Title: Soap bubble theorems in Convex Geometry and Geometric Measure Theory.

Abstract: The celebrated Alexandrov theorem (1958) asserts that a compact and embedded smooth hypersurface with constant mean curvature in the Euclidean space must be a sphere. In this talk I will discuss two recent extensions of this result to singular varieties. The first extension deal with sets of finite perimeter with constant (possibly anisotropic) distributional mean curvature, which correspond to the critical points of the (possibly anisotropic) isoperimetric functional. The second extension deal with arbitrary convex bodies.

]]>Title: The Feynman-Kac Formula for generalized Kolmogorov Backward Equations

Abstract: Feynman-Kac formula is an important link between parabolic partial differential equations and stochastic processes, specifically Ito processes. We will introduce the Feynman-Kac formula along with its proof and some examples of application.

]]>Speaker: Yanzhao Cao

Affiliation: Auburn University

Title: Uncertainty quantification of deep neural networks

Abstract: In this talk, I will first give a mathematical introduction to deep learning. Then I will talk about a recent work on uncertainty quantification (UQ) of deep learning. Uncertainty quantification of deep neural networks (DNN) is a very important issue in deep learning. In our UQ for DNN framework, the DNN architecture is the neural ordinary differential equations (Neural-ODE), which formulates the evolution of potentially huge hidden layers in the DNN as a discretized ordinary differential equation (ODE) system. To characterize the randomness caused by the uncertainty of models and noises of data, we add a multiplicative Brownian motion noise to the ODE as a stochastic diffusion term, which changes the ODE to a stochastic differential equation (SDE), and the deterministic DNN becomes a stochastic neural network (SNN) In the SNN, the drift parameters serve as the prediction of the network, and the stochastic diffusion governs the randomness of network output, which serves to quantify the epistemic uncertainty of deep learning. I will present results on convergence as well as numerical experiments.

]]>Title: Almost-periodic Response Solutions for a forced quasi-linear Airy equation

Abstract: The problem of response solutions for PDEs has been widely studied in many contexts. In this seminar I will prove the existence of almost-periodic solutions for quasi-linear perturbations of the Airy equation. Such solutions turn out to be analytic in both time and space. To prove the result one needs to combine the so called Craig-Wayne approach combined with a KAM reducibility scheme and pseudodifferential calculus on $T^\infty$.

This is a joint work with R. Montalto and M. Procesi.

]]>AFFILIATION: Lawrence-Berkeley

TITLE: A Graph-Based Machine Learning System for the Forward Property Predictions and Inverse Structural Design of Molecules

ABSTRACT: Completing the round-trip tour between molecular structure and property relationship involves solving the forward prediction problem as well as the inverse design problem. In this talk, I will first present an algorithmic pipeline built around a graph representation of molecules that leads to predictive models of molecular properties with very fast time-to-accuracy. Specifically, we use Gaussian process regression models and their low-rank approximations on top of a Marginalized Graph Kernel covariance function to solve various molecular property prediction and dataset denoising tasks. I will then explain how such predictive models can be used as a surrogate to drive an inverse design protocol, which transforms molecules by graph rewriting as guided by the predicted likelihood of obtaining desired properties. Last, I will introduce the open-source Python package GraphDot. The package contains robust implementations of the aforementioned methods and beyond. It uses just-in-time GPU code generation and compilation to achieve an acceleration of up to four orders of magnitude against similar CPU packages. It allows easy experimentation of sophisticated experimentation of molecular graph representations and kernel design.

]]>Animals, including humans, have had to evolve in order to balance the risk of catching infectious diseases from each other against the need for social contact and useful group organization. In her presentation, Professor Fefferman will use mathematical modeling to show how some of the behavioral patterns we see in nature achieve this balance really efficiently. She will try to apply some of these lessons to human society today.

]]>TITLE: Minimal surfaces and Plateau problem

ABSTRACT: In calculus of variations, Plateau problem is the problem of finding the surface that minimises area among all surfaces spanning a given curve. This was posed Lagrange in 1760 and solved independently by Douglas and Radó in 1930. In 1936 Douglas was awarded the Fields Medal for his efforts. After an introduction to the field of minimal surfaces, I will talk about Plateau problem and its proof.

]]>Title: The geometry of constant mean curvature surfaces in Euclidean space.

Abstract: I will begin by reviewing classical geometric properties of constant mean curvature surfaces, H>0, in R^3. I will then talk about several more recent results for surfaces embedded in R^3 with constant mean curvature, such as curvature and radius estimates for simply-connected surfaces embedded in R^3 with constant mean curvature. Finally I will show applications of such estimates including a characterisation of the round sphere as the only simply-connected surface embedded in R^3 with constant mean curvature and area estimates for compact surfaces embedded in a flat torus with constant mean curvature and finite genus. This is joint work with Meeks.

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]]>Speaker: Juan Pablo Borthagaray

Affiliation: Universidad de la Republica, Uruguay

Title: Local error estimates for nonlocal problems

Abstract: The integral fractional Laplacian of order $s \in (0,1)$ is a nonlocal operator. It is known that solutions to the Dirichlet problem involving such an operator exhibit an algebraic boundary singularity regardless of the domain regularity. This, in turn, deteriorates the global regularity of solutions and as a result the global convergence rate of the numerical solutions.

In this talk we shall discuss regularity of solutions on bounded Lipschitz domains. For finite element discretizations, we derive local error estimates in the $H^s$-seminorm and show optimal convergence rates in the interior of the domain by only assuming meshes to be shape-regular. These estimates quantify the fact that the reduced approximation error is concentrated near the boundary of the domain. We illustrate our theoretical results with several numerical examples.

The talk is based on joint work with Dmitriy Leykekhman (University of Connecticut) and Ricardo Nochetto (University of Maryland).

]]>Title: Aggregation-diffusion equation: symmetry, uniqueness and non-uniqueness of steady states

Abstract: The aggregation-diffusion equation is a nonlocal PDE that arises in the collective motion of cells. Mathematically, it is driven by two competing effects: local repulsion modeled by nonlinear diffusion, and long-range attraction modeled by nonlocal interaction. In this talk, I will discuss several qualitative properties of its steady states and dynamical solutions. Using continuous Steiner symmetrization techniques, we show that all steady states are radially symmetric up to a translation. (joint work with Carrillo, Hittmeir and Volzone). In a recent work, we further investigate whether they are unique within the radial class, and show that for a given mass, uniqueness/non-uniqueness of steady states are determined by the power of the degenerate diffusion, with the critical power being m = 2. (joint work with Delgadino and Yan.)

]]>AFFILIATION: University of Bath, UK

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]]>AFFILIATION: University of Bath, UK

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Speaker: Marcus Sarkis

Affiliation: Worcester Polytechnic Institute

Title: Robust Model Reductions for the Transmission Problem with High-Contrast and Heterogeneous Coefficients

Abstract:

The goal of this talk is to present finite element discretizations for second-order elliptic problems with heterogeneous and possibly with high-contrast coefficients. Based on a class of adaptive domain decomposition preconditioners named Balancing Domain Decomposition with Constraints--BDDC, Variational Multiscale Methods--VMS and Localized Orthogonal Decomposition Methods--LOD, we design robust discretizations and establish optimal a priori error energy estimates without assuming regularity on the solution.

]]>AFFILIATION: University of Bath, UK

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]]>Speaker: Franziska Weber

Affiliation: CMU

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AFFILIATION: University of Bath, UK

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]]>Title: Permanental Random Point Process and BEC

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]]>Speaker: Guannan Zhang

Affiliation: ORNL

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]]>AFFILIATION: University of Bath, UK

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]]>TIME: 3:35 PM - 4:3f PM

ROOM: Ayres 405

HOST: Jan Rosinski ]]>

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Speaker: Victor DeCaria

Affiliation: ORNL

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]]>AFFILIATION: University of Bath, UK

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]]>AFFILIATION: University of Bath, UK

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