# Fall 2019 Colliquia

### September 13, 2019

#### Jehanzeb Chaudhry, University of New Mexico (Host: James Adler)

##### Efficient Distribution Estimation and Uncertainty Quantification for Elliptic Problems on Domains with Stochastic Boundaries

**Time: **3:00pm **Location: **Bromfield-Pearson 101 **Reception:** 4:00pm **Abstract:** We study the problem of uncertainty quantification for the numerical solution of elliptic partial differential equation boundary value problems posed on domains with stochastically varying boundaries. We introduce simple transformations that map a family of domains with stochastic boundaries to a fixed reference domain. We exploit the transformations to carry out a priori and a posteriori error analyses and to derive an efficient Monte Carlo sampling procedure. Time permitting, we'll also talk about error estimation for non-overlapping and overlapping domain decomposition methods.

### October 25, 2019

#### Norbert Wiener Lecture, Mary Wheeler, Professor, The University of Texas at Austin

##### Phase Field Modeling for Diffusive Networks and Stimulation in Porous Media

**Time:** 3:00pm **Location: **Bromfield-Pearson 101 **Reception:** 4:00pm **Abstract: **The phase field method has emerged in recent years as a powerful variational approach to model crack propagation in elastic porous media. The most important feature of this method lies in the fact that it can handle fracture nucleation, propagation, merging, branching, kinking and curvilinear paths without any post-processing or additional computations; these complex fracture paths arise naturally as part of the numerical solution of an underlying partial differential equation. Unlike other computational fracture mechanics approaches, the phase field method does not require modeling the discontinuities in the medium or tracking the crack tip Instead, it introduces a diffusive zone that interpolates between the fracture zone and the intact material. This smears out the sharp crack interfaces that typically introduce singularities in the numerical computations.

In this presentation, a novel computational framework is introduced for simulation of multiphase flow, geomechanics, and fracture propagation in porous media based on Biot's model for poroelasticity. Here, state-of-the-art numerical modeling of natural and hydraulic fractures using a diffusive adaptive finite element phase field approach is employed. Since realistic porous media contains many natural fractures, not only is it important to stimulate hydraulic fractures but also to study the interaction between natural and hydraulic fractures. The enriched Galerkin finite element methods (EG) are employed to model multiphase flow with local mass conservation and dynamic mesh adaptivity. Geomechanics approximated by a continuous Galerkin finite element method is coupled to multiphase flow by applying an iteratively coupled scheme. Numerical examples are presented that demonstrate the effectiveness of this framework for different propagation scenarios, by varying the degrees of physics, and the capabilities to perform high-fidelity simulations on complex fracture networks.

### November 1, 2019

#### Ming-Jun Lai, University of Georgia (Host: Xiaozhe Hu)

##### Recent Development on Matrix Completion

**Time:** 3:00pm **Location:** Bromfield-Pearson 101 **Reception:** 4:00pm **Abstract:** Starting from the Netflex problem in 2006, I will survey the development of matrix completion in theory and computation. In particular, I will explain a few negative and a positive results on when a given incomplete matrix M can be uniquely completed. Also, I will show that the completion of a matrix is not only dependent on the number of entries, but also dependent on the values of entries. Next I shall explain several algorithms to complete matrices and present a comparison of their performance.

### November 12, 2019

#### Giovanni Gallavotti, Universita` degli Studi di Roma "La Sapienza" (Host: Boris Hasselblatt)

##### Reversibility and nonequilibrium ensembles in stationary Navier Stokes equation

**Time:** 3:00pm **Location:** Bromfield-Pearson 007 **Reception:** 4:00pm **Abstract:** In the '70s Ruelle proposed that "generically" chaotic motions would generate a uniquely determined statistics, i.e., a unique probability distribution on the possible states of the system, provided initial data were chosen with probability 1 with respect to (any) absolutely continuous measure. This means assigning to each measurable set in phase space the frequency of visit to it, hence a stationary probability distribution. After recalling the proposal and pointing out that in the case of equilibrium states of isolated mechanical systems it is a bold generalization of the ergodic hypothesis the natural question is whether the impressive developments that followed since Boltzmann and Maxwell in the last century the new proposal could lead to a deeper understanding of nonequilibrium phenomena and to a formulation of a theory of ensembles analogous to that for the equilibrium ensembles (e.g., canonical or microcanonical ensembles) to describe the stationary states out of equilibrium, which led and still leads also to impressive mathematical results. Since Ruelle's initial motivation was turbulence theory it seems appropriate here to analyze the example of the Navier–Stokes equation. The Navier–Stokes equation will be considered for an incompressible fluid in a periodic box and subject to a stirring force constant in time and acting at large scale (that is, at the scale of the container): the simplest geometry. Stationary states depend (therefore) on a single parameter R=Reynolds number=inverse of viscosity and form a family E of probability distributions on the velocity fields. The possibility will be discussed of existence of other equations whose stationary states have exactly the same distributions through a homogeneization mechanismanalogous to that for the equivalence of equilibrium states of different ensembles (and will be proposed to be similar to the equivalence in the thermodynamic limit which, in the Navier–Stokes case, will correspond to the ultraviolet regularization N → [infinity].

### November 22, 2019

#### Robert Kropholler

##### Fibering and incoherence for free-by-free groups

**Time: **3:00pm **Location:** Bromfield-Pearson 101 **Reception:** 4:00pm **Abstract:** We will survey recent results on virtual fibering of groups and discuss recent work on free-by-free groups and more generally finitely generated-by-free groups. This is joint work with Genevieve Walsh.

### December 4, 2019

#### Kelly Delp, Cornell and ICERM, (Host: Genevieve Walsh)

##### Playing with Surfaces: Spheres, Monkey Pants, and Zippergons

**Time:** 3:00pm **Location:** Bromfield-Pearson 007 **Reception: **4:00pm **Abstract:** In the Spring of 2010 I began a collaboration with William Thurston, which was inspired by clothing design, and involved building smooth surfaces from flat, rigid material. This project was an application of the following fact, which clothing makers regularly apply: the Gaussian curvature of a singular surface can be defined as a signed measure via the Gauss-Bonnet Theorem. I'll describe an explicit example of how to make well-fitting clothing for a sphere, based on an octahedral pattern, and report on my current adventures in making patterns for surfaces with long meandering seams. This talk will be accessible to a wide mathematical audience, including undergraduates.

### December 10, 2019

#### Kim Ruane, Tufts University

**Time:** 4:00pm **Location:** Nelson Auditorium (Anderson, 112) **Reception:** 5:00pm, Burden Lounge **Abstract:** I will discuss my experience teaching a math course through the Tufts University Prison Initiative Program (TUPIT). I taught a quantitative reasoning course at MCI Concord to 26 incarcerated men in the Spring/Summer of 2019. I will begin by describing the structure and material of the course. I will discuss the many pedagogical challenges I faced as a teacher in an unusual environment with an incredibly diverse group of students. It was simultaneously the most difficult yet rewarding challenge of my teaching career. I will also give an overview of the activities and mission of TUPIT because this is an amazing program. I prefer for this to be a discussion rather than a lecture so I am happy to answer any questions you might have. *Co-Sponsored by the Center for the Enhancement of Learning and Teaching, Tufts Department of Education, Tufts Math Society, and Tufts Prison Initiative of Tisch College*