Wednesdays at 4:30 PM in 384H.
A system of stochastic differential delay equations models the motion of a light-sensitive robot, exploring an inhomogeneous landscape. It can be approximated by a stochastic differential equation without delay and, in this form, studied using a multiscale expansion. The resulting dynamics depends on the value of the delay parameter, which can be positive or negative---I will explain what this means and how it is realized experimentally. The results for a single robot are applied to predict an aggregation-deaggregation transition in many-robot systems, in a very good agreement with experiments. A nontechnical summary can be found at https://physics.aps.org/articles/v9/13
We discuss two renormalization approaches to the problem of KPZ universality. The first one is based on information regarding optimal actions and is related to the Airy sheet process. The second approach is defined in pure geometrical terms. We shall discuss some results and conjectures related to the second renormalization scheme.
In this work, we perform an asymptotic analysis of a nonlocal reaction-diffusion equation, with a fractional laplacian as the diffusion term and with a nonlocal reaction term. Such equation models the evolutionary dynamics of a phenotypically structured population. We perform a rescaling considering large time and small effect of mutations, but still with algebraic law. We prove that asymptotically the phenotypic distribution density concentrates as a Dirac mass which evolves in time. This work extends an approach based on Hamilton-Jacobi equations with constraint, that has been developed to study models from evolutionary biology, to the case of fat-tailed mutation kernels. However, unlike previous works within this approach, the WKB transformation of the solution does not converge to a viscosity solution of a Hamilton-Jacobi equation but to a viscosity supersolution of such equation which is minimal in a certain class of supersolutions.
Optimal transport (Wasserstein metric) nowadays play important roles in data science. In this talk, we brief review its development and applications in machine learning. In particular, we will focus its induced differential structure. We will introduce the Wasserstein natural gradient in parametric models. The metric tensor in probability density space is pulled back to the one on parameter space. We derive the Wasserstein gradient flows and proximal operator in parameter space. We demonstrate that the Wasserstein natural gradient works efficiently in several statistical machine learning problems, including Boltzmann machine, generative adversary models (GANs) and variational Bayesian statistics.
Parametric uncertainty, considered broadly to include uncertainty in system parameters and driving forces (source terms and initial and boundary conditions), is ubiquitous in mathematical modeling. The method of distributions, which comprises PDF and CDF methods, quantifies parametric uncertainty by deriving deterministic equations for either probability density function (PDF) or cumulative distribution function (CDF) of model outputs. Since it does not rely on finite-term approximations (e.g., a truncated Karhunen-Loeve transformation) of random parameter fields, the method of distributions does not suffer from the ``curse of dimensionality''. On the contrary, it is exact for a class of nonlinear hyperbolic equations whose coefficients lack spatio-temporal correlation, i.e., exhibit an infinite number of random dimensions.
Understanding and tailoring light-matter interactions lies at the heart of many modern solid-state technologies, ranging from imaging, to solar photovoltaics, data storage and quantum computations. This inspires research in artificial composite materials with properties acquired by design, often termed metamaterials. In this talk, I will demonstrate how metamaterial principles along with simple photonic concepts, such as surface plasmon polaritons, phonons and excitons, guided waves and optical bandgaps, can be utilized and combined to introduce novel material functionalities that surpass natural bounds. These include optical magnetism, unattainable with natural materials, ultra-lightweight mirrors with van der Waals heterostructures, and actively tunable dielectric response. Controlling the flow of light via its interaction with structured matter has further implications in photonic thermal management, with applications in nanoscale cooling, thermal circuitry, and waste heat recovery. In the second part of this talk, I will discuss how the framework of fluctuational electrodynamics helps us understand and control the flow of heat. Our recent results show that, by borrowing ideas from optoelectronics, we can design nanoscale thermal switches, the thermal counterpart to the electronic field-effect transistor. Thermal management is also crucial in energy recycling, in particular thermophotovoltaics, an emerging technology where conversion of heat to electricity is mediated by emission of electromagnetic radiation. I will discuss fundamental limits (e.g. blackbody limit), current limitations (e.g. intrinsic semiconductor processes such as non-radiative recombination), and how these are mitigated using near-field radiative heat transfer.materials
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