# Events for 12/01/2021 from all calendars

## Probability Seminar

**Time: ** 10:00AM - 11:00AM

**Location: ** Zoom/BLOC 302

**Speaker: **Paul Jung, KAIST

**Title: ***A Generalization of Hierarchical Exchangeability on Trees to Directed Acyclic Graphs*

**Abstract: **A random array indexed by the paths of an infinitely-branching rooted tree of finite depth is hierarchically exchangeable if its joint distribution is invariant under rearrangements that preserve the tree structure underlying the index set. Austin and Panchenko (2014) prove that such arrays have de Finetti-type representations, and moreover, that an array indexed by a finite collection of such trees has an Aldous-Hoover-type representation. Motivated by issues in Neural Networks and Bayesian nonparametric models used in probabilistic programming languages, we generalize hierarchical exchangeability to a new kind of partial exchangeability for random arrays which we call DAG-exchangeability. In our setting a random array is indexed by N^|V| for some DAG G=(V,E), and its exchangeability structure is governed by the edge set E. We will present a representation theorem for such arrays which generalizes the Aldous-Hoover and Austin-Panchenko representation theorems. Based on joint work with Jiho Lee (KAIST Mathematics), Sam Staton (Oxford CS), and Hongseok Yang (KAIST CS)

## Number Theory Seminar

**Time: ** 11:00AM - 12:00PM

**Location: ** BLOC 302

**Speaker: **Riad Masri, Texas A&M University

**Title: ***The distribution of short orbits of singular moduli*

**Abstract: **The values of the modular j-function at CM points are algebraic numbers called singular moduli. There is a natural action of the class group G(d) of discriminant d on the set of singular moduli. For each d, fix a singular modulus j(d,0) and a subgroup H(d) < G(d). Let Av(d,0) denote the average of the numbers in the orbit H(d)*j(d,0). Under a mild growth condition on |H(d)|, we will compute the distribution of Av(d,0) as |d| approaches infinity. Our growth condition is expressed as a function of the best progress towards the Lindelöf hypothesis for modular L-functions. This is joint work with Annika Mauro and Tanis Nielsen.

## Numerical Analysis Seminar

**Time: ** 1:00PM - 2:00PM

**Location: ** zoom

**Speaker: **Amirreza Khodaddian, Leibniz University of Hannover

**Title: ***Uncertainty quantification in phase-field fracture problem*

**Abstract: **Phase-field fracture is a very active research field with numerous applications. The model is used to describe the crack propagation in brittle materials (e.g., concrete and ceramics), ductile materials such as metals and steel, and hydraulic fracture (extracting oil and natural gases). Moreover, the uncertainty arises from the heterogeneity of the material structure and spatial fluctuation of the material properties. The challenging part is the multiphysics fracture framework since we should deal with different subdomains (each has different PDEs), which significantly increases the computational costs. In brittle fracture, the computational model is based on the coupling of the elasticity equation (to model displacement) and the phase-field fracture (modeling the crack propagation). In hydraulic fracture, these equations are additionally coupled with the Darcy-type flow to model fluid pressure. Here, mechanical and geo-mechanical parameters have an influential effect on the model simulations; however, most of these parameters can not be estimated experimentally. Bayesian inversion is an efficient and reliable probabilistic model to estimate the material parameters when a synthetic/measured reference observation is available. In this talk, for phase-field fracture models, we employ Markov chain Monte Carlo (MCMC) techniques to estimate the posterior distributions of the parameters. In Bayesian inversion, hundreds or thousands of (PDE-based) forward runs are necessary. The simulations are computationally expensive since, in order to achieve reliable accuracy, a significant degree of freedom is needed. Next, we develop multiscale techniques, specifically non-intrusive global/local approach in which a fine-scale problem is solved in the fracture region and a linearized coarse problem in the remaining domain. The global/local setting is coupled with Bayesian inversion to identify the material parameters and model the crack pattern. The results show a significant computational cost reduction compared to the full mode

## Groups and Dynamics Seminar

**Time: ** 3:00PM - 4:00PM

**Location: ** online

**Speaker: **Matthieu Joseph, Ecole Normale Superieure, Lyon

**Title: ***Allosteric actions of surface groups*

**Abstract: **In a recent work, I introduced the notion of allosteric actions: a minimal action of a countable group on a compact space, with an ergodic invariant measure, is allosteric if it is topologically free but not essentially free. In the first part of my talk I will explain some properties of allosteric actions, and their links with Invariant Random Subgroups (IRS) and Uniformly Recurrent Subgroups (URS). In the second part, I will explain a recent result of mine: the fundamental group of a closed hyperbolic surface admits allosteric actions.

## Topology Seminar

**Time: ** 4:00PM - 4:50PM

**Location: ** Zoom

**Speaker: **Carissa Slone, University of Kentucky

**Title: *** Characterizing 2-slices over C_2 and K_4*

**Abstract: **The slice filtration focuses on producing certain irreducible spectra, called slices, from a genuine G-spectrum X over a finite group G. We have a complete characterization of all 1-, 0-, and (-1)-slices for any such G. We will characterize 2-slices over $C_2$ and expand this characterization to $K_4 = C_2 \times C_2$.