Abstract: The last decade has witnessed tremendous advances in the fabrication of two-dimensional (2D) materials with novel electronic structures. Celebrated examples of such materials include graphene and black phosphorus.
The surface conductivity in these systems in the infrared frequency regime permits the propagation of fine-scale electromagnetic waves called surface plasmon-polaritons (SPPs).
In this talk, I will discuss macroscopic consequences of the optical conductivity of 2D materials via solutions of classical Maxwell's equations. I will formally discuss the following topics:
(I) Edges of anisotropic 2D materials act as induced sources of SPPs.
(II) Periodic structures made of 2D materials intercalated in conventional dielectrics may allow for the propagation
of homogenized, slowly varying waves with nearly no phase delay (epsilon-near-zero behavior).
(III) The curvature of 2D materials may generate further confinement of SPPs.
(IV) Nonlinearities of the 2D material and the ambient media cause non-intuitive dispersion of SPPs.
Part of this work is jointly with: A. Andreeva (U. Minnesota), E. Kaxiras (Harvard), T. Low (U Minnesota), M. Luskin (U. Minnesota), M. Maier (Texas A&M), A. Mellet (U MD)
Abstract: In this talk, I will review our recent works on numerical methods and analysis for solving the Dirac equation in the nonrelativistic limit regime, involving a small dimensionless parameter which is inversely proportional to the speed of light. In this regime, the solution is highly oscillating in time and the energy becomes unbounded and indefinite, which bring significant difficulty in analysis and heavy burden in numerical computation. We begin with four frequently used finite difference time domain (FDTD) methods and the time splitting Fourier pseudospectral (TSFP) method and obtain their rigorous error estimates in the nonrelativistic limit regime by paying particularly attention to how error bounds depend explicitly on mesh size and time step as well as the small parameter. Then we consider a numerical method by using spectral method for spatial derivatives combined with an exponential wave integrator (EWI) in the Gautschi-type for temporal derivatives to discretize the Dirac equation. Rigorous error estimates show that the EWI spectral method has much better temporal resolution than the FDTD methods for the Dirac equation in the nonrelativistic limit regime. Based on a multiscale expansion of the solution, we present a multiscale time integrator Fourier pseudospectral (MTI-FP) method for the Dirac equation and establish its error bound which uniformly accurate in term of the small dimensionless parameter. Numerical results demonstrate that our error estimates are sharp and optimal. Finally, these methods and results are then extended to the nonlinear Dirac equation in the nonrelativistic limit regime. This is a joint work with Yongyong Cai, Xiaowei Jia, Qinglin Tang and Jia Yin.
Abstract: Every organism transmits the information for making a similar organism across bottleneck stages that are considered generational boundaries. The bottleneck stage is minimally a single cell, which has two interdependent but distinct stores of information. One store is the well-understood linear DNA sequence that is replicated during cell divisions. The other is a three-dimensional arrangement of molecules that cycles during development such that it is nearly recreated at the start of each generation. Together they form the cell code for making an organism â a union of âgeneticâ and âepigeneticâ information stores that coevolve. I will discuss the implications of this perspective for our understanding of living systems and the beginnings of a framework for the joint consideration of all the information that is transmitted across generations to perpetuate life.
Abstract: What to do when the size and complexity of your model essentially prevent you from using it? Well, get a smaller and simpler model... At the heart of this dimension reduction process is the notion of parameter importance which, ultimately, is part of the modeling process itself. Global Sensitivity Analysis (GSA) aims at efficiently identifying important and non-important parameters; non-importance is important! We will present in this talk advances and challenges in GSA; these will include how to deal with correlated variables, how to treat time-dependent problems and stochastic problems and how to analyze the robustness of GSA itself at low cost. The role played by surrogate models will also be discussed. The discussion will be illustrated by an application from neurovascular modeling. Joint work with Alen Alexanderian, Tim David, Joey Hart and Ralph Smith.
Abstract: I will present the construction of solutions of the 3D Navier-Stokes equations whose initial vorticity is supported on curves (vortex filaments). This is the first instance for 3D Navier-Stokes where the stability of asymptotically (microscopically) self-similar solutions can be proved. This is joint work with J. Bedrossian and B. Harrop-Griffiths.
Abstract: We discuss a distributed Lagrange multiplier formulation of the Finite Element Immersed Boundary Method for the numerical approximation of the interaction between fluids and solids. The discretization of the problem leads to a mixed problem for which a rigorous stability analysis is provided. Optimal convergence estimates are proved for the finite element space discretization. The model, originally introduced for the coupling of incompressible fluids and solids, can be extended to include the simulation of compressible structures.
Abstract: We show that a class of spaces of vector fields whose semi-norms involve the magnitude of âdirectionalâ difference quotients is in fact equivalent to the class of fractional Sobolev spaces. The equivalence can be considered a Korn-type characterization of fractional Sobolev spaces. We use the result to better understand the energy space associated to a strongly coupled system of nonlocal equations related to a nonlocal continuum model via peridynamics. Moreover, the equivalence permits us to apply classical Sobolev embeddings in the process of proving that weak solutions to the nonlocal system enjoy both improved differentiability and improved integrability.
Abstract: We present a positive and asymptotic preserving numerical scheme for solving linear kinetic, transport equations that relax to a diffusive equation in the limit of infinite scattering. The proposed scheme is developed using a standard spectral angular discretization and a classical micro-macro decomposition. The three main ingredients are a semi-implicit temporal discretization, a dedicated finite difference spatial discretization, and realizability limiters in the angular discretization. Under mild assumptions, the scheme becomes a consistent numerical discretization for the limiting diffusion equation when the scattering cross-section tends to infinity. The scheme also preserves positivity of the particle concentration on the space-time mesh and therefore fixes a common defect of spectral angular discretizations. The scheme is tested on well-known benchmark problems with a uniform material medium as well as a medium composed from different materials that are arranged in a checkerboard pattern.
Abstract: We introduce a second-order time discretization method for stiff kinetic equations. The method is asymptotic-preserving (AP) -- can capture the Euler limit without numerically resolving the small Knudsen number; and positivity-preserving -- can preserve the non-negativity of the solution which is a probability density function for arbitrary Knudsen numbers. The method is based on a new formulation of the exponential Runge-Kutta method and can be applied to a large class of stiff kinetic equations including the BGK equation (relaxation type), the Fokker-Planck equation (diffusion type), and even the full Boltzmann equation (nonlinear integral type). Furthermore, we show that when coupled with suitable spatial discretizations the fully discrete scheme satisfies an entropy-decay property. Various numerical results are provided to demonstrate the theoretical properties of the method.
Abstract: We will describe generalizations of matrix algebras which suggest a considerable broadening of the spectral analysis toolkit and their applications. Including such classical notions such as Parseval identities, Fourier transforms, Spectral decomposition and Singular Value decomposition.
Abstract: I will present certain developments related to the macroscopic limit of the Cucker-Smale flocking model. The main focus will be on the multi-D euclidean space results from the recent joint work with Raphael Danchin, Piotr Mucha and Bartosz WrÃ³blewski.
Abstract: Several important integral operators in harmonic analysis will be presented in an organized way. The fundamental question about these operators is the boundedness on Lebesgue spaces. We will show that the difficulty to establish the boundedness will change dramatically once we move from the classic type of operators to their curved and discrete analogs. It is very interesting to see that harmonic analysis can interact with many other fields of mathematics such as PDE, ergodic theory, number theory, combinatorics, and even algebraic geometry.
Abstract: Self-interacting systems of particles/agents arise in many areas of science, such as particle systems in physics, flocking and swarming models in biology, and opinion dynamics in social science. A natural question is to learn the laws of interaction between the particles/agents from data consisting of trajectories. In the case of distance-based interaction laws, we present efficient regression algorithms to estimate the interaction kernels, and we develop a nonparametric statistical learning theory addressing identifiability, consistency and optimal rate of convergence of the estimators. Especially, we show that despite the high-dimensionality of the systems, optimal learning rates can still be achieved. (Joint work with Mauro Maggioni, Sui Tang and Ming Zhong).
Abstract: Visual objects are often known up to some ambiguity, depending on the methods used to acquire them. The first-order approximation to any transformation is, by definition, affine, and the affine approximation to changes between images has been used often in computer vision. Thus it is beneficial to deal with objects known only up to an affine transformation. For example, feature points on a planar transform projectively between different views, and the projective transformation can in many cases be approximated by an affine transformation. More generally, given two visual objects in a containing Euclidean space R^k, one may study vision group actions between these two objects often with an underlying signature which are equivalent under some symmetry or minimal distortion action with respect to a suitable metric inherited by this action. For example, Euclidean groups, similarity, Equi-Affine, projections, camera rotations and video groups. The study of the space of ordered configurations of n distinct points in R^k up to similarity transformations was pioneered by Kendall who coined the name shape space. For different groups of transformations (rigid, similarity, linear, affine, projective for example) one obtains different shape spaces. Moreover, while these formulations allow often global optimal optimization, e.g. using convex objectives , many of the problems above require efficient approximation methods which work locally. This framework has applications to biological structural molecule reconstruction problems, to recognition tasks and to matching features across images with minimal distortionâ This talk will discuss various work with collaborators around this circle of ideas.
Abstract: A PDE/ODE system can evolve in 3 time scales when the fast scales are associated with 2 small parameters that tend to zero at different rates. We investigate the limiting dynamics when the fast dynamics is generated by 2 skew-seft-adjoint operators and the initial time derivative is uniformly bounded regardless of the small parameters. To find the subspace that the limiting dynamics resides, we rely on matrix perturbation theory.
Abstract: Yves Couder and coworkers in Paris discovered that droplets walking on a vibrating fluid bath exhibit several features previously thought to be exclusive to the microscopic, quantum realm. These walking droplets propel themselves by virtue of a resonant interaction with their own wavefield, and so represent the first macroscopic realization of a pilot-wave system of the form proposed for microscopic quantum dynamics by Louis de Broglie in the 1920s. New experimental and theoretical results allow us to rationalize the emergence of quantum-like behavior in this hydrodynamic pilot-wave system in a number of settings, and explore its potential and limitations as a quantum analog.
Abstract: A key question in population biology is understanding the conditions under which the species of an ecosystem persist or go extinct. Theoretical and empirical studies have shown that persistence can be facilitated or negated by both biotic interactions and environmental fluctuations. We study the dynamics of n interacting species that live in a stochastic environment. Our models are described by n dimensional piecewise deterministic Markov processes. These are processes (X(t), r(t)) where the vector X denotes the density of the n species and r(t) is a finite state space process which keeps track of the environment. In any fixed environment the process follows the flow given by a system of ordinary differential equations. The randomness comes from the changes or switches in the environment, which happen at random times. We give sharp conditions under which the populations persist as well as conditions under which some populations go extinct exponentially fast. As an example we look at the competitive exclusion principle from ecology and show how the random switching can `rescue' species from extinction. The talk is based on joint work with Dang H. Nguyen (University of Alabama).
Abstract: Crumples in a sheet of paper, wrinkles on curtains, cracks in metallic alloys, and defects in superconductors are examples of patterns in materials. A thorough understanding of the underlying phenomenon behind the pattern formation provides a different prospective on the properties of the existing materials and contributes to the development of new ones. In my talk I will address the issue of modelling pattern formation via nonconvex energy minimization problems, regularized by higher order terms. Two particular examples of such models will be described in greater details: formation of vortices in Ginzburg-Landau model of superconductors, as well as emergence of patterns in phase transitions in shape-memory alloys. I will discuss the issue of well-posedness of such modelling, which reduces to the question of the existence of minimizers in certain functional classes. I will also provide some examples qualitative properties of minimizers via sharp energy bounds.
Abstract: In a recent paper, Bedrossian, Masmoudi and Mouhot proved the stability of equilibria satisfying the Penrose condition for the Vlasov-Poisson equation (with screened potential) on the whole space. We shall discuss a joint work with Nguyen and Rousset where we propose a new proof of this result, based on a lagrangian approach.
Abstract: Computational modeling can be used to reveal insights into the mechanisms and potential side effects of a new drug. Here we will focus on two major diseases: diabetes, which affects 1 in 10 people in North America, and hypertension, which affects 1 in 3 adults. For diabetes, we are interested in a class of relatively novel drug treatment, the SGLT2 inhibitors (sodium-glucose co-transporter 2 inhibitors). E.g., Dapagliflozin, Canagliflozin, and Empagliflozin. We conduct simulations to better understand any side effect these drugs may have on our kidneys (which are the targets of SGLT2 inhibitors). Interestingly, these drugs may have both positive and negative side effects. For hypertension, we want to better understand the sex differences in the efficacy of some of the drug treatments. Women generally respond better to ARBs (angiotensin receptor blockers) than ACE inhibitors (angiontensin converting enzyme inhibitors), whereas the opposite is true for men. We have developed the first sex-specific computational model of blood pressure regulation, and applied that model to assess whether the "one-size-fits-all" approach to blood pressure control is appropriate with regards to sex.
Abstract: Random matrix statistics emerge in a broad class of strongly correlated systems, with evidence suggesting they can play a universal role comparable to the one Gaussian and Poisson distributions do classically. Indeed, observational studies have identified these statistics among heavy nucleii, Riemann zeta zeros, pedestrians, land divisions, parked cars, perched birds, and other forms of traffic. Noticing that these latter real-world systems all operate in a decentralized manner, we investigate the simplest possible games that admit Coulomb gas dynamics as a Nash equilibrium and investigate their basic features, many of which are atypical or even new for the literature on many player games. For example, there is a nonlocal-to-local transition in the population argument of the N-Nash system of PDEs and, perhaps most significantly, there is a sensitivity of local limit behavior to the chosen model of player information. If there is time, this talk will also discuss some future research directions based on the many questions these results raise.
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