Difference between revisions of "Probability Seminar"

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(Wednesday, May 1, VV B115, Bálint Vető, University of Bonn)
(Replacing page with '__NOTOC__ == Fall 2013 == Thursdays in 901 Van Vleck Hall at 2:25 PM, unless otherwise noted. If you would like to receive announcements about upcoming seminars, please visi...')
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== Spring 2013 ==
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== Fall 2013 ==
  
  
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[[Past Seminars]]
 
[[Past Seminars]]
  
== Thursday, January 31, Bret Larget, UW-Madison ==
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== Thursday, September 12, TBA ==
  
Title: Approximate conditional independence of separated subtrees and phylogenetic inference
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Title: TBA
  
 
Abstract:
 
Abstract:
Bayesian methods to reconstruct evolutionary trees from aligned DNA
 
sequence data from different species depend on Markov chain Monte
 
Carlo sampling of phylogenetic trees from a posterior distribution.
 
The probabilities of tree topologies are typically estimated with the
 
simple relative frequencies of the trees in the sample. When the
 
posterior distribution is spread thinly over a very large number of
 
trees, the simple relative frequencies from finite samples are often
 
inaccurate estimates of the posterior probabilities for many trees. We
 
present a new method for estimating the posterior distribution on the
 
space of trees from samples based on the approximation of conditional
 
independence between subtrees given their separation by an edge in the
 
tree. This approximation procedure effectively spreads the estimated
 
posterior distribution from the sampled trees to the larger set of
 
trees that contain clades (sets of species in subtrees) that have been
 
sampled, even if the full tree is not part of the sample.  The
 
approximation is shown to be accurate for many data sets and is
 
theoretically justified. We also explore a consequence of this result
 
that may lead to substantial increases in computational efficiency for
 
sampling trees from posterior distributions. Finally, we present an
 
open problem to compare rates of convergence between the simple
 
relative frequency approach and the approximation approach.
 
 
==Thursday, February 14, Jean-Luc Thiffeault, UW-Madison==
 
 
Title: Biomixing and large deviations
 
 
Abstract: As fish, micro-organisms, or other bodies move through a fluid, they
 
stir their surroundings.  This can be beneficial to some fish, since
 
the plankton they eat depends on a well-stirred medium to feed on
 
nutrients.  Bacterial colonies also stir their environment, and this
 
is even more crucial for them since at small scales there is no
 
turbulence to help mixing.  I will discuss a simple model of the
 
stirring action of moving bodies through a fluid.  An attempt will be
 
made to explain existing data on the displacements of small particles,
 
which exhibits probability densities with exponential tails.  A
 
large-deviation approach helps to explain some of the data, but
 
mysteries remain.
 
 
==  <span style="color:#FF0000"> Tuesday, March 5, 2:30pm VV B341</span>, Janosch Ortmann, University of Toronto==
 
 
Title: Product-form Invariant Measures for Brownian Motion with Drift Satisfying a Skew-symmetry Type Condition
 
 
Abstract: Motivated by recent developments on positive-temperature polymer models we propose a generalisation of reflected Brownian motion (RBM) in a polyhedral domain. Our process is obtained by replacing the singular drift on the boundary by a continuous one which depends, via a potential U, on the position of the process relative to the domain. We show that our generalised process has an invariant measure in product form, under a certain skew-symmetry condition that is independent of the choice of potential. Applications include TASEP-like particle systems, generalisations of Brownian motion with rank-dependent drift and diffusions connected to the generalised Pitman transform.
 
 
==Thursday, March 14, Brian Rider, Temple University==
 
 
Title: Universality for the stochastic Airy operator
 
 
Abstract: The stochastic Airy operator (SAO) has the form second derivative plus shifted white noise potential. Its reason for being is that it describes the Tracy-Widom laws extended to "general beta" (from the original beta=1,2,4 laws tied to real, complex, and quaternion symmetries). More to the point, SAO is known to be the operator limit for certain random tridiagonal matrices which realize, for example, log-gas distributions on the line with quadratic potential (the "beta Hermite ensembles"), scaled to the edge of their spectrum.  Here we show that SAO characterizes edge universality for a more general class of log-gases, defined by more general polynomial potentials beyond the quadratic case.  Joint work with M. Krishnapur and B. Virag.
 
 
==Thursday, March 21, Timo Seppalainen (UW Madison) ==
 
 
Title:  Limits of ratios of partition functions for the log-gamma polymer
 
 
Abstract: For the model known as the directed polymer in a random medium, the definition of weak disorder is that normalized
 
partition functions converge to a positive limit.  In strong disorder this limit vanishes.  In the log-gamma polymer we
 
can show that ratios of point-to-point and point-to-line partition functions converge to gamma-distributed limits.
 
One consequence of this is that the quenched polymer measure converges to a random walk in a correlated random environment.
 
This RWRE can be regarded as a positive temperature analogue of the competition interface of last-passage percolation,
 
or the second class particle.
 
 
== Thursday, April 11, [http://math.arizona.edu/~klin/index.php Kevin Lin], University of Arizona==
 
 
Title: Stimulus-response reliability of dynamical networks
 
 
Abstract: A network of dynamical systems (e.g., neurons) driven by a
 
fluctuating time-dependent signal is said to be reliable if,
 
upon repeated presentations of the same signal, it gives
 
essentially the same response each time.  As a system's
 
degree of reliability may constrain its ability to encode
 
and transmit information, a natural question is how network
 
conditions affect reliability; this question is of interest
 
in e.g. computational neuroscience.  In this talk, I will
 
report on a body of work aimed at discovering network
 
conditions and dynamical mechanisms that affect the
 
reliability of networks, within a class of idealized neural
 
network models.  I will discuss a general condition for
 
reliability, and survey some specific mechanisms for
 
reliable and unreliable behavior in concrete models.
 
 
== <span style="color:#FF0000"> Tuesday, April 16, 2:30pm, VV B341</span> [http://www.mathstat.concordia.ca/faculty/lpopovic/ Lea Popovic], Concordia University==
 
 
Title: Stochastically induced bistability in chemical reaction systems
 
 
Abstract: We study a stochastic two-species interacting population system, in which species interact within each compartment according to some nonlinear dynamics. In addition we have another mechanism (e.g. migration between compartments, or splitting of compartments) which  yield unbiased perturbative changes to species amounts. If each compartment has a large but bounded capacity, then certain combination of these two mechanisms can lead to stochastically induced bistability. In fact,  depending on the relative rates between the mechanisms, there are two ways in which bistability can occur, with distinct signatures. This problem is motivated by dynamics of certain biochemical processes such as gene expression, where the numbers of species interacting are small enough that the randomness inherent in chemical reaction processes can no longer be ignored.
 
This is joint work with J. McSweeney.
 
 
== Thursday, April 18, [http://www.math.uiuc.edu/~rdeville/ Lee DeVille], University of Illinois==
 
 
Title: Emergent metastability for dynamical systems on networks
 
 
Abstract:  We will consider stochastic dynamical systems defined on networks that exhibit the phenomenon of collective metastability---by this we mean network dynamics where none of the individual nodes' dynamics are metastable, but the configuration is metastable in its collective behavior.  We will concentrate on the case of SDE with small white noise for concreteness.  We also present some specific results relating to stochastic perturbations of the Kuramoto system of coupled nonlinear oscillators.  Along the way, we show that there is a non-standard spectral problem that appears naturally, and that the important features of this spectral problem are determined by a certain homology group.
 
 
== Thursday, April 25,  [http://math.berkeley.edu/~rezakhan/ Fraydoun Rezakhanlou], UC - Berkeley==
 
 
Title: Stochastic Poincare--Birkhoff Theorem
 
 
Abstract: The Poincare--Birkhoff Theorem , also called Poincare Last Geometric
 
Theorem, is a landmark result in area preserving
 
dynamics. It was formulated by Poincare based on his investigations in
 
celestial mechanics.
 
The theorem may be easily stated: a periodic twist map of an infinite closed
 
strip, or closed annulus, has at least
 
two geometrically distinct fixed points.
 
V.I. Arnold realized that the correct generalization to higher dimensions
 
concerned symplectic maps, not volume preserving maps. He then formulated
 
the higher dimensional analogue of the Poincare--Birkhoff Theorem: the
 
Arnold Conjecture.
 
A parallel generalization of classical the Poincare--Birkhoff Theorem is
 
to investigate whether it holds in the stochastic setting.
 
That is, maps are now stochastic with respect to some probability
 
measure. In this talk, I discuss a variant of the Poincare--Birkhoff Theorem
 
for stationary area preserving dynamics, and hopefully opens
 
the way to a stochastic Arnold Conjecture. (Joint work with Alvaro Pelayo.)
 
 
 
==  <span style="color:#FF0000"> Wednesday, May 1, VV B115, </span> [http://www-wt.iam.uni-bonn.de/~vetob/ Bálint Vető], University of Bonn ==
 
 
Title: Stationary Solution of the 1D KPZ Equation
 
 
Abstract: The KPZ equation is believed to describe a variety of surface growth phenomena that appear naturally, e.g. crystal growth, facet boundaries, solidification fronts, paper wetting or burning fronts. In the recent years, serious efforts were made to describe the solution with different types of initial data. In the present work, we derive an explicit solution for the equation with stationary, i.e. two-sided Brownian motion initial condition. Our approach to the solution for the KPZ equation is via its representation as the free energy of a certain directed random polymer model. By providing integral formulas for the action of Macdonald difference operators, we characterize explicitly the free energy of another polymer model by giving a Fredholm determinant formula which is suitable for asymptotic analysis. In the large time limit of the solution, we recover the distribution obtained for the limiting fluctuations of the height function of the stationary totally asymmetric simple exclusion process (TASEP).
 

Revision as of 13:12, 11 June 2013


Fall 2013

Thursdays in 901 Van Vleck Hall at 2:25 PM, unless otherwise noted. If you would like to receive announcements about upcoming seminars, please visit this page to sign up for the email list.


Past Seminars

Thursday, September 12, TBA

Title: TBA

Abstract: