawler, Bramson and Griffeath proved that with high
probability the resulting random set of n occupied sites is close to a
ball. We show that its fluctuations from circularity are, with high
probability, at most logarithmic in the radius of the ball, answering
a question posed by Lawler in 1995 and confirming a prediction made by
chemical physicists in the 1980's. Joint work with David Jerison and
Scott Sheffield. 


Nov 17   Michael Kelly (UCSD)
Title:   Bounding The Rate of Adaptation In A Large Asexually
Reproducing Population With Fast Mutation Rates. 
Abstract:   We consider a model of asexually reproducing individuals. The birth and death rates of the individuals are affected by a fitness parameter. The rate of mutations that cause the
fitnesses to change is proportional to the population size, N. The mutations may be either
beneficial or deleterious. In a paper by Yu, Etheridge and Cuthbertson (2009) it was shown
that the average rate at which the mean fitness increases in this model is bounded below by
log^{1d} N for any d > 0. We achieve an upper bound on the average rate at which the mean
fitness increases of O(log N / log log N). 


Nov 24   Thanksgiving Holiday: no seminar. 

Dec 1   Benoit Collins (RIMS & University of Ottawa)
Title:   Applications of Random Matrix Theory to Quantum Information Theory via Free Probability 
Abstract:   I will first describe a generalization of a result by Haagerup and Thorbjornsen on the asymptotic norm of noncommutative polynomials of random matrices, in the case of unitary matrices. Then I will show how such results help us refine our understanding of the outputs of random quantum channels. In particular one obtains optimal estimates for the minimum output entropy of a large class of typical quantum channels. The first part of this talk is based on joint work with Camille Male, and the second part is based on joint work with Serban Belinschi and Ion Nechita. 

Winter Quarter Schedule
Jan 12   Pat Fitzsimmons (UCSD)
Title:   Two results on Gaussian measures. 
Abstract:   1. I'll show that Hunt's hypothesis (H) fails for Leonard Gross' infinite dimensional Brownian
motion, by exhibiting a subset of the state space of the motion that is hit exactly once for certain starting points.
2. It is well known that a Lebesgue measurable additive function from R to R is necessarily continuous (and linear). I'll show how D. Stroock's recent proof
of L. Schwartz's "Borel graph theorem" can be adapted to show that a "universally Gaussian measurable" and additive map from one Banach space to another
is automatically continuous (and linear). 


Jan 19   WeiKuo Chen (University of California, Irvine)
Title:   Chaos problem in the SherringtonKirkpatrick model. 
Abstract:   In physics, the main objective in spin glasses is to understand
the strange magnetic properties of alloys. Yet the models invented to
explain the observed phenomena are of a rather fundamental nature in
mathematics. In this talk, we will focus on one of the most important mean
field models, called the SherringtonKirkpatrick model, and discuss its
disorder and temperature chaos problems. Using the Guerra replica symmetry
breaking bound and the GhirlandaGuerra identities, we will present
mathematically rigorous results and proofs for these problems. Most part
of the talk is the joint work with Dmitry Panchenko (Texas A&M). 


Feb 2   Brian Rider (University of Colorado, Boulder)
Title:   Spiking the random matrix hard edge. 
Abstract:   The largest eigenvalue of a rank one perturbation of random hermitian matrix is known to exhibit a phase transition. If the perturbation is small, one sees the famous TracyWidom law; if the perturbation is large, the result is simple Gaussian fluctuations. Further, there is a scaling window about a critical value of the perturbation which leads to a new one parameter family of limit laws. The same phenomena exists for random sample covariance matrices in which one of the population eigenvalues is "spiked", or takes a value other than one. BloemendalVirag have shown how this picture persists in the context of the general beta ensembles, giving new formulations of the discovered critical limit laws (among other things). Yet another route, explained here, is to go through the random matrix hard edge, perturbing the smallest eigenvalues in the sample covariance setup. A limiting procedure then recovers all the alluded to limit distributions. (Joint work with Jose Ramirez.) 


Feb 9   Tonci Antunovic (UC Berkeley)
Title:   Some path properties of Brownian motion with variable drift. 
Abstract:   If B is a Brownian motion and f is a function in the Dirichlet space, then by CameronMartin theorem, the process Bf has the same almost sure path properties as B. In this talk we will present some properties of the image and the zero set of Brownian motion perturbed by certain less regular drifts f (examples include Hilbert curves and Cantor functions). Based on joint works with Krzysztof Burdzy, Yuval Peres, Julia Ruscher and Brigitta Vermesi. 


Mar 8   Maria Eulalia Vares (Universidade Federal do Rio de Janeiro)
Title:   First passage percolation and escape stragegies 
Abstract:   Consider first passage percolation on Z^d with passage
times given by i.i.d. random variables with common distribution F. Let t_\pi(u,v) be the time from u to v for a path
\pi and t(u,v) the minimal time among all such paths from u to v. We ask whether or not there exist points x,y in Z^d
and a semiinfinite path pi=(y_0=y,y_1,...) such that t_\pi(y,y_{n+1}) < t(x,y_n) for all n. Necessary and sufficient conditions
on F are given for this to occur. Based on joint work with Enrique D. Andjel. 


Mar 15   Kay Kirkpatrick (UIUC)
Title:   BoseEinstein condensation and quantum manybody systems 
Abstract:   Near absolute zero, a gas of quantum particles can condense into an
unusual state of matter, called BoseEinstein condensation (BEC), that
behaves like a giant quantum particle. The rigorous connection has
recently been made between the physics of the microscopic manybody
dynamics and the mathematics of the macroscopic model, the cubic
nonlinear Schrodinger equation (NLS). I'll discuss recent progress
with Gerard Ben Arous and Benjamin Schlein on a central limit theorem
for the quantum manybody systems, a step towards large deviations for
BoseEinstein condensation. 

Spring Quarter Schedule
April 19   Michael Kozdron (University of Regina & MSRI)
Title:   On the convergence of looperased random walk to SLE(2) in the natural parametrization. 
Abstract:   The SchrammLoewner evolution is a oneparameter family of random growth processes in the complex plane introduced by Oded Schramm in 1999. In the past decade, SLE has been successfully used to describe the scaling limits of various twodimensional lattice models. One of the first proofs of convergence was due to Lawler, Schramm, and Werner who gave a precise statement that the scaling limit of looperased random walk is SLE with parameter 2. However, their result was only for curves up to reparameterization. There is reason to believe that the scaling limit of looperased random walk is SLE(2) with the very specific natural time parameterization that was recently introduced by Lawler and Sheffield, and further studied by Lawler, Zhou, and Rezaei. I will describe several possible choices for the parameterization of the discrete curve that should all give the natural time parameterization in the limit, but with the key difference being that some of these discrete time parameterizations are easier to analyze than the others. This talk is based on joint work in progress with Tom Alberts and Robert Masson. 


Apr 26   Nathan Ross (UC Berkeley)
Title:   A probabilistic approach to local limit theorems 
Abstract:   We discuss a new method for obtaining a local limit theorem (LLT) from a
known distributional limit theorem. The method rests on a simple analytic
inequality (essentially due to Hardy, Landau, and Littlewood) which can be
applied directly after quantifying the smoothness of the distribution of
interest. These smoothness terms are nontrivial to handle and so we also
provide new (probabilistic) tools for this purpose. We illustrate our
approach by showing LLTs with rates for the magnetization in the
CurieWeiss model at high temperature and for some counts in an
ErdosRenyi random graph. This is joint work with Adrian Roellin. 


May 3   Ioana Dumitriu (University of Washington)
Title:   Random Regular Graphs: how local structure determines Wignermatrix eigenstatistics 
Abstract:   Fueled partly by the recent explosive progress in proving
universality for random matrix ensembles, spectral graph theory has also
made significant advances in the past couple of years.
For the ErdosRenyi random graph model, p >> log n / n, shortscale
semicircle laws, spacings in the bulk, extremal eigenvalue distributions,
and complete delocalization of eigenvectors have been shown by
ErdosKnowlesYauYin ('11'12) via a treatment similar to the Wigner
matrix case (and based on "hard" analysis).
Regular random graphs, being less Wignerlike, are harder to tackle with
such methods, but recent combinatorial treatments have yielded much
success. Through a careful estimation of probabilities and a reduction to
the ErdosRenyi model, TranVuWang ('10) have been able to establish
shortscale semicircle laws for the d >> log n regime in the case of
dregular graphs. Also, a certain type of delocalization has been
established for an extended growth regime for d by BrooksLindenstrauss
('09). Somewhat complementarily, for a polylogarithmic regime of d, D.Pal
proved both shortscale semicircle laws, and a slightly weaker
delocalization ('09) using the local treelike structure of the graph.
I will sketch the method of proof, and explain how the same ideas can be
used to obtain similar results for bipartite, biregular graphs
(D.Johnson, '12), global fluctuations for the permutation model
(D.JohnsonPalPaquette, '11), and global fluctuations for the uniform
model in polylogarithmic d regime (Johnson '12).



May 17   Ruth Williams (UCSD)
Special Time: 4pm
Title:   Resource Sharing in Stochastic Networks 
Abstract:   Stochastic networks are used as models for complex
processing systems involving dynamic interactions subject to uncertainty. Applications arise in hightech manufacturing,
the service industry, telecommunications, computer systems and bioengineering. The control and analysis of such networks
present challenging mathematical problems. In this talk, a concrete application will be used to illustrate a general approach
to the study of stochastic processing networks based on deriving more tractable approximate models. Specically,
we will consider a model of Internet congestion control in which processing can involve the simultaneous
use of several resources (or links), a phenomenon that is not well understood. Elegant fluid and diffusion
approximations will be derived and used to study the performance of this model. A key insight from this
analysis is a geometric representation of the consequences of using a "fair" policy for the sharing of resources.
The talk will conclude with a summary of the current status and description of open problems associated
with approximate models for general stochastic processing networks. 


May 24   Konstantinos Spiliopoulos (Brown University & Boston University)
Location: AP&M 7421
Title:   Large Deviations and Monte Carlo Methods for Problems with Multiple Scales 
Abstract:   The need to simulate rare events occurs in many application areas, including telecommunication, finance, insurance, computational physics and chemistry. However, virtually any simulation problem involving rare events will have a number of mathematical and computational challenges. As it is well known, standard Monte Carlo sampling techniques perform very poorly in that the relative errors under a fixed computational effort grow rapidly as the event becomes more rare. In this talk, I will discuss large deviations, rare events and Monte Carlo methods for systems that have multiple scales and that are stochastically perturbed by small noise. Depending on the type of interaction of the fast scales with the strength of the noise we get different behavior, both for the large deviations and for the corresponding Monte Carlo methods. Using stochastic control arguments we identify the large deviations principle for each regime of interaction. Furthermore, we derive a control (equivalently a change of measure) that allows to design asymptotically efficient importance sampling schemes for the estimation of associated rare event probabilities and expectations of functionals of interest. Standard Monte Carlo methods perform poorly in these kind of problems in the small noise limit. In the presence of multiple scales one faces additional difficulties and straightforward adaptation of importance sampling schemes for standard small noise diffusions will not produce efficient schemes. We resolve this issue and demonstrate the theoretical results by examples and simulation studies. Applications of these results in chemistry problems and in mathematical finance will also be discussed. 


June 7   Tom Alberts (Caltech)
Title:   Diffusions of Multiplicative Cascades 
Abstract:   A multiplicative cascade is a randomization of any measure on the unit interval,
constructed from an iid collection of random variables indexed by the dyadic intervals. Given an arbitrary initial measure I will describe a method for constructing a
continuous time, measure valued process whose value at each time is a cascade of the
initial one. The process also has the Markov property, namely at any given time it
is a cascade of the process at any earlier time. It has the further advantage of being
a martingale and, under certain extra conditions, it is also continuous. I will discuss
applications of this process to models of tree polymers and onedimensional random
geometry.
Joint work with Ben Rifkind (University of Toronto). 




