Probability Seminar: Difference between revisions

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[[Probability | Back to Probability Group]]


= Fall 2020 =
[[Past Seminars]]


<b>Thursdays in 901 Van Vleck Hall at 2:30 PM</b>, unless otherwise noted.
= Fall 2023 =
<b>We  usually end for questions at 3:20 PM.</b>
<b>Thursdays at 2:30 PM either in 901 Van Vleck Hall or on Zoom</b>


<b> IMPORTANT: </b> In Fall 2020 the seminar is being run online. [https://uwmadison.zoom.us/j/91828707031?pwd=YUJXMUJkMDlPR0VRdkRCQVJtVndIdz09 ZOOM LINK]
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 join [https://groups.google.com/a/g-groups.wisc.edu/forum/#!forum/probsem our group].
== September 14, 2023: [https://www.mathjunge.com/ Matthew Junge] (CUNY) ==
'''The frog model on trees'''
== September 17, 2020, [https://www.math.tamu.edu/~bhanin/ Boris Hanin] (Princeton and Texas A&M) ==


'''Pre-Talk: (1:00pm)'''
The frog model describes random activation and spread. Think combustion or an epidemic. I have studied these dynamics on ''d''-ary trees for ten years. I will discuss our progress and what remains to be done.


'''Neural Networks for Probabilists'''
== September 21, 2023: [https://yierlin.me/ Yier Lin] (U. Chicago) ==
'''Large Deviations of the KPZ Equation and Most Probable Shapes'''


Deep neural networks are a centerpiece in modern machine learning. They are also fascinating probabilistic models, about which much remains unclear. In this pre-talk I will define neural networks, explain how they are used in practice, and give a survey of the big theoretical questions they have raised. If time permits, I will also explain how neural networks are related to a variety of classical areas in probability and mathematical physics, including random matrix theory, optimal transport, and combinatorics of hyperplane arrangements.


'''Talk: (2:30pm)'''
The KPZ equation is a stochastic PDE that plays a central role in a class of random growth phenomena. In this talk, we will explore the Freidlin-Wentzell LDP for the KPZ equation through the lens of the variational principle. Additionally, we will explain how to extract various limits of the most probable shape of the KPZ equation using the variational formula. We will also discuss an alternative approach for studying these quantities using the method of moments. This talk is based in part on joint works with Pierre Yves Gaudreau Lamarre and Li-Cheng Tsai.


'''Effective Theory of Deep Neural Networks'''  
== September 28, 2023: [https://warwick.ac.uk/fac/sci/statistics/staff/academic-research/rosati/ Tommaso Rosati] (U. Warwick) ==
'''The Allen-Cahn equation with weakly critical initial datum'''


Deep neural networks are often considered to be complicated "black boxes," for which a full systematic analysis is not only out of reach but also impossible. In this talk, which is based on ongoing joint work with Sho Yaida and Daniel Adam Roberts, I will make the opposite claim. Namely, that deep neural networks with random weights and biases are exactly solvable models. Our approach applies to networks at finite width n and large depth L, the regime in which they are used in practice. A key point will be the emergence of a notion of "criticality," which involves a finetuning of model parameters (weight and bias variances). At criticality, neural networks are particularly well-behaved but still exhibit a tension between large values for n and L, with large values of n tending to make neural networks more like Gaussian processes and large values of L amplifying higher cumulants. Our analysis at initialization has many consequences also for networks during after training, which I will discuss if time permits.
We study the 2D Allen-Cahn with white noise initial datum. In a weak coupling regime, where the nonlinearity is damped in relation to the smoothing of the initial condition, we prove Gaussian fluctuations. The effective variance that appears can be described as the solution to an ODE. Our proof builds on a Wild expansion of the solution, which is controlled through precise combinatorial estimates. Joint work with Simon Gabriel and Nikolaos Zygouras.


== September 24, 2020, [https://people.ucd.ie/neil.oconnell Neil O'Connell] (Dublin) ==
== October 5, 2023:  ==
'''Abstract, title: TBA'''


'''Some new perspectives on moments of random matrices'''
== October 12, 2023: No Seminar ==


The study of `moments' of random matrices (expectations of traces of powers of the matrix) is a rich and interesting subject, with fascinating connections to enumerative geometry, as discovered by Harer and Zagier in the 1980’s. I will give some background on this and then describe some recent work which offers some new perspectives (and new results). This talk is based on joint work with Fabio Deelan Cunden, Francesco Mezzadri and Nick Simm.
== October 19, 2023: ==


== October 1, 2020, [https://marcusmichelen.org/ Marcus Michelen] (UIC) ==
== October 26, 2023: Yuchen Liao (UW - Madison) ==
'''Abstract, title: TBA'''


'''Roots of random polynomials near the unit circle'''
== November 2, 2023: [http://homepages.math.uic.edu/~couyang/ Cheng Ouyang] (U. Illinois Chicago) ==
'''Abstract, title: TBA'''


It is a well-known (but perhaps surprising) fact that a polynomial with independent random coefficients has most of its roots very close to the unit circle.  Using a probabilistic perspective, we understand the behavior of roots of random polynomials exceptionally close to the unit circle and prove several limit theorems; these results resolve several conjectures of Shepp and Vanderbei.  We will also discuss how our techniques provide a heuristic, probabilistic explanation for why random polynomials tend to have most roots near the unit circle.  Based on joint work with Julian Sahasrabudhe.
== November 9, 2023: [https://scottandrewsmith.github.io/ Scott Smith] (Chinese Academy of Sciences) ==
'''Abstract, title: TBA'''


== October 8, 2020, [http://sites.harvard.edu/~sus977/index.html Subhabrata Sen] (Harvard) ==
== November 16, 2023: ==
'''Abstract, title: TBA'''


'''Large deviations for dense random graphs: beyond mean-field'''
== November 23, 2023: No Seminar ==
'''No seminar. Thanksgiving.'''


In a seminal paper, Chatterjee and Varadhan derived an Erdős-Rényi random graph, viewed as a random graphon. This directly provides LDPs for continuous functionals such as subgraph counts, spectral norms, etc. In contrast, very little is understood about this problem if the underlying random graph is inhomogeneous or constrained.
== November 30, 2023: [http://web.mit.edu/youngtak/www/homepage.html Youngtak Sohn] (MIT) ==
'''Abstract, title: TBA'''


In this talk, we will explore large deviations for dense random graphs, beyond the “mean-field” setting. In particular, we will study large deviations for uniform random graphs with given degrees, and a family of dense block model
== December 7, 2023: Minjae Park (U. Chicago) ==
random graphs. We will establish the LDP in each case, and identify the rate function. In the block model setting, we will use this LDP to study the upper tail problem for homomorphism densities of regular sub-graphs. Our results establish that this problem exhibits a symmetry/symmetry-breaking transition, similar to one observed for Erdős-Rényi random graphs.
'''Abstract, title: TBA'''
 
Based on joint works with Christian Borgs, Jennifer Chayes, Souvik Dhara, Julia Gaudio and Samantha Petti.
 
== October 15, 2020, [https://math.cornell.edu/philippe-sosoe Philippe Sosoe] (Cornell) ==
 
Title: '''Concentration in integrable polymer models'''
 
I will discuss a general method, applicable to all known integrable stationary polymer models, to obtain nearly optimal bounds on the
central moments of the partition function and the occupation lengths for each level of the polymer system. The method was developed
for the O'Connell-Yor polymer, but was subsequently extended to discrete integrable polymers. As an application, we obtain
localization of the OY polymer paths along a straight line on the scale O(n^{2/3+o(1)}).
 
Joint work with Christian Noack.
 
==October 22, 2020, [http://www.math.toronto.edu/balint/ Balint Virag] (Toronto) ==
 
Title: '''The heat and the landscape'''
 
Abstract: The directed landscape is the conjectured universal scaling limit of the
most common random planar metrics. Examples are planar first passage
percolation, directed last passage percolation, distances in percolation
clusters, random polymer models, and exclusion processes. The limit laws of distances of objects are given by the KPZ fixed point.
 
We show that the KPZ fixed point is characterized by the Baik Ben-Arous
Peche statistics well-known from random matrix theory.
 
This provides a general and elementary method for showing convergence to
the KPZ fixed point. We apply this method to two models related to
random heat flow: the O'Connell-Yor polymer and the KPZ equation.
 
Note: there will be a follow-up talk with details about the proofs at 11am, Friday, October 23.
 
==October 29, 2020, [https://www.math.wisc.edu/node/80 Yun Li] (UW-Madison) ==
 
Title: '''TBA'''
 
Abstract: TBA
 
== November 5, 2020, [http://sayan.web.unc.edu/ Sayan Banerjee] (UNC at Chapel Hill) ==
 
Title: '''TBA'''
 
Abstract: TBA
 
== November 12, 2020, [https://cims.nyu.edu/~ajd594/ Alexander Dunlap] (NYU Courant Institute) ==
 
Title: '''A forward-backward SDE from the 2D nonlinear stochastic heat equation'''
 
Abstract: I will discuss a two-dimensional stochastic heat equation in the weak noise regime with a nonlinear noise strength. I will explain how pointwise statistics of solutions to this equation, as the correlation length of the noise is taken to 0 but the noise is attenuated by a logarithmic factor, can be related to a forward-backward stochastic differential equation (FBSDE) depending on the nonlinearity. In the linear case, the FBSDE can be explicitly solved and we recover results of Caravenna, Sun, and Zygouras. Joint work with Yu Gu (CMU).
 
== November 19, 2020, [https://statistics.wharton.upenn.edu/profile/dingjian/ Jian Ding] (University of Pennsylvania) ==
 
Title: '''TBA'''
 
Abstract: TBA
 
== December 3, 2020, [https://www.math.wisc.edu/people/faculty-directory Tatyana Shcherbina] (UW-Madison) ==
 
Title: '''TBA'''
 
Abstract: TBA
 
== December 10, 2020, [https://www.ewbates.com/ Erik Bates] (UW-Madison) ==
 
Title: '''TBA'''
 
Abstract: TBA
 
 
[[Past Seminars]]

Latest revision as of 13:28, 25 September 2023

Back to Probability Group

Past Seminars

Fall 2023

Thursdays at 2:30 PM either in 901 Van Vleck Hall or on Zoom

We usually end for questions at 3:20 PM.

September 14, 2023: Matthew Junge (CUNY)

The frog model on trees

The frog model describes random activation and spread. Think combustion or an epidemic. I have studied these dynamics on d-ary trees for ten years. I will discuss our progress and what remains to be done.

September 21, 2023: Yier Lin (U. Chicago)

Large Deviations of the KPZ Equation and Most Probable Shapes


The KPZ equation is a stochastic PDE that plays a central role in a class of random growth phenomena. In this talk, we will explore the Freidlin-Wentzell LDP for the KPZ equation through the lens of the variational principle. Additionally, we will explain how to extract various limits of the most probable shape of the KPZ equation using the variational formula. We will also discuss an alternative approach for studying these quantities using the method of moments. This talk is based in part on joint works with Pierre Yves Gaudreau Lamarre and Li-Cheng Tsai.

September 28, 2023: Tommaso Rosati (U. Warwick)

The Allen-Cahn equation with weakly critical initial datum

We study the 2D Allen-Cahn with white noise initial datum. In a weak coupling regime, where the nonlinearity is damped in relation to the smoothing of the initial condition, we prove Gaussian fluctuations. The effective variance that appears can be described as the solution to an ODE. Our proof builds on a Wild expansion of the solution, which is controlled through precise combinatorial estimates. Joint work with Simon Gabriel and Nikolaos Zygouras.

October 5, 2023:

Abstract, title: TBA

October 12, 2023: No Seminar

October 19, 2023:

October 26, 2023: Yuchen Liao (UW - Madison)

Abstract, title: TBA

November 2, 2023: Cheng Ouyang (U. Illinois Chicago)

Abstract, title: TBA

November 9, 2023: Scott Smith (Chinese Academy of Sciences)

Abstract, title: TBA

November 16, 2023:

Abstract, title: TBA

November 23, 2023: No Seminar

No seminar. Thanksgiving.

November 30, 2023: Youngtak Sohn (MIT)

Abstract, title: TBA

December 7, 2023: Minjae Park (U. Chicago)

Abstract, title: TBA

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