Seminar Information
Thursdays, 10am AP&M 6402
University of California, San Diego
La Jolla, CA 92093-0112
Map
Organizer: Todd Kemp
E-mail: tkemp@math.ucsd.edu
Previous years' webpages:
2014-2015
2013-2014
2012-2013
2011-2012
2010-2011
2009-2010
2008-2009
2007-2008
2006-2007
2005-2006
2004-2005
2003-2004
2002-2003
2000-2001
1999-2000
1998-1999
|
|
Fall 2015 Schedule
| Oct 8 | | Joscha Diehl, Technical University of Berlin
| Title: | | Weakly asymmetric Ginzburg-Landau lattice model
and stochastic Burgers. |
| Abstract: | |
The Ginzburg-Landau lattice model is a system
of stochastic differential equations that is known to converge, if properly
rescaled, to a linear SPDE. When introducing an asymmetry in the model
that vanishes in the limit, we prove convergence to the (semilinear)
stochastic Burgers equation. The theory for this stochastic PDE is
non-trivial, and we apply the theory of an energy solution. The latter
was introduced by Goncalves-Jara and under a slight reformulation
Gubinelli-Perkowski were recently able to show well-posedness for this
equation. |
|
| |
| Oct 15 | | Eviatar Procaccia, Texas A&M University
| Title: | | The boy who cried Wulff. |
| Abstract: | |
We consider a Gibbs distribution over random walk paths on the square lattice, proportional to the cardinality
of the path's boundary. We show that in the zero temperature limit, the paths condensate around an asymptotic shape.
This limit shape is characterized as the minimizer of the functional, mapping open connected subsets of the plane
to the sum of their principle eigenvalue and perimeter (with respect to some norm). A prime novel feature of this
limit shape is that it is not in the class of Wulff shapes. |
|
| |
| Oct 22 | | David Renfrew, UCLA
| Title: | | Spectral properties of large Non-Hermitian Random Matrices. |
| Abstract: | |
The study of the spectrum of non-Hermitian random matrices with independent, identically distributed entries was
introduced by Ginibre and Girko. I will present two generalizations of the iid model where the independence
and identical distribution assumptions are relaxed. |
|
| |
| Oct 29 | | Xue-Mei Li, University of Warwick
| Title: | | Limits theorems on Random ODEs on Manifolds and Examples. |
| Abstract: | |
We explain limit theorems associated with a family of random
ordinary differential equations on manifolds, driven by randomly
perturbed vector fields. After rescaling, the differentiable random
curves converge to a Markov process whose Markov generator can be written
explicitly in Hormander form. We also give rates of convergence in the Wasserstein distance.
Example 1. A unit speed geodesic, which chooses a direction randomly and uniformly
at every instant of order 1/epsilon, converges to a Brownian motion as epsilon tends to 0.
Furthermore their horizontal lifts converge to the Horizontal Brownian motion.
Examples 2. Inspired by the problem of the convergence of Berger's spheres to a S^(1/2),
we introduce a family of Interpolation equations on a Lie group G.
These are stochastic differential equations on a Lie group driven by diffusion vector fields
in the direction of a subgroup H rescaled by 1/epsilon, and a drift vector field in a transversal direction.
If there is a reductive structure, we identify a family of slow variables which, after rescaling,
converges to a Markov process on G. Furthermore, the projection of the limiting Markov process
to the orbit manifolds G/H is Markov. The limits can be identified in terms of the eigenvalue
of a second order differential operator on the subgroup and the Ad(H) invariant
decomposition of the Lie algebra. |
|
| |
| Nov 5 | | Steven Heilman, UCLA
| Title: | | Low Correlation Noise Stability of Euclidean Sets. |
| Abstract: | |
The noise stability of a Euclidean set is a well-studied quantity. This quantity uses the Ornstein-Uhlenbeck semigroup
to generalize the Gaussian perimeter of a set. The noise stability of a set is large if two correlated Gaussian random
vectors have a large probability of both being in the set. We will first survey old and new results for maximizing the
noise stability of a set of fixed Gaussian measure. We will then discuss some recent results for maximizing the
low-correlation noise stability of three sets of fixed Gaussian measures which partition Euclidean space. Finally,
we discuss more recent results for maximizing the low-correlation noise stability of symmetric subsets of Euclidean
space of fixed Gaussian measure. All of these problems are motivated by applications to theoretical computer science. |
|
| |
| Nov 12 | | Lionel Levine, Cornell University
| Title: | | Threshold state of the abelian sandpile. |
| Abstract: | |
A sandpile on a graph is an integer-valued function on the vertices.
It evolves according to local moves called topplings. Some sandpiles
stabilize after a finite number of topplings, while others topple
forever. For any sandpile s_0 if we repeatedly add a grain of sand at
an independent random vertex, we eventually reach a "threshold state''
s_T that topples forever. Poghosyan, Poghosyan, Priezzhev and Ruelle
conjectured a precise value for the expected amount of sand in s_T in
the limit as s_0 tends to negative infinity. I will outline how this
conjecture was proved here
by means of a Markov renewal theorem. |
|
| |
| Nov 26 | | Thanksgiving Holiday: no seminar. |
| |
| |
| Dec 4, 3:00pm | | Tianyi Zheng, Stanford University
| Title: | | Random walk parameters and the geometry of groups |
| Abstract: | | The first characterization of groups by an asymptotic description
of random walks on their Cayley graphs dates back to Kesten's criterion of amenability. I will first review some connections between
the random walk parameters and the geometry of the underlying groups. I will then discuss a flexible construction that gives solution
to the inverse problem (given a function, find a corresponding group) for large classes of speed, entropy and return probability of
simple random walks on groups of exponential volume growth. Based on joint work with Jeremie Brieussel. |
|
"Winter" 2016 Schedule
| Jan 7 | | Alex Hening, University of Oxford
| Title: | | The free path in a high velocity random flight process
associated to a Lorentz gas in an external field. |
| Abstract: | |
We investigate the asymptotic behavior of the free path of a
variable density random flight model in an external field as the initial
velocity of the particle goes to infinity. The random flight models we
study arise naturally as the Boltzmann-Grad limit of a random Lorentz gas
in the presence of an external field. By analyzing the time duration of
the free path, we obtain exact forms for the asymptotic mean and variance
of the free path in terms of the external field and the density of
scatterers. As a consequence, we obtain a diffusion approximation for the
joint process of the particle observed at reflection times and the amount
of time spent in free flight. This is based on joint work with Doug
Rizzolo and Eric Wayman. |
|
| |
| Jan 25, 10:00am | | Fabrice Baudoin, Purdue University
| Title: | | Topics in Stochastic Analysis
|
| Abstract: | | Starting from basic principles, we will present some recent developments in the theory of rough paths and in the
theory of sub-Riemannian diffusions. The first part of the talk will be devoted to the theory of rough paths.
This theory was developed in the 1990s by T. Lyons, and allows to give a sense to solutions of differential equations
driven by irregular paths. The theory itself has nothing to do with probability theory but has had a tremendous
impact on several recent developments in stochastic analysis; it served as an inspiration to Hairer's regularity
structure theory, for which he was awarded the Fields medal in 2014. In the second part of the talk, we will address
several problems in the geometric analysis of some sub-Riemannian manifolds, which can (surprisingly) be solved using
diffusion semigroups techniques. |
|
| |
| Jan 28 | | Prasad Tetali, Georgia Tech
| Title: | | Displacement convexity of entropy and discrete curvature |
| Abstract: | | Inspired by the recent developments and mature understanding of the
notion of lower-boundedness of Ricci curvature in continuous settings (such as Riemannian manifolds), several independent groups of
researchers have proposed intriguing analogs of such a curvature in discrete settings (such as graphs). The proposals depend on the
perspective being probabilistic, analytical or combinatorial. In this talk, I will briefly mention a few of these approaches,
consequences, and state some open problems. |
|
| |
| Feb 4 | | Michael Anshelevich, Texas A&M University
| Title: | | The exponential homomorphism in non-commutative probability |
| Abstract: | | The wrapping transformation is easily seen to intertwine convolutions
of probability measures on the real line and the circle. It is also easily seen to not transform additive free convolution into
the multiplicative one. However, we show that on a large class L of probability measures on the line, wrapping does transform
not only the free but also Boolean and monotone convolutions into their multiplicative counterparts on the circle. This allows
us to prove various identities between multiplicative convolutions by simple applications of the additive ones. The restriction
of the wrapping to L has several other unexpected nice properties, for example preserving the number of atoms. This is joint
work with Octavio Arizmendi. |
|
| |
| Feb 18 | | Solesne Bourguin, Boston University
| Title: | | Portmanteau inequalities on the Poisson space |
| Abstract: | | In this talk, we present some results originating
from a new general inequality obtained by combining the Chen-Stein method with Malliavin calculus on the Poisson space,
such as multidimensional Poisson approximations, mixed limit theorems, as well as a characterization of asymptotic
independence for U-statistics. Applications to stochastic geometry through limit theorems involving the joint
convergence of vectors of subgraph-counting statistics exhibiting both a Poisson and a Gaussian behavior will
also be discussed. |
|
| |
| Feb 25 | | Tai Melcher, University of Virginia
| Title: | | Small-time asymptotics of subRiemannian Hermite functions |
| Abstract: | | As in the Riemannian setting, a subRiemannian heat kernel
is controlled by the geometry of the underlying manifold. In particular, the asymptotic behavior of the kernel can reveal certain
geometric and topological data. We study the logarithmic derivatives of subRiemannian heat kernels in some cases and show that,
under appropriate scaling, they converge to their analogues on stratified groups. This gives one quantification of the now standard
idea that stratified groups play the role of the tangent space to subRiemannian manifolds.
This is joint work with Joshua Campbell. |
|
| |
| Mar 10 | | Steve Zelditch, Northwestern University
| Title: | | Large N limit of heat kernel measure on positive Hermitian matrices
and random metrics. |
| Abstract: | | Heat kernel measure K(t, I, A) dA on positive Hermitian NxN
matrices is a probability measure whose large N limit is important for
several different types of problems in mathematical physics. My talk
introduces a new application: to random Kahler metrics on any Kahler manifold.
The pair correlation function of random metrics is explicitly calculated for
each N. The large N asymptotics are closely related to zero sets of random
holomorphic functions. |
|
Spring 2016 Schedule
| Apr 14 | | Tom Alberts, University of Utah
| Location: | | AP&M 7421 |
| Title: | | Random Geometry in the Spectral Measure of the Circular Beta Ensemble |
| Abstract: | | The Circular Beta Ensemble is a family of random unitary matrices
whose eigenvalue distribution plays an important role in statistical physics. The spectral measure is a canonical way of
describing the unitary matrix that takes into account the full operator, not just its eigenvalues. When the matrix
is infinitely large (i.e. an operator on some infinite-dimensional Hilbert space) the spectral measure is supported
on a fractal set and has a rough geometry on all scales. This talk will describe the analysis of these fractal properties.
Joint work with Raoul Normand and Balint Virag. |
|
| |
| May 19 | | Tom Liggett, UCLA
| Title: | | k-dependent q-colorings of the integers |
| Abstract: | | In 2008, Oded Schramm asked the following question:
For what values of k and q does there exist a stationary,
k-dependent sequence of random variables with values
in {1,2,...,q} assigning different values to consecutive
integers? Schramm proved a number of results related
to this question, and speculated about what the answer
might be in general. As it turns out, the truth is quite
different from his informed guess. This is joint work
with A. E. Holroyd. |
|
| |
| May 26 | | Amber Puha, California State University San Marcos
| Location: | | AP&M 5402 |
| Title: | | Analysis of Processor Sharing Queues via Relative Entropy |
| Abstract: | | Processor sharing is a mathematical idealization of round-robin
scheduling algorithms commonly used in computer time-sharing. It is a fundamental example of a non-head-of-the-line service
discipline. For such disciplines, it is typical that any Markov description of the system state is infinite dimensional.
Due to this, measure-valued stochastic processes are becoming a key tool used in the modeling and analysis of stochastic
network models operating under various non-head-of-the-line service disciplines.
In this talk, we discuss a new approach to studying the asymptotic behavior of fluid model solutions (formal functional
law of large numbers limits) for critically loaded processor sharing queues. For this, we introduce a notion of relative
entropy associated with measure-valued fluid model solutions. This approach is developed with idea that similar notions
involving relative entropy may be helpful for understanding the asymptotic behavior of critical fluid model solutions
for stochastic networks operating under protocols naturally described by measure-valued processes. |
|
| |
| June 2 | | Georg Menz, UCLA
| Title: | | The log-Sobolev inequality for unbounded spin systems |
| Abstract: | | The log-Sobolev inequality (LSI) is a very useful tool for analyzing
high-dimensional situations. For example, the LSI can be used for
deriving hydrodynamic limits, for estimating the error in stochastic
homogenization, for deducing upper bounds on the mixing times of Markov
chains, and even in the proof of the Poincaré conjecture by
Perelman. For most applications, it is crucial that the constant in the
LSI is uniform in the size of the underlying system. In this talk, we
discuss when to expect a uniform LSI in the setting of unbounded spin
systems.
|
|
|