Applied Mathematics Seminar, Spring 2023

- Please direct
**questions or comments**about the seminar to**Will Feldman**(`feldman (at) math.utah.edu`

), or**Akil Narayan**(`akil (at) sci.utah.edu`

), or to**Kshiteej Deshmukh**(`kjdeshmu (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.

**Wednesday, January 18 at 4pm. In-person, LCB 219**

Speaker: Alexander B. Watson,
Mathematics Department, University of Minnesota Twin Cities

**Title: **Twisted bilayer graphene at the moiré scale

**Abstract: ** 2D materials consist of a single sheet of atoms. The first 2D material, graphene, a single sheet of carbon atoms, was isolated in 2005. In recent years, attention has shifted to materials created by stacking 2D materials with a relative twist. Such materials are known as moiré materials because of the approximate periodicity of their atomic structures over long distances, known as a moiré pattern. In 2018, experiments showed that, when twisted to the so-called "magic" angle (approximately 1 degree), twisted bilayer graphene exhibits exotic quantum phenomena such as superconductivity. I will present the first rigorous justification of the Bistritzer-MacDonald moiré-scale PDE model of twisted bilayer graphene, which played a critical role in identifying the magic angle, from a microscopic tight-binding model. I will then discuss generalizations of this work accounting for atomic relaxation and vibration (phonons), mathematical questions posed by moir\é materials as fundamentally aperiodic/incommensurate systems, and other related work.

**Thursday, January 26 at 1pm. In-person, JWB 335**

Speaker: Raghav Venkatraman,
Courant Institute, NYU

**Title: **The robustness of ENZ devices

**Abstract: **"ENZ" devices are a class of electromagnetic devices that operate at a frequency at which one of their components has dielectric permittivity close to zero. Such devices have curious properties that have made them valuable in creating entirely new kinds of waveguides and resonators. While their analytical study in the physics literature so far has been limited to the idealized "epsilon =0" limit, the robustness of the associated effects to epsilon merely small have only been explored numerically, and an analytical understanding is lacking.
We will discuss a few different examples in which we can get complete analytical information in the setting where epsilon is merely small.
This is joint work with Bob Kohn, based on conversations with Nader Engheta.

**March 13. In-person**

Speaker: Ying Wu,
Computer, Electrical and Mathematical Science and Engineering Division, KAUST

**Title: **Controlling acoustic wave propagation via rotational motion

**Abstract: **The propagation of acoustic waves in time-varying and moving media has recently gained much attention and is expected to offer intriguing applications. In this talk, I present our recent work on acoustic wave propagation in spinning media, such as air or water. I begin by reviewing the theoretical foundation built upon the Mie scattering framework, where we discuss both the wave equation and the boundary conditions in detail. Our study is limited to the linear regime, and we observe peculiar scattering features.

I then demonstrate three examples to showcase the potential of our work. Firstly, we introduce a generalized scattering cancellation theory (SCT) that can cloak spinning objects and make them invisible to static observers. The technique is based on rotating shells with varying angular velocity, and our work extends the realm of SCT one step closer to practical realization involving moving objects. Secondly, we show that a spinning cylindrical column of fluid in a static fluid environment possesses an intrinsic spin angular momentum. We investigate the torque and force it experiences in evanescent acoustic fields and observe that the resulting discontinuity can scatter sound in unusual ways, such as generating a negative radiation force, despite having no imaginary part in its parameters associated with intrinsic absorption. Finally, we discuss an acoustic analogue of an optical fiber based on rotation in our third example. Our work offers a new route to guiding acoustic waves and has potential applications in acoustic communications. Overall, we hope that our work will inspire further research on the propagation of acoustic waves in time-varying and/or moving media, and contribute to the development of novel applications in the field.

**March 27. Online**

Speaker: A. Martina Neuman,
Department of Computational Mathematics, Science and Engineering, Michigan State University

**Title: **Superiority of GNN over NN in generalizing bandlimited functions

**Abstract: **Graph Neural Network (GNN) with its ability to integrate graph information has been widely used for data analyses. However, the expressive power of GNN has only been studied for graph-level tasks but not for node-level tasks, such as node classification, where one tries to interpolate missing nodal labels from the observed ones. In this paper, we study the expressive power of GNN for the said classification task, which is in essence a function interpolation problem. Explicitly, we derive the number of weights and layers needed for a GNN to interpolate a band-limited function in R^d. Our result shows that, the number of weights needed to eps-approximate a bandlimited function using the GNN architecture is much fewer than the best known one using a fully connected neural network (NN) - in particular, one only needs O((log 1/eps)^d) weights using a GNN trained by O((log 1/eps)^d) samples to eps-approximate a discretized bandlimited signal in R^d. The result is obtained by drawing a connection between the GNN structure and the classical sampling theorems, making our work the first attempt in this direction.

**Friday, April 7 at 4pm. In-person, LCB 222**

Speaker: Guosheng Fu,
Department of Applied and Computational Mathematics and Statistics, University of Notre Dame

**Title: **High order spatial discretization for variational time implicit schemes: Wasserstein gradient flows and reaction-diffusion systems

**Abstract: ** We design and compute first-order implicit-in-time variational schemes with high-order spatial discretization for initial value gradient flows in generalized optimal transport metric spaces. We first review some examples of gradient flows in generalized optimal transport spaces from the Onsager principle. We then use a one-step time relaxation optimization problem for time-implicit schemes, namely generalized Jordan-Kinderlehrer-Otto schemes. Their minimizing systems satisfy implicit-in-time schemes for initial value gradient flows with first-order time accuracy. We adopt the first-order optimization scheme ALG2 (Augmented Lagrangian method)
and high-order finite element methods in spatial discretization to compute the one-step optimization problem. This allows us to derive the implicit-in-time update of initial value gradient flows iteratively. We remark that the iteration in ALG2 has a simple-to-implement point-wise update based on optimal transport and Onsager's activation functions. The proposed method is unconditionally stable for convex cases. Numerical examples are presented to demonstrate the effectiveness of the methods in two-dimensional PDEs, including Wasserstein gradient flows, Fisher--Kolmogorov-Petrovskii-Piskunov equation, and two and four species reversible reaction-diffusion systems.
This is a joint work with Stanley Osher from UCLA and Wuchen Li from U. South Carolina.

**April 10. Online**

Speaker: Matthew Grasinger,
Materials and Manufacturing Directorate, Air Force Research Laboratory, Ohio

**Title: **Local rotations and interfacial phenomena in polymer fluids and solids

**Abstract: ** The first part of the talk concerns polymer network models which allow one to model the constitutive relationships of a broader polymer network from the behavior of a single polymer chain. These network models have been used to characterize multiscale phenomena in a variety of contexts such as rubber elasticity, soft multifunctional materials, and biological materials. For decades, a myriad of polymer network models have been developed with different representative volume elements (RVEs) consisting of chains crosslinked in many different ways. Here we show how a simple, intuitive assumption for how the RVE rotates relative to applied loading unifies many of the disparate polymer network models and recovers arguably the most successful model for elasticity, the 8-chain model. We then briefly explore some interesting implications of the modified modelling approach for stimuli-responsive, biological, and semi-crystalline polymer networks.
The second part concerns the infusion of polymer resins into fiber reinforcements towards the manufacturing of aerospace composites. The lattice Boltzmann method is used to simulate the flow through fiber bundles while considering wetting interactions between the resin and the fibers. The effects of the wetting contact angle on the character of the flow front, mechanisms for void formation, and void transport are explored.

**April 17. In person**

Speaker: Aaron Zeff Palmer,
Department of Mathematics, UCLA

**Title: **The sharp interface limit of a mean-field game phase transition model

**Abstract: **Mean-field games may exhibit phase transitions where the players exhibit qualitatively distinct behaviors in different parameter regimes. The 'Ising game' is a prototype inspired by the well-known 'Ising model' of statistical physics. A phase transition appears when the 'Ising game' admits two constant equilibrium solutions in the super-critical parameter regime. In this talk, we consider a macroscopic limit for which Nash-equilibrium solutions to the 'Ising game' concentrate on the two constant equilibria. A sharp interface forms and minimizes an energy functional related to an anisotropic space-time minimal surface. To prove this, we reformulate The 'Ising game' equilibria as critical points of an energy functional with non-local spatial interactions, a kinetic energy of the time gradient, and a double well potential. Our Gamma-convergence argument combines tools from both local and non-local phase transition models and handles the novel initial and terminal conditions from the mean-field game. Joint work with W. Feldman and I. Kim.

**April 24. In person**

Speaker: Julianne Chung,
Department of Mathematics, Emory University

**Title: **A journey to the world of computational inverse problems

**Abstract: **In this talk, we take a journey to the world of computational inverse problems, where we highlight important connections to mathematics, statistics, machine learning, and applications. The main goal of an inverse problem is to extract some underlying parameters or information from available and noisy observations. However, there are enormous computational challenges when solving state-of-the-art inverse problems of interest.

We present new tools for tackling these challenges. We describe generalized hybrid projection methods, which are iterative methods for solving large-scale inverse problems, and we show how approximations provided by the iterative method can be used for subsequent uncertainty quantification. Then, for problems where training or calibration data are readily available, we describe recent advances in exploiting machine learning techniques for estimating regularization parameters. Examples from atmospheric inverse modeling and image processing are discussed.

`feldman (at) math.utah.edu`

),
`akil (at) sci.utah.edu`

), and
`kjdeshmu (at) math.utah.edu`

).
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