Difference between revisions of "Probability Seminar"

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(Tuesday, April 16, 2:30pm, B341 Lea Popovic, Concordia University)
(April 23, 2020, Martin Hairer (Imperial College))
 
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__NOTOC__
 
__NOTOC__
  
== Spring 2013 ==
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= Spring 2020 =
  
 +
<b>Thursdays in 901 Van Vleck Hall at 2:30 PM</b>, unless otherwise noted.
 +
<b>We  usually end for questions at 3:20 PM.</b>
  
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 [https://www-old.cae.wisc.edu/mailman/listinfo/apseminar this page] to sign up for the email list.
+
If you would like to sign up for the email list to receive seminar announcements then please send an email to
 +
[mailto:join-probsem@lists.wisc.edu join-probsem@lists.wisc.edu]
  
 +
 +
== January 23, 2020, [https://www.math.wisc.edu/~seppalai/ Timo Seppalainen] (UW Madison) ==
 +
'''Non-existence of bi-infinite geodesics in the exponential corner growth model
 +
'''
  
[[Past Seminars]]
+
Whether bi-infinite geodesics exist has been a significant open problem in first- and last-passage percolation since the mid-80s.  A non-existence proof  in the case of directed planar last-passage percolation with exponential weights was posted by Basu, Hoffman and Sly in  November 2018. Their proof utilizes estimates from integrable probability.    This talk describes an independent proof completed 10 months later that relies on couplings, coarse graining, and control of geodesics through planarity and increment-stationary last-passage percolation. Joint work with Marton Balazs and Ofer Busani (Bristol).
 +
 
 +
== January 30, 2020, [https://www.math.wisc.edu/people/vv-prof-directory Scott Smith] (UW Madison) ==
 +
'''Quasi-linear parabolic equations with singular forcing'''
 +
 
 +
The classical solution theory for stochastic ODE's is centered around Ito's stochastic integral.  By intertwining ideas from analysis and probability, this approach extends to many PDE's, a canonical example being multiplicative stochastic heat equations driven by space-time white noise.  In both the ODE and PDE settings, the solution theory is beyond the scope of classical deterministic theory because of the ambiguity in multiplying a function with a white noise.  The theory of rough paths and regularity structures provides a more quantitative understanding of this difficulty, leading to a more refined solution theory which efficiently divides the analytic and probabilistic aspects of the problem, and remarkably, even has an algebraic component.
 +
 
 +
In this talk, we will discuss a new application of these ideas to stochastic heat equations where the strength of the diffusion is not constant but random, as it depends locally on the solution.  These are known as quasi-linear equations.  Our main result yields the deterministic side of a solution theory for these PDE's, modulo a suitable renormalization.  Along the way, we identify a formally infinite series expansion of the solution which guides our analysis, reveals a nice algebraic structure, and encodes the counter-terms in the PDE.  This is joint work with Felix Otto, Jonas Sauer, and Hendrik Weber.
 +
 
 +
== February 6, 2020, [https://sites.google.com/site/cyleeken/ Cheuk-Yin Lee] (Michigan State) ==
 +
'''Sample path properties of stochastic partial differential equations: modulus of continuity and multiple points'''
 +
 
 +
In this talk, we will discuss sample path properties of stochastic partial differential equations (SPDEs). We will present a sharp regularity result for the stochastic wave equation driven by an additive Gaussian noise that is white in time and colored in space. We prove the exact modulus of continuity via the property of local nondeterminism. We will also discuss the existence problem for multiple points (or self-intersections) of the sample paths of SPDEs. Our result shows that multiple points do not exist in the critical dimension for a large class of Gaussian random fields including the solution of a linear system of stochastic heat or wave equations.
 +
 
 +
== February 13, 2020, [http://www.jelena-diakonikolas.com/ Jelena Diakonikolas] (UW Madison) ==
 +
'''Langevin Monte Carlo Without Smoothness'''
  
== Thursday, January 31, Bret Larget, UW-Madison ==
+
Langevin Monte Carlo (LMC) is an iterative algorithm used to generate samples from a distribution that is known only up to a normalizing constant. The nonasymptotic dependence of its mixing time on the dimension and target accuracy is understood mainly in the setting of smooth (gradient-Lipschitz) log-densities, a serious limitation for applications in machine learning. We remove this limitation by providing polynomial-time convergence guarantees for a variant of LMC in the setting of non-smooth log-concave distributions. At a high level, our results follow by leveraging the implicit smoothing of the log-density that comes from a small Gaussian perturbation that we add to the iterates of the algorithm and while controlling the bias and variance that are induced by this perturbation.
 +
Based on joint work with Niladri Chatterji, Michael I. Jordan, and Peter L. Bartlett.
  
Title: Approximate conditional independence of separated subtrees and phylogenetic inference
+
== February 20, 2020, [https://math.berkeley.edu/~pmwood/ Philip Matchett Wood] (UC Berkeley) ==
 +
'''A replacement principle for perturbations of non-normal matrices'''
  
Abstract:
+
There are certain non-normal matrices whose eigenvalues can change dramatically when a small perturbation is added. However, when that perturbation is an iid random matrix, it appears that the eigenvalues become stable after perturbation and only change slightly when further small perturbations are added.  Much of the work is this situation has focused on iid random gaussian perturbations. In this talk, we will discuss work on a universality result that allows for consideration of non-gaussian perturbations, and that shows that all perturbations satisfying certain conditions will produce the same limiting eigenvalue measureInterestingly, this even allows for deterministic perturbations to be considered. Joint work with Sean O'Rourke.
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 sampleThe
 
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==
+
== February 27, 2020, No seminar ==
 +
''' '''
  
Title: Biomixing and large deviations
+
== March 5, 2020, [https://www.ias.edu/scholars/jiaoyang-huang Jiaoyang Huang] (IAS) ==
 +
''' Large Deviation Principles via Spherical Integrals'''
  
Abstract: As fish, micro-organisms, or other bodies move through a fluid, they
+
In this talk, I'll explain a framework to study the large deviation principle for matrix models and their quantized versions, by tilting the measures using the asymptotics of spherical integrals obtained by Guionnet and Zeitouni. As examples, we obtain
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==
+
1) the large deviation principle for the empirical distribution of the diagonal entries of $UB_NU^*$, for a sequence of $N\times N$ diagonal matrices $B_N$ and unitary/orthogonal Haar distributed matrices $U$;
  
Title: Product-form Invariant Measures for Brownian Motion with Drift Satisfying a Skew-symmetry Type Condition
+
2) the large deviation upper bound for the empirical eigenvalue distribution of $A_N+UB_NU^*$, for two sequences of $N\times N$ diagonal matrices $A_N, B_N$, and their complementary lower bounds at "good" probability distributions;
  
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.
+
3) the large deviation principle for the Kostka number $K_{\lambda_N \eta_N}$, for two sequences of partitions $\lambda_N, \eta_N$ with at most $N$ rows;
  
==Thursday, March 14, Brian Rider, Temple University==
+
4) the large deviation upper bound for the Littlewood-Richardson coefficients $c_{\lambda_N \eta_N}^{\kappa_N}$, for three sequences of partitions $\lambda_N, \eta_N, \kappa_N$ with at most $N$ rows, and their complementary lower bounds at "good" probability distributions.
  
Title: Universality for the stochastic Airy operator
+
This is a joint work with Belinschi and Guionnet.
  
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.
+
== March 12, 2020, No seminar ==
 +
''' '''
  
==Thursday, March 21, Timo Seppalainen (UW Madison) ==
+
== March 19, 2020, Spring break ==
 +
''' '''
  
Title: Limits of ratios of partition functions for the log-gamma polymer
+
== March 26, 2020, CANCELLED, [https://math.cornell.edu/philippe-sosoe Philippe Sosoe] (Cornell) ==
 +
''' '''
  
Abstract: For the model known as the directed polymer in a random medium, the definition of weak disorder is that normalized
+
== April 2, 2020, CANCELLED, [http://pages.cs.wisc.edu/~tl/ Tianyu Liu] (UW Madison)==
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==
+
== April 9, 2020, CANCELLED, [http://stanford.edu/~ajdunl2/ Alexander Dunlap] (Stanford) ==
 +
''' '''
  
Title: Stimulus-response reliability of dynamical networks
+
== April 16, 2020, CANCELLED, [https://statistics.wharton.upenn.edu/profile/dingjian/ Jian Ding] (University of Pennsylvania) ==
 +
''' '''
  
Abstract: A network of dynamical systems (e.g., neurons) driven by a
+
== April 22-24, 2020, CANCELLED, [http://frg.int-prob.org/ FRG Integrable Probability] meeting ==
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==
+
3-day event in Van Vleck 911
  
Title: Stochastically induced bistability in chemical reaction systems
+
== April 23, 2020, CANCELLED, [http://www.hairer.org/ Martin Hairer] (Imperial College) ==
  
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.
+
[https://www.math.wisc.edu/wiki/index.php/Colloquia Wolfgang Wasow Lecture] at 4pm in Van Vleck 911
This is joint work with J. McSweeney.
 
  
== Thursday, April 18, [http://www.math.uiuc.edu/~rdeville/ Lee DeVille], University of Illinois==
+
== April 30, 2020, [http://willperkins.org/ Will Perkins] (University of Illinois at Chicago) ==
 +
''' '''
  
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: TBA
 
  
Abstract: TBA
 
  
== Wednesday, May 1, [http://www-wt.iam.uni-bonn.de/~vetob/ Bálint Vető], University of Bonn ==
 
  
Title: TBA
 
  
Abstract: TBA
+
[[Past Seminars]]

Latest revision as of 13:51, 24 March 2020


Spring 2020

Thursdays in 901 Van Vleck Hall at 2:30 PM, unless otherwise noted. We usually end for questions at 3:20 PM.

If you would like to sign up for the email list to receive seminar announcements then please send an email to join-probsem@lists.wisc.edu


January 23, 2020, Timo Seppalainen (UW Madison)

Non-existence of bi-infinite geodesics in the exponential corner growth model

Whether bi-infinite geodesics exist has been a significant open problem in first- and last-passage percolation since the mid-80s. A non-existence proof in the case of directed planar last-passage percolation with exponential weights was posted by Basu, Hoffman and Sly in November 2018. Their proof utilizes estimates from integrable probability. This talk describes an independent proof completed 10 months later that relies on couplings, coarse graining, and control of geodesics through planarity and increment-stationary last-passage percolation. Joint work with Marton Balazs and Ofer Busani (Bristol).

January 30, 2020, Scott Smith (UW Madison)

Quasi-linear parabolic equations with singular forcing

The classical solution theory for stochastic ODE's is centered around Ito's stochastic integral. By intertwining ideas from analysis and probability, this approach extends to many PDE's, a canonical example being multiplicative stochastic heat equations driven by space-time white noise. In both the ODE and PDE settings, the solution theory is beyond the scope of classical deterministic theory because of the ambiguity in multiplying a function with a white noise. The theory of rough paths and regularity structures provides a more quantitative understanding of this difficulty, leading to a more refined solution theory which efficiently divides the analytic and probabilistic aspects of the problem, and remarkably, even has an algebraic component.

In this talk, we will discuss a new application of these ideas to stochastic heat equations where the strength of the diffusion is not constant but random, as it depends locally on the solution. These are known as quasi-linear equations. Our main result yields the deterministic side of a solution theory for these PDE's, modulo a suitable renormalization. Along the way, we identify a formally infinite series expansion of the solution which guides our analysis, reveals a nice algebraic structure, and encodes the counter-terms in the PDE. This is joint work with Felix Otto, Jonas Sauer, and Hendrik Weber.

February 6, 2020, Cheuk-Yin Lee (Michigan State)

Sample path properties of stochastic partial differential equations: modulus of continuity and multiple points

In this talk, we will discuss sample path properties of stochastic partial differential equations (SPDEs). We will present a sharp regularity result for the stochastic wave equation driven by an additive Gaussian noise that is white in time and colored in space. We prove the exact modulus of continuity via the property of local nondeterminism. We will also discuss the existence problem for multiple points (or self-intersections) of the sample paths of SPDEs. Our result shows that multiple points do not exist in the critical dimension for a large class of Gaussian random fields including the solution of a linear system of stochastic heat or wave equations.

February 13, 2020, Jelena Diakonikolas (UW Madison)

Langevin Monte Carlo Without Smoothness

Langevin Monte Carlo (LMC) is an iterative algorithm used to generate samples from a distribution that is known only up to a normalizing constant. The nonasymptotic dependence of its mixing time on the dimension and target accuracy is understood mainly in the setting of smooth (gradient-Lipschitz) log-densities, a serious limitation for applications in machine learning. We remove this limitation by providing polynomial-time convergence guarantees for a variant of LMC in the setting of non-smooth log-concave distributions. At a high level, our results follow by leveraging the implicit smoothing of the log-density that comes from a small Gaussian perturbation that we add to the iterates of the algorithm and while controlling the bias and variance that are induced by this perturbation. Based on joint work with Niladri Chatterji, Michael I. Jordan, and Peter L. Bartlett.

February 20, 2020, Philip Matchett Wood (UC Berkeley)

A replacement principle for perturbations of non-normal matrices

There are certain non-normal matrices whose eigenvalues can change dramatically when a small perturbation is added. However, when that perturbation is an iid random matrix, it appears that the eigenvalues become stable after perturbation and only change slightly when further small perturbations are added. Much of the work is this situation has focused on iid random gaussian perturbations. In this talk, we will discuss work on a universality result that allows for consideration of non-gaussian perturbations, and that shows that all perturbations satisfying certain conditions will produce the same limiting eigenvalue measure. Interestingly, this even allows for deterministic perturbations to be considered. Joint work with Sean O'Rourke.

February 27, 2020, No seminar

March 5, 2020, Jiaoyang Huang (IAS)

Large Deviation Principles via Spherical Integrals

In this talk, I'll explain a framework to study the large deviation principle for matrix models and their quantized versions, by tilting the measures using the asymptotics of spherical integrals obtained by Guionnet and Zeitouni. As examples, we obtain

1) the large deviation principle for the empirical distribution of the diagonal entries of $UB_NU^*$, for a sequence of $N\times N$ diagonal matrices $B_N$ and unitary/orthogonal Haar distributed matrices $U$;

2) the large deviation upper bound for the empirical eigenvalue distribution of $A_N+UB_NU^*$, for two sequences of $N\times N$ diagonal matrices $A_N, B_N$, and their complementary lower bounds at "good" probability distributions;

3) the large deviation principle for the Kostka number $K_{\lambda_N \eta_N}$, for two sequences of partitions $\lambda_N, \eta_N$ with at most $N$ rows;

4) the large deviation upper bound for the Littlewood-Richardson coefficients $c_{\lambda_N \eta_N}^{\kappa_N}$, for three sequences of partitions $\lambda_N, \eta_N, \kappa_N$ with at most $N$ rows, and their complementary lower bounds at "good" probability distributions.

This is a joint work with Belinschi and Guionnet.

March 12, 2020, No seminar

March 19, 2020, Spring break

March 26, 2020, CANCELLED, Philippe Sosoe (Cornell)

April 2, 2020, CANCELLED, Tianyu Liu (UW Madison)

April 9, 2020, CANCELLED, Alexander Dunlap (Stanford)

April 16, 2020, CANCELLED, Jian Ding (University of Pennsylvania)

April 22-24, 2020, CANCELLED, FRG Integrable Probability meeting

3-day event in Van Vleck 911

April 23, 2020, CANCELLED, Martin Hairer (Imperial College)

Wolfgang Wasow Lecture at 4pm in Van Vleck 911

April 30, 2020, Will Perkins (University of Illinois at Chicago)





Past Seminars