# Math Colloquium - Fall 2022

Colloquia are held on Fridays at 11:30 a.m. in Cullimore Lecture Hall I, unless noted otherwise.

Colloquia are held on Fridays at 11:30 a.m. in Cullimore Lecture Hall I, unless noted otherwise.

To join the Applied Mathematics Colloquium mailing list visit https://groups.google.com/a/njit.edu/forum/?hl=en#!forum/math-colloquium/join (Google Profile required). To join the mailing list without a Google Profile, submit the seminar request form.

**Helen Wilson**, University College London

Hosted by: Linda Cummings

*Please note, this seminar will be held virtually via WebEx only. Please contact math@njit.edu for access information*

**Mathematical Modelling of Suspensions**

Materials made from a mixture of liquid and solid are, instinctively, very obviously complex. From dilatancy (the reason wet sand becomes dry when you step on it) to extreme shear-thinning (quicksand) or shear-thickening (cornflour oobleck) there is a wide range of behaviours to explain and predict. I'll discuss the seemingly simple case of solid spheres suspended in a Newtonian fluid, which still has plenty of surprises up its sleeve.

**Wooyoung Choi**, NJIT

Hosted by: Anand Oza

**New Directions and Challenges in Modeling Nonlinear Waves in Shallow Water**

Nonlinear waves in shallow water have been often described by weakly nonlinear and weakly dispersive long wave models. Considering that the number of extreme events increases, strongly nonlinear long wave models have recently attracted much attention. In this talk, a high-order strongly nonlinear long wave model will be introduced and its well-posedness and solitary wave solution will be discussed. To solve numerically the high-order long wave model, it is found that finite difference schemes could be unstable due to discretization errors in approximating high-order derivatives and, therefore, a more accurate numerical scheme is required. Here a pseudo-spectral method is adopted and some numerical solutions of the high-order model will be presented. Additional challenges in practical applications of the high-order long wave model will be also discussed.

**Jean-Luc Thiffeault**, University of Wisconsin Madison

Hosted by: Michael Booty

**Shake Your Hips: An Active Particle with a Fluctuating Propulsion Force**

The active Brownian particle (ABP) model describes a microswimmer, synthetic or living, whose direction of swimming is a Brownian motion. The swimming is due to a propulsion force, and the fluctuations are often assumed thermal in origin. We present a 2D model where the fluctuations arise from nonthermal noise in a propelling force acting at a single point, such as that due to a flagellum. We carefully take the overdamped limit and find several modifications to the traditional ABP model. Since the fluctuating force causes a fluctuating torque, the diffusion tensor describing the process has a coupling between translational and rotational degrees of freedom. An anisotropic particle also exhibits a noise-induced drift. We show that these effects have measurable consequences for the long-time diffusivity of active particles, in particular adding a contribution that is independent of where the force acts. This is joint work with Jiajia Guo.

**Saleh Tanveer**, Ohio State University

Hosted by: Wooyoung Choi and Mike Siegel

**Study of a Reduced Model for Two Fluid Shear Flow**

Mathematical reduction of complicated equations describing any phenomenon is common in Applied Mathematics. Instead of ad-hoc modeling, it is desirable to systematically reduce complicated models to simpler more tractable equations using asymptotic arguments, particularly when there are many parameters involved. Such reductions elucidate parameter dependence of phenomena in far greater detail than possible for the original model.

In this vein, we consider a nonlocal thin film model Kaligirou et al (2016) model for two fluid shear flows in a channel. Shear flows involving two fluids are quite common in nature, whether they occur in the context of flow in a coated pipe, wind-water interaction or other geo or astrophysical context. We use tools from analysis, numerical calculations and asymptotics in this study. We present results on whether or not the 2+1 model equations allow smooth solutions for all time, or if there is singularity in finite time. Furthermore, we discuss existence of different branches of steady traveling wave solutions in one dimension that initially bifurcate from a flat state and present stability and bifurcation properties, including Hopf-bifurcation to time periodic states. We also highlight surprising singular effects of small slip on an otherwise stably stratified flow.

*The work presented is in collaboration with D. Papageorgiou, Imperial.

**Shuwen Lou**, Loyola

Hosted by: Eliza Michalopoulou

**A Model of Distorted Brownian Motion with Varying Dimension**

We will begin with an introduction to standard Brownian motion and its applications in STEM fields since early 1900s. Then we will introduce what a "distorted Brownian motion" is, and what we mean by "varying dimension". The last 1/3 of the talk will be focused on our main results: heat kernel estimates of such a model. The term "heat kernel" comes from the link between Brownian motion and heat equations in PDE. No background in probability is required to understand the first half of the talk.

**Yue Yu**, Lehigh

Hosted by: Roy Goodman

**Learning Nonlocal Operators for Heterogeneous Material Modeling**

Constitutive modeling based on the continuum mechanics theory has been a classical approach for modeling the mechanical responses of materials. However, when constitutive laws are unknown or when defects and/or high degrees of heterogeneity present, these classical models may become inaccurate. In this talk we propose to use data-driven modeling which directly utilizes high-fidelity simulation and experimental measurements on displacement fields, to predict a material's response. Two nonlocal operator regression approaches will be presented, to obtain homogenized and heterogeneous surrogates for material modeling, respectively.

In the first approach, we propose to consider nonlocal models as coarse-grained, homogenized models to accurately capture the system's global behavior, and learn optimal kernel functions directly from data. By combining machine learning identifiability theory, known physics, and nonlocal theory, this combination guarantees that the resulting model is mathematically well-posed, physically consistent, and converging as the data resolution increases. As an application, we employ this ``linear operator regression’’ technique to model wave propagation in metamaterials.

In the second approach, we further propose to leverage the linear operator regression to nonlinear models by combining it with neural networks, and model the heterogeneous material response as mapping between loading conditions and its resultant displacement fields. To this end, we develop deep integral neural operator architectures, which is resolution-independent and naturally embeds the material micromechanical properties and defects in the integrand. As an application, we learn soft tissue models directly from digital image correlation (DIC) displacement tracking measurements, and show that the learnt model substantially outperforms conventional constitutive models.

**Marcus Roper**, UCLA

Hosted by: Anand Oza

**Title: TBA**

Abstract: TBA

**Amin Doostmohammadi**, Niels Bohr Institute

Hosted by: Yuan-Nan Young

**Title: TBA**

Abstract: TBA

**Javier Gomez Serrano**, Brown University

Hosted by: Michael Siegel

**Title: TBA**

Abstract: TBA

**James Kelly**, U.S. Naval Research Lab

Hosted by: Simone Marras (Mechanical Engineering)

**Title: TBA**

Abstract: TBA

**No Colloquium - Thanksgiving Break**

**Javier Diez**, CIFICEN-CONICET-CICPBA, Instituto de Física Arroyo Seco, Universidad Nacional del Centro de la Provincia de Buenos Aires

Hosted by: Lou Kondic

**Title: TBA**

Abstract: TBA

**Stephanie Chaillat-Loseille**, ENSTA Paris - UMA Laboratoire POems

Hosted by: Travis Askham

**Title: TBA**

Abstract: TBA

*Updated: October 3, 2022*