Dynamics and Probability Seminar Spring 2023

The Dynamics and Probability seminar will meet on Tuesday afternoons at 2:30 in DSB A128, or on Zoom when the speaker is not present in Victoria. All are welcome. Please contact Anthony Quas for the zoom link. The list of seminars is as follows. For further information, please contact me at aquas(a)uvic.ca or Gourab Ray at gourabray(a)uvic.ca

Date Speaker Title
10/1/2023 Sebastian Ferrando (Toronto Metropolitan) Non-probabilistic supermartingales
17/1/2023 Gourab Ray (UVic) Characterizing nonamenability through stochastic domination and finitary factors
24/1/2023 Grigory Terlov (Illinois) Stein’s method for conditional central limit theorem
31/1/2023 Will Perkins (Georgia Tech) The (symmetric) Ising perceptron: progress and problems
7/2/2023 Peleg Michaeli (Carnegie Mellon) Fast construction on a restricted budget
14/2/2023 Yin-Ting Liao (Irvine)
28/2/2023 Ahmed Bou-Rabee (Cornell)
7/3/2023 Máté Wierdl (Memphis)
16/3/2023 Pablo Shmerkin (UBC)
21/3/2023 Neha Bansal (UBC Okanagan)
28/3/2023 Emilio Corso (UBC)
4/4/2023 Natalie Behague (UVic)

Date: 10/1/2023
Speaker: Sebastian Ferrando (Toronto Metropolitan University)
Title: Non-probabilistic supermartingales
Abstract: We provide a brief motivational overview of recent developments of extensions of stochastics tools to deal with uncertainty. These are: Peng's nonlinear expectations and Ito's calculus without probabilities. We then describe a non-probabilistic version of a supermartingale theory closely motivated by financial considerations of no-arbitrage. The basic object replacing the classical filtered probability space is a structured trajectory set which allows the definition of conditional outer integrals as well as null sets. The conditional outer integrals are non linear functionals that allow to circumvent the linearity of the classical conditional expectations in proofs and definitions. Integrability notions emerge in our setting through non-classical conditional integral operators that lead to the special case of non-probabilistic martingales. One can define non-probabilistic supermartingales and prove analogous of classical results like: Doob's optional sampling theorem, Dubin's upcrossing inequalities and Doob's a.e. convergence for non-negative supermartingales. All constructions and results have a hedging and superhedging interpretation and there is a direct way in which the new results generalize the classical case. Null sets appearing in the results have a financial interpretation and are handled in a more concrete way than in the classical theory.
Date: 17/1/2023
Speaker: Gourab Ray (UVic)
Title: Characterizing nonamenability through stochastic domination and finitary factors
Abstract: Take an Ising model with very low temperature. What is the largest p such that the Ising model dominates Bernoulli percolation with parameter p ? We will show that the answer to this question depends drastically on the geometry of the graph. We also obtain similar results for for two Ising models at very low, but close temperatures. A process is a finitary factor of iid if it can be written as a measurable and equivariant function of an iid process. As an application of the domination results, we show that the very low temperature Ising model on a nonamenable graph is a finitary factor of iid. This is in stark contrast with the amenable setting, where it is known through a celebrated result of Van Den Berg and Steif that the low temperature Ising model is not a finitary factor of iid. Joint work with Yinon Spinka.
Date: 24/1/2023
Speaker: Grigory Terlov (Illinois)
Title: Stein’s method for conditional central limit theorem
Abstract: It is common in probability theory and statistics to study distributional convergences of sums of random variables conditioned on another such sum. In this talk I will present a novel approach using Stein’s method for exchangeable pairs that allows to derive a conditional central limit theorem of the form $(X_n|Y_n = k)$ with explicit rate of convergence as well as its extensions to a multidimensional setting. We will apply these results to particular models including pattern counts in a random binary sequence and subgraph counts in Erdös-Rényi random graph. This talk is based on joint work with Partha S. Dey.
Date: 31/1/2023
Speaker: Will Perkins (Georgia Tech)
Title: The (symmetric) Ising perceptron: progress and problems
Abstract: The Perceptron model was proposed as early as the 1950's as a toy model of a one-layer neural network. The basic model consists of a set of solutions (either the Hamming cube or the sphere of dimension n) and a set of constraints given by independent n-dimensional Gaussian vectors. The constraints are that the inner product of a solution vector with each constraint vector scaled by sqrt{n} must lie in some interval on the real line. Probabilistic questions about the model include the satisfiability threshold (or the "storage capacity") and questions about the typical structure of the solution space. Algorithmic questions include the tractability of finding a solution (the learning problem in the neural network interpretation). I will describe the model, the main problems, and recent progress.
DATE: 7/2/2023
Speaker: Peleg Michaeli (Carnegie Mellon)
Title: Fast construction on a restricted budget
Abstract: We introduce a model of a controlled random process. In this model, the vertices of a hypergraph are ordered randomly and then revealed, one by one, to an algorithm. The algorithm must decide, immediately and irrevocably, whether to keep each observed vertex. Given the total number of observed vertices ("time"), the algorithm's goal is to succeed - asymptotically almost surely - in completing a hyperedge by keeping ("purchasing") the smallest possible number of vertices. We analyse this model in the context of random graph processes, where the corresponding hypergraph defines a natural graph property, such as minimum degree, connectivity, Hamiltonicity and the containment of fixed-size subgraphs. Joint work with Alan Frieze and Michael Krivelevich.
Date: 14/2/2023
Speaker: Yin-Ting Liao (Irvine)
Date: 28/2/2023
Speaker: Ahmed Bou-Rabee (Cornell)
Date: 7/3/2023
Speaker: Máté Wierdl (Memphis)
Date: 14/3/2023
Speaker: Pablo Shmerkin (UBC)
Date: 21/3/2023
Speaker: Neha Bansal (UBC Okanagan)
Date: 28/3/2023
Speaker: Emilio Corso (UBC)
Date: 4/4/2023
Speaker: Natalie Behague (UVic)

For (some) previous semesters, see
Spring 2006
Fall 2006
Spring 2007
Spring 2008
Spring 2009
Spring 2010
Spring 2011
Spring 2012
Spring 2013
Spring 2014
Spring 2015
Autumn 2020
Spring 2021
Spring 2022