# Applied Math Colloquium - Spring 2020

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

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

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.

**No Colloquium**

**Carlos Pérez**, Pontificia Universidad Católica de Chile

Hosted by: Catalin Turc

**Integral Equation Methods for Inverse Design of Metasurfaces**

Metasurfaces are planar metamaterial slabs of subwavelength thickness composed by a large number of optical elements that can be engineered to manipulate the direction, amplitude, phase, and polarization of light. Inverse metasurface designs typically rely on the use of fast but low-order ray-optics (locally-periodic) approximations. Accurate full-wave metasurface simulations, in turn, pose significant challenges to well-established electromagnetic PDE solvers as they lead to intractably large computational domains and/or unreasonably long computing times.

In this talk, I will present two boundary integral equation approaches that partially overcome the aforementioned issues. I will first present a windowed Green function method based on a second-kind single-trace formulation for full-wave simulation of all-dielectric metasurfaces. Finally, I will present a novel integral equation framework that allows to compute corrections and assess the accuracy of a popular ray-optics approximation used in inverse metasurface design.

This is joint work with Raphäel Pestourie (Harvard), Steven G. Johnson (MIT).

** Bryan Quaife, **Florida State University

Hosted by: Yuan-Nan Young

**Coupling Between Flow and Porous Structures**

In 1855, Henry Darcy experimentally established Darcy's law to relate the flow rate and pressure drop through the permeability. However, in many applications, the permeability is dynamically evolving with the flow. For example, in groundwater flow, the porous structure is altered by sedimentation, deposition, and erosion. As another example, depending on the tension of an elastic membrane, it can be permeable to a solvent, but not the solute. I will describe recent work on both erosion in groundwater flow, and semi-permeability of elastic membranes. If time permits, I will also discuss a related clamped eigenvalue problem.

** Roy Goodman, **NJIT

Hosted by: David Shirokoff

**Transfer Entropy for Network Reconstruction in a Simple Model**

Dynamics on a network are ubiquitous in areas as diverse as genetics, meteorology, and the social sciences. By a network, we simply mean a system of interacting agents. A common problem is to reconstruct a network given the behavior of its agents, that is, given a set of time series describing the states of each node on the network, determine which other nodes are influencing that node. Information theory provides one set of tools for doing so in a model-independent manner. A popular tool in this game is called transfer entropy. Transfer entropy measures the reduction in uncertainty in predicting the state of one agent at a time (t+1) from its state at time t given the additional knowledge of the state of a second j at time t, compared with predicting it the knowledge of the first agent alone. This can be thought of as measuring the transfer of information from the second agent to the first. We analyze a simple model in which we derive asymptotic formulae relating the transfer entropy between two nodes to the strength of the connection between the nodes. Higher-order terms in this formula show the subtle effect of network topology on the transfer entropy. This allows for more accurate network reconstruction in this particular model and points to possible sources of errors when transfer entropy is used to reconstruct networks in a more general setting.

** Alejandro Rodriguez, **Princeton

Hosted by: David Shirokoff

**Bringing to Light the Fundamental Limits of Optical Control at the Nanoscale**

Nanophotonics deals with the myriad ways of manipulating light by nano-structuring materials at or below the scale of the electromagnetic wavelength. From sub-wavelength metallic resonators to periodic crystals, different combinations of material and geometry can be used to confine light to ultra small domains (thousandths of times smaller than its free space wavelength) and over long times (billions of times longer than the optical period), greatly enhancing its interaction with matter. Applications include new kinds of antennas, lasers, super-lenses, on-chip interconnects, and cloaking devices. In this talk, I will survey recent developments in the area of photonic computational (inverse) design methods, also known as large-scale optimization. Complementing the strengths of physically intuitive design paradigms, these computer-aided design techniques are beginning to probe the fundamental limits of the possible: new nonlinear micron-scale cavities that convert light from infrared to visible wavelengths, textured thin films that enhance single-photon extraction from electronic defects, and heated objects whose radiation greatly surpass (by hundreds even thousands of times) the predictions of the famous Stefan–Boltzmann blackbody limits. How does one know whether these designs are approaching optimal performance (as physically allowed by Maxwell's equations)? Concurrently, I will survey recently developed "dual optimization" methods whose main aim is to shed light on the question of optical bounds. Preliminary work suggests that while there is still plenty of room at the bottom for novel designs, inverse techniques are already producing devices that come within factors of unity of what is possible.

** Rongjie Lai, **RPI

Hosted by: Brittany Hamfeldt

**Understanding Manifold-Structure Data via Geometric Modeling and Learning**

Analyzing and inferring the underlying global intrinsic structures of data from its local information is critical in many fields. Different from image and signal processing which handle functions on flat domains with well-developed tools for processing and learning, manifold-structured data sets are even more challenging due to their complicated geometry. For example, a geometric object can have very different coordinate representations due to various embeddings, transformations or representations (imagine the same human body shape can have different poses as its nearly isometric embedding ambiguities). In this talk, I will first discuss geometric modeling based methods for 3D shape analysis. This approach uses geometric PDEs to adapt the intrinsic manifolds structure of data. It extracts various invariant descriptors to characterize and understand data through solutions of differential equations on manifolds. Inspired by recent developments of deep learning, I will also discuss our recent work of a new way of defining convolution on manifolds. This geometric way of defining parallel transport convolution (PTC) provides a natural combination of modeling and learning on manifolds. PTC allows for the construction of compactly supported filters and is also robust to manifold deformations. It enables further applications of comparing, classifying and understanding manifold-structured data by combining with recent advances in deep learning.

** Misha Tsodyks, **Institute for Advanced Study

Hosted by: Victor Matveev

**Mathematical Models of Human Memory**

Human memory is a multi-staged phenomenon of extreme complexity, which results in highly unpredictable behavior in real-life situations. Psychologists developed classical paradigms for studying memory in the lab, which produce easily quantifiable measures of performance at the cost of using artificial content, such as lists of randomly assembled words. I will introduce a set of simple mathematical models describing how information is maintained and recalled in these experiments. Surprisingly, they provide a very good description of experimental data obtained with internet-based memory experiments on large number of human subjects. Moreover, more detailed mathematical analysis of the models leads to some interesting ideas for future experiments with potentially very surprising results.

** Son-Young Yi, **University of Texas at El Paso

Hosted by: Yassine Boubendir

**The Immersed Interface Hybridized Difference Method for Interface Problems**

Interface problems arise in a variety of disciplines, including mathematical biology, material science, and fluid mechanics, to name a few.

Conventional numerical methods tend to lose accuracy near the interface unless they use interface-fitted meshes to resolve the low-regularity solution around the interface. In practice, however, structured meshes such as Cartesian grids that are independent of the interface are preferable. In response to this, a new class of numerical methods called the Immersed Interface Hybridized Difference Method (IHDM) had been introduced for solving elliptic and parabolic interface problems. I will describe the IHDM for elliptic interface problems in detail first, then discuss how to extend the method to parabolic interface problems. I will also present some numerical results to demonstrate the accuracy and efficiency of the IHDM.

**No Colloquium - Spring Break**

** Sumit Mukherjee, **Columbia University

Hosted by: Zuofeng Shang

**Title: TBA**

Abstract: TBA

** Alex Gittens, **RPI

Hosted by: Catalin Turc

**Title: TBA**

Abstract: TBA

**No Colloquium - Good Friday**

** Kui Ren, **Columbia University

Hosted by: Christina Frederick

**Title: TBA**

Abstract: TBA

** Anne Gelb, **Dartmouth College

Hosted by: David Shirokoff

**Title: TBA**

Abstract: TBA

** Selim Esedoglu, **University of Michigan

Hosted by: Roy Goodman

**Title: TBA**

Abstract: TBA

*Updated: January 28, 2020*