The talk aims at providing an introduction into some basic problems occurring in the ergodic theory of uncountable group actions and a setup and a few tools on how to resolve these issues. This part of the talk shall be accessible to anyone with a graduate-level background in probability and analysis. Towards the end of…
This talk will focus on using the Euclidean Signature to determine whether two smooth planar curves are congruent under the Special Euclidean group. Work done by Emilio Musso and Lorenzo Nicolodi emphasizes that signatures must be used with caution by constructing 1-parameter families of non-congruent curves with degenerate vertices (curve segments of constant curvature) with identical signatures. We address the claim…
Archetypal analysis is an unsupervised learning method that uses a convex polytope to summarize multivariate data. For fixed k, the method finds a convex polytope with k vertices, called archetype points, such that the polytope is contained in the convex hull of the data and the mean squared distance between the data and the polytope…
Given a non-zero square-integrable function $g$ and $\Lambda=\{(a_k, b_k)\}_{k=1}^N \subset \mathbb{R}^2$ let $\mathcal{G}(g, \Lambda)=\{e^{2\pi i b_k \cdot}g(\cdot - a_k)\}_{k=1}^N.$ The Heil-Ramanathan-Topiwala (HRT) Conjecture is the question of whether $\mathcal{G}(g, \Lambda)$ is linearly independent. For the last two decades, very little progress has been made in settling the conjecture. In the first part of the talk,…
We consider the problem of inferring the basal sliding coefficient field for an uncertain Stokes ice sheet forward model from surface velocity measurements. The uncertainty in the forward model stems from unknown (or uncertain) auxiliary parameters (e.g., rheology parameters). This inverse problem is posed within the Bayesian framework, which provides a systematic means of quantifying uncertainty in the solution. To account…
I give an introductory talk about geometric inverse problems and ray transforms (no proofs are involved). I mainly focus on the Euclidean X-ray transform of scalar fields and vector fields, but also introduce the basic properties of the X-ray transform of higher order tensor fields. I give some unique continuation results for the normal operator…
Quasiconvex functions, a major generalization of convex functions, naturally arise in calculus of variations, optimal control and differential games in L-infinity. This connection with HJ equations, representation formulas, obstacle problems, and reach-avoid problems will be discussed. Zoom meeting ID: 802 764 2791
Let $U_1,\ldots,U_n$ be independent random vectors which are uniformly distributed on the unit sphere. The random hyperplanes $U_1^\perp,\ldots,U_n^\perp$ dissect the space into a collection of random cones. A uniform random cone $S_n$ from this collection is called the Schläfli random cone. In a classical paper of Cover and Efron (1967) it was proved that the…
In this talk I will present a Bayesian approach to quantify the uncertainty of model parameters and hemodynamic predictions in a one-dimensional fluid-dynamics model of the pulmonary system by integrating mouse imaging data and hemodynamic data. The long-term aim is to devise a calibrated patient-specific model. I emphasize an often neglected, though the important source…
In this talk, we will discuss a recent global existence result on the Poiseuille flow of nematic liquid crystals via full Ericksen-Leslie model. The existing results on the Ericksen-Leslie model for the liquid crystals mainly focused on the parabolic and elliptic type models by omitting the kinetic energy term. In this recent progress, we established…
I will discuss recent progress on dimension-free estimates for the Hardy--Littlewood maximal functions in the continuous and discrete settings. Website: https://sites.google.com/view/paw-seminar Host: Paata Ivanisvili  pivanis@ncsu.edu
A nonlinear analogue of the Rademacher type of a Banach space was introduced in classical work of Enflo. The key feature of Enflo type is that its definition uses only the metric structure of the Banach space, while the definition of Rademacher type relies on its linear structure. I will speak about the joint work with…
We will discuss Marstrand's classical theorem concerning the interplay between density of a measure and the Hausdorff dimension of the measure's support in the context of finite-dimensional Banach spaces. This is joint work with David Bate and Tatiana Toro. Website: https://sites.google.com/view/paw-seminar Host: Paata Ivanisvili  pivanis@ncsu.edu
Free-boundary problems are partial differential equations in which the unknown function and its domain must be simultaneously determined. They arise ubiquitously as mathematical models for phenomena in many fields, most notably in physics, biology and finance. Free boundary problems are typically highly nonlinear and nonlocal in nature, making their analysis challenging. I will discuss two fundamental…
A central question in machine learning theory is whether an algorithm enjoys good generalization, which is the ability to correctly predict new examples from prior observations. While classical wisdom advocates for methods with fewer parameters than data points in order to avoid overfitting, modern machine learning algorithms are severely over-parameterized and perfectly fit training data.…
Thin liquid films flowing down vertical fibers exhibit complex and interesting interfacial dynamics, including droplet formation and traveling wave patterns. Such dynamics play a crucial role in the design of heat and mass exchangers for many engineering applications, including cooling and desalination systems. Recent experiments present a wealth of new dynamics that illustrate the need…