Applied Mathematics Seminar, Fall 2021

## Mondays 4:00 PM - 5:00 PM MT, Hybrid: In-Person, LCB 219 and Online (zoom information will be provided before the seminars)

• This seminar can be taken for credit: Students can get 1-3 credits by registering to the Applied Math Seminar class Math 7875 Section 010 for Fall 2021. Students should talk to the seminar organizer before taking it for a credit. Grading is based on attendance and giving a talk by presenting an applied-mathematics paper (not necessarily your own). Student talks will be appropriately labeled to distinguish them from visitor talks. The seminar organizer is available to review your slides, for dry-runs etc.
• Please direct questions or comments about the seminar (or its class) to Yekaterina Epshteyn (epshteyn (at) math.utah.edu) or to Akil Narayan (akil (at)sci.utah.edu) or to Jody Reimer (reimer (at) math.utah.edu)
• Talks are announced through the applied-math mailing list. Please ask the seminar organizers for information about how to subscribe to this list.

September 20. In-Person
Speaker: Zachary Boyd, Department of Mathematics, Brigham Young University
Title: Metrics and learning on directed graphs, and some open problems relating to robustness
Abstract: Numerous data science and machine learning problems can be phrased in terms of directed graphs, yet most recent theories and algorithms target undirected graphs specifically, inevitably throwing away valuable information and risking serious loss of meaning in the results. I will explain recent techniques for (1) measuring (symmetric) distances in directed graphs and (2) cluster detection/partitioning using mean exit time to encode directionally coherent structure. Various applications of these ideas will be given, including amplification of community structure, data exploration, and dynamical anomaly detection. Finally, I will give an informal survey of robustness in the face of noisy and incomplete graph structures, especially as they arise in the social sciences, and describe some open problems in this domain.

September 27. Online
Speaker: David Morison, Department of Physics and Astronomy, University of Utah
Title: Order to Disorder in Quasiperiodic Composites
Abstract: We introduce a novel class of two phase composites that are structured by deterministic Moiré patterns, and display exotic behavior in their bulk electrical, magnetic, diffusive, thermal, and optical properties as system parameters are varied. The dependence of classical transport coefficients on mixture geometry is distilled into the spectral properties of an operator analogous to the Hamiltonian in quantum physics. As the system is tuned with a small change in the twist angle, there is a marked transition in the microstructure from periodic to quasiperiodic, and the transport properties switch from those of ordered to randomly disordered materials. Corresponding spectral properties, such as eigenvalue and field localization characteristics, exhibit behavior analogous to an Anderson transition in wave phenomena, with band gaps and mobility edges — even though there are no wave scattering or interference effects at play here. Our findings establish a parallel between quantum transport in solids and classical transport in composite materials with periodic or quasiperiodic microstructure.

October 4. Online
Speaker: Yuming Paul Zhang, Department of Mathematics, UCSD
Title: Optimal Estimates on the Propagation of Reactions with Fractional Diffusion
Abstract: We study the reaction-fractional-diffusion equation u_t+(-\Delta)^s u=f(u) with ignition and monostable reactions f, and s\in (0,1). We obtain the first optimal bounds on the propagation of front-like solutions in the cases where no traveling fronts exist. Our results cover most of these cases, and also apply to propagation from localized initial data. This is a joint work with A. Zlatos.

October 18. In-Person. Joint Applied Math/Stochastics Seminar
Speaker: Bao Wang, Department of Mathematics & SCI, University of Utah
Title: How Differential Equations and Random Graph Insights Benefit Deep Learning
Abstract: We will present recent results on developing new deep learning algorithms leveraging differential equations and random graph insights. First, we will present a new class of continuous-depth deep neural networks that were motivated by the ODE limit of the classical momentum method, named heavy-ball neural ODEs (HBNODEs). HBNODEs enjoy two properties that imply practical advantages over NODEs: (i) The adjoint state of an HBNODE also satisfies an HBNODE, accelerating both forward and backward ODE solvers, thus significantly accelerate learning and improve the utility of the trained models. (ii) The spectrum of HBNODEs is well structured, enabling effective learning of long-term dependencies from complex sequential data. Second, we will extend HBNODE to graph learning leveraging diffusion on graphs, resulting in new algorithms for deep graph learning. The new algorithms are more accurate than existing deep graph learning algorithms and more scalable to deep architectures, and also suitable for learning at low labeling rate regimes. Moreover, we will present a fast multipole method-based efficient attention mechanism for modeling graph nodes interactions. Third, if time permits, we will discuss building an efficient and reliable overlay network for decentralized federated learning based on the random graph theory.

October 25. In-Person
Speaker: Pania Newell, Department of Mechanical Engineering, University of Utah
Title: Exploring Fracture in Heterogenous Porous Materials Across Multiple Length Scales
Abstract: Many materials surrounding us from man-made materials such as cement, concrete, and ceramics to natural materials such as biological tissue, rocks, and soil are considered porous materials. Due to their unique properties, such as lightweight, heat resistance, sound absorption, thermal conductivity, electrical resistivity, porous materials are appealing to various engineering and scientific applications. Heterogeneous porous media are defined as materials whose structures are characterized by multiple periodicities over several disparate length scales. Such hierarchical structure poses challenges in understanding damage and fracture in these materials. This talk will address how computational tools enable us to enhance our fundamental understanding of fracture propagation mechanisms in such materials across multiple scales. At the nanoscale, molecular dynamics simulation provides information about mechanical properties, such as fracture energy release rate for various pore morphology. At microscale, the impact of the pore shape and size on fracture pattern is investigated through a two-scale homogenization method coupled with the state-of-the-art phase- field fracture technique. The results of this hierarchical coupling approach highlight the importance of higher-order parameters associated with pore shape and size on fracture strength and pattern at the continuum scale.

November 1. In-Person. Joint Applied Math/MathBio Seminar
Speaker: Henry Fu, Department of Mechanical Engineering, University of Utah
Title: Physical constraints on the propulsion of microrobots
Abstract: Currently, artificially propelled magnetic micro- and nanoparticles are of interest for potential applications such as hyperthermic therapy, drug delivery, and microsurgery. Here, we focus on propulsion via rotation of rigid magnetic particles by an external magnetic field, which is attractive since it can be scaled down to micron sizes, and since magnetic fields permeate through many media including biological tissue. I will discuss our work exploring the fundamental physical constraints on this type of propulsion. By analogy with biological swimming, which is constrained by the kinematic reversibility of Stokes flow, it was expected that magnetically rotated micropropulsion required chiral geometries, like that of a helical bacterial flagellum. In contrast, we have shown that propulsion by magnetic rotation is possible for achiral geometries. Still, the propulsion of the simplest geometries, such as spheres, would seem to be prohibited by fore-aft symmetry along the rotation axis. We have found that in nonlinearly viscoelastic fluids, a symmetry breaking propulsion is possible for rotating microspheres. Finally, most models of magnetically rotating microrobots assume a permanent magnetic moment, but actual magnetic materials have moments which depend on the externally applied field. We show that such magnetic response leads to limits on the propulsion velocity achievable by rotating propulsion.

November 2. Special PDE/Geometric Analysis Seminar. LCB 215, 3:30-4:30pm *Please note the room and date*
Speaker: Peter McGrath, Department of Mathematics, North Carolina State University
Title: Bending energy minimizers with prescribed genus and isoperimetric ratio.
Abstract: The Bending (or Willmore) energy of a surface immersed in Euclidean three-space is the integral of the square of the surface's mean curvature and has been studied since the early 1800's—by Germain and Poisson—in the study of elastic membranes. In the 1970's, Biologist Canham proposed to model red blood cells by surfaces minimizing bending energy with a constrained isoperimetric ratio. I will discuss the recent proof—by R. Kusner and myself—of the existence of a smooth surface with minimum bending energy amongst the class of surfaces of any prescribed genus and isoperimetric ratio.

November 8. In-Person/Hybrid
Speaker: Mihai Putinar, Department of Mathematics, University of California at Santa Barbara
Title: Christoffel-Darboux analysis
Abstract: The Christoffel-Darboux kernel is a respectable concept of mathematical analysis. For a century and a half the asymptotics of this reproducing kernel proved to be essential in constructive function theory and spectral analysis. Recently, central questions of Christoffel-Darboux analysis turned out to be relevant in some inverse problems of statistical or spectral flavor. The seminar will focus on three such developments of the last decade: shape reconstruction from moments, spectral analysis of Koopman's operator associated with a dynamical system and the detection of outliers in large point distributions.

November 22. In-Person.
Speaker: Will Feldman, Department of Mathematics, University of Utah
Title: Interface energies in periodic media
Abstract: I will discuss some singular phenomena which can occur for Allen-Cahn and curvature motions in periodic media.

November 29. In-Person.
Speaker: Kshiteej Deshmukh, Department of Mathematics, University of Utah
Title: Multiband homogenization of metamaterials in real-space
Abstract: We consider the problem of wave scattering from interfaces between metamaterials and obtain a macroscopic model with effective conditions to be applied at the interface to find the scattering coefficients. Rational function approximations of the exact dispersion relation are used to obtain a multiband homogenized model posed in real space and time. The homogenized macroscopic model provides predictions of wave transmission that match well with the fine-scale solution. Compared to asymptotic expansions, they are much faster to compute, easy to apply, and valid over a broad range of frequencies, while accounting for the microstructure away from the interface.

December 6.
Speaker: Farhan Abedin, Department of Mathematics, University of Utah
Title: Inverse Iteration for the Monge-Ampere Eigenvalue Problem
Abstract: I will discuss an iterative method for solving the Monge-Ampere eigenvalue problem. This is joint work with Jun Kitagawa (Michigan State University).

Seminar organizers: Yekaterina Epshteyn (epshteyn (at) math.utah.edu), Akil Narayan (akil (at)sci.utah.edu) and Jody Reimer (reimer (at)math.utah.edu).

155 South 1400 East, Room 233, Salt Lake City, UT 84112-0090, T:+1 801 581 6851, F:+1 801 581 4148