Seminar Information
Thursdays, 10:30am 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:
2017-2018
2016-2017
2015-2016
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
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Fall 2018 Schedule
| Sept 27 | | Nikhil Srivastava, UC Berkeley
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Concentration for Sums of Random Matrices with Markov Dependence
There are many well-known concentration results for sums of independent random matrices, e.g. those of Rudelson, Ahlswede-Winter, Tropp, and Oliveira.
We move beyond the independent setting, and prove a Chernoff-type bound for sums of matrix-valued random variables sampled via a random walk on an
reversible Markov chain, confirming a conjecture due to Wigderson and Xiao. Our proof is based on a new multi-matrix extension of the Golden-Thompson
inequality which follows from complex interpolation methods.
(Joint work with A. Garg, Y. Lee, and Z. Song.)
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| Oct 25 | | Li-Cheng Tsai, Columbia University
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Lower-tail large deviations of the KPZ equation
Regarding time as a scaling parameter, we prove the one-point, lower tail Large Deviation Principle (LDP) of the KPZ equation, with an explicit rate function.
This result confirms existing physics predictions. We utilize a formula from [Borodin Gorin 16] to convert LDP of the KPZ equation to calculating an exponential
moment of the Airy point process, and analyze the latter via stochastic Airy operator and Riccati transform.
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| Nov 8 | | Joshua Frisch, Caltech
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Proximal actions, Strong amenability, and infinite conjugacy class groups.
A topological dynamical system (i.e. a group acting by homeomorphisms on a compact topological space) is said to be proximal
if for any two points $p$ and $q$ we can simultaneously push them together i.e. there is a sequence $g_n$ such that
$\lim g_n(p)=\lim g_n (q)$. In his paper introducing the concept of proximality Glasner noted that whenever $\mathbb{Z}$ acts proximally
that action will have a fixed point. He termed groups with this fixed point property "strongly amenable" and showed that non-amenable
groups are not strongly amenable and virtually nilpotent groups are strongly amenable. In this talk I will discuss recent work precisely
characterizing which (countable) groups are strongly amenable.
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| Nov 15 | | Noah Forman, University of Washington
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The diffusion analogue to a tree-valued Markov chain
In '99, David Aldous conjectured that a certain natural "random walk" on the space of binary combinatorial trees
should have a continuum analogue, which would be a diffusion on the Gromov-Hausdorff space of continuum trees.
This talk discusses ongoing work by F-Pal-Rizzolo-Winkel that has recently verified this conjecture with a
path-wise construction of the diffusion. This construction combines our work on dynamics of certain projections
of the combinatorial tree-valued random walk with our previous construction of interval-partition-valued diffusions.
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"Winter" 2019 Schedule
| Jan 10 | | Tom Kurtz, University of Wisconsin
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Population models as partial observations of genealogical models
Classical models of biological populations, for example, Markov branching processes, typically model population size and possibly
the distribution of types and/or locations of individuals in the population. The intuition behind these models usually includes
ideas about the relationships among the individuals in the population that cannot be directly recovered from the model.
This loss of information is even greater if one employs large population approximations such as the diffusion approximations popular
in population genetics. "Lookdown" constructions provide representations of population models in terms of countable systems of
particles in which each particle has a "type" which may record both spatial location and genetic type and a "level" which incorporates
the lookdown structure which in turn captures the genealogy of the population. The original population model can then be viewed as the
result of partial observation of the more complex model. We will exploit ideas from filtering of Markov processes to make the idea of
partial observation clear and to justify the lookdown construction.
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| Jan 17 | | Yi Sun, Columbia University
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Gaussian fluctuations for products of random matrices
This talk concerns singular values of $M$-fold products of i.i.d. right-unitarily invariant $N\times N$ random matrix ensembles.
As $N\to\infty$, the height function of the Lyapunov exponents converges to a deterministic limit by work of
Voiculescu and Nica-Speicher for $M$ fixed and by work of Newman and Isopi-Newman for $M$ tending to infinity with $N$.
In this talk, I will show for a variety of ensembles that fluctuations of these height functions about their mean converge
to explicit Gaussian fields which are log-correlated for $M$ fixed and have a white noise component for $M$ tending to infinity with $N$.
These ensembles include rectangular Ginibre matrices, truncated Haar-random unitary matrices, and right-unitarily invariant matrices
with fixed singular values. I will sketch our technique, which derives a central limit theorem for global fluctuations via certain
conditions on the multivariate Bessel generating function, a Laplace-transform-like object associated to the spectral measures
of these matrix products. This is joint work with Vadim Gorin.
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| Jan 31 | | Haosui Duanmu, UC Berkeley
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Nonstandard analysis and its application to Markov processes
Nonstandard analysis, a powerful machinery derived from mathematical logic,
has had many applications in probability theory as well as stochastic processes.
Nonstandard analysis allows construction of a single object - a hyperfinite probability
space - which satisfies all the first order logical properties of a finite probability space,
but which can be simultaneously viewed as a measure-theoretical probability space
via the Loeb construction. As a consequence, the hyperfinite/measure duality has
proven to be particularly useful in porting discrete results into their continuous settings.
In this talk, for every general-state-space continuous-time Markov process satisfying appropriate
conditions, we construct a hyperfinite Markov process which has all the basic order logical properties
of a finite Markov process to represent it. We show that the mixing time and the hitting time agree with
each other up to some multiplicative constants for discrete-time general-state-space reversible Markov
processes satisfying certain condition. Finally, we show that our result is applicable
to a large class of Gibbs samplers and Metropolis-Hasting algorithms.
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| Jan 31 | | Brian Hall, University of Notre Dame
Joint with Functional Analysis
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Eigenvalues of random matrices in the general linear groups
I will consider random matrices in the general linear group GL(N;C) distributed according to a heat kernel measure.
This may also be described as the distribution of Brownian motion in GL(N;C) starting at the identity. Numerically,
the eigenvalues appear to cluster into a certain domain $\Sigma_t$ as $N$ tends to infinity. A natural candidate for
the limiting eigenvalue distribution is the “Brown measure” of the limiting object, which is Biane’s "free multiplicative Brownian motion."
I will describe recent work with Driver and Kemp in which we compute this Brown measure. The talk will be self contained and will have
lots of pictures.
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| Feb 28 | | Hanbaek Lyu, UCLA
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Stable network observables via dynamic embedding of motifs
We propose a novel framework for constructing and computing various stable network observables. Our approach is based on sampling a random homomorphism
from a small motif of choice into a given network. Integrals of the law of the random homomorphism induces various network observables, which include
well-known quantities such as homomorphism density and average clustering coefficient. We show that these network observables are stable with respect
to renormalized cut distance between networks. For their efficient computation, we also propose two Markov chain Monte Carlo algorithms and analyze
their convergence and mixing times. We demonstrate how our techniques can be applied to network data analysis, especially for hypothesis testing and
hierarchical clustering, through analyzing both synthetic and real world network data.
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March 7
APM 2402 | | Xin Sun, Columbia University
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Conformal embedding and percolation on the uniform triangulation
Following Smirnov's proof of Cardy's formula and Schramm's discovery of SLE, a thorough understanding of the scaling limit of
critical percolation on the regular triangular lattice the has been achieved. Smirnorv's proof in fact gives a discrete
approximation of the conformal embedding which we call the Cardy embedding. In this talk I will present a joint project
with Nina Holden where we show that the uniform triangulation under the Cardy embedding converges to the Brownian disk
under the conformal embedding. Moreover, we prove a quenched scaling limit result for critical percolation on uniform triangulations.
Time permitting, I will also explain how this result fits in the the larger picture of random planar maps and Liouville quantum gravity.
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Spring 2019 Schedule
Apr 4
10:30am | | Mark Meckes, Case Western University
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Quenched central limit theorem in a corner growth setting
We consider point-to-point directed paths in a random environment on the two-dimensional integer lattice. For a general independent environment under mild
assumptions we show that the quenched energy of a typical path satisfies a central limit theorem as the mesh of the lattice goes to zero. The proofs rely
on concentration of measure techniques and some combinatorial bounds on families of paths. This is joint work with Christian Gromoll and Leonid Petrov.
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Apr 4
1pm | | Elizabeth Meckes, Case Western University
Department Colloquium
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Eigenvalues of random matrices: convergence of spectral measures and eigenvalue rigidity
The behavior of the eigenvalues of large random matrices is generally very predictable, on multiple scales. Macroscopically, results like the semi-circle
law describe the overall shape of the eigenvalue distributions. Indeed, for many natural ensembles of random matrices, we can describe in great detail
the way the distribution of the eigenvalues converges to some limiting deterministic probability measure. On a microscopic scale, we often see the
phenomenon of eigenvalue rigidity, in which individual eigenvalues concentrate strongly at predicted locations. I will describe some general approaches
to these phenomena, with many examples: Wigner matrices, Wishart matrices, random unitary matrices, truncations of random unitary matrices,
Brownian motion on the unitary group, and others.
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Apr 11
4pm | | Roman Vershynin, UC Irvine
Department Colloquium
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Mathematics of Deep Learning
Deep learning is a rapidly developing area of machine learning, which uses artificial neural networks to
perform learning tasks. Although mathematical description of neural networks is simple, theoretical explanation
of spectacular performance of deep learning remains elusive. Even the most basic questions about it
remain open. For example, how many different functions can a neural network compute? Jointly with Pierre
Baldi (UCI CS) we discovered a general capacity formula for all fully connected boolean networks. The formula predicts,
counterintuitively, that shallow networks have greater capacity than deep ones. So, mystery remains.
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Apr 25
10:30am | | Boguslaw Zegarlinski, Imperial College London
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Coercive inequalities for Markov generators on nilpotent Lie groups
I will review and present some new results on construction, long time behaviour
and coercive inequalities for Markov semigroups on nilpotent Lie groups.
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May 30
10:30am | | Mike Cranston, UC Irvine
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Some properties of the Riemann zeta distribution
An alternative to selecting an integer uniformly from $1$ to $N$ and letting $N$ go to infinity is to select an
integer according to the Riemann zeta distribution: the probability of selecting $n$ is $1/\zeta(s)n^s$, and
letting $s$ go to $1$. We will explain several results that arise naturally due to the multiplicative
property of this distribution.
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June 6
10am | | Larry Goldstein, USC
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Dickman Approximation in Quickselect sorting and Probabilistic Number Theory
The generalized Dickman distribution ${\cal D}_\theta$ with parameter $\theta>0$ is the unique solution to the distributional equality
$W=_d W^*$, where
\begin{align*}
W^*=_d U^{1/\theta}(W+1),
\end{align*}
with $W$ non-negative with probability one, $U \sim {\cal U}[0,1]$ independent of $W$, and $=_d$ denoting equality in distribution. Members of this family appear
in the study of algorithms, number theory, stochastic geometry, and perpetuities.
The Wasserstein distance $d(\cdot,\cdot)$ between such a $W$ with finite mean, and $D \sim {\cal D}_\theta$ obeys
\begin{align*} d(W,D) \le (1+\theta)d(W^*,W).
\end{align*}
The specialization of this bound to the case $\theta=1$ and coupling constructions yield for $n \ge 1$ that
\begin{align*}
d_1(W_n,D) \le \frac{8\log (n/2)+10}{n} \quad \mbox{where } \quad W_n=\frac{1}{n}C_n-1,
\end{align*}
and $C_n$ is the number of comparisons made by the Quickselect algorithm to find the smallest element of a list of $n$ distinct numbers.
Joint with Bhattacharjee, using Stein's method, bounds for Wasserstein type distances can also be computed between ${\cal D}_\theta$ and weighted sums arising
in probabilistic number theory of the form
\begin{align*}
S_n=\frac{1}{\log(p_n)} \sum_{k=1}^n X_k \log(p_k)
\end{align*}
where $(p_k)_{k \ge 1}$ is an enumeration of the prime numbers in increasing order and $X_k$ is, for instance, Geometric with parameter $1-1/p_k$.
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June 6
11am | | Amber Puha, Cal State San Marcos
Joint Stochastic Systems Seminar
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Asymptotic Behavior of a Critical Fluid Model for a Multiclass Processor Sharing Queue via Relative Entropy
Queueing systems operating under the processor sharing discipline are relevant for studying time-sharing in computer and communication systems. Measure-valued processes,
which track the residual service times of all jobs in the system, have been used to describe the dynamics of such systems. However, exact analysis of these infinite-dimensional
stochastic processes is rarely possible. As a tool for approximate analysis of such systems, it has been proved that a fluid model arises as a functional law of large numbers
limit of a multi-class processor sharing queue. This talk will focus on the asymptotic behavior of such a fluid model in the interesting regime of critical loading, where
the average inflow of work to the system is equal to the capacity of the system to process that load.
Using an approach involving a certain relative entropy functional, we show that critical fluid model solutions converge to a set of invariant states as time goes to infinity,
uniformly for all initial conditions lying in certain relatively compact sets. This generalizes an earlier single-class result of Puha and Williams to the more complex multiclass
setting. In particular, several new challenges are overcome, including formulation of a suitable relative entropy functional and identifying a convenient form of the time
derivative of the relative entropy applied to trajectories of critical fluid model solutions.
This is joint work with Justin A. Mulvany (USC) and Ruth J. Williams (UCSD).
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