Probability, Differential Geometry and Mathematical Physics
Department of Mathematics and Statistics
Texas Tech University
This is an expository talk about a subject about which the speaker is currently learning. The theory of regularity structures was invented by Martin Hairer in 2014 (for which he won the Fields Medal). In short, many equations in physics such as Kardar-Parisi-Zhang equation, parabolic Anderson model, Phi4 model from quantum field theory, and even Navier-Stokes equations, were studied under the forcing by white noise (typically white in both space and time) which was so singular that the most fundamental question about the existence of its solution was not proven rigorously. In fact, solutions in the classical sense (or even weak sense) actually do not exist. The theory of regularity structures was a first (along with the theory of paracontrolled distributions due to Gubinelli et al.) systematic way to actually prove the existence of a limiting solution (after much work of renormalization, computing Wick products, and writing tree diagrams akin to Feynman diagrams). Its impact has been immense and has generated waves of new results in many related fields (even stochastic quantization of Yang-Mills).
Please virtually attend this seminar via this zoom link Wednesday the 14th at 3 PM.Constant Mean Curvature (CMC) surfaces constitute a classical subject in Differential Geometry and are mathematical models in many disciplines of science. In this talk, I will present a recent work on the existence of CMC 2-spheres in an arbitrary Riemannian 3-sphere. This is a joint work with Da Rong Cheng.
Please virtually attend Dr. Zhou's talk on Wednesday at 3 PM via this zoom link.N/A
This week's PDGMP seminar details available at this pdf
Please virtually attend Dr. Aulisa's presenation at 3 PM on Wednesday the 24th at this zoom link.
The subject of the talk are two new biological problems that we model using Einstein paradigm of the generalized Brownian motion of two-components “prey-predator” systems coupled by chemotactic dynamics. We develop a thought experiment to build a model of chemotactic interaction between coupled biological systems.
To this end we propose the length of the free transport of a predator (jump without mutual "collision") to be a random variable. We hypothesize that the expected value of free jumps is proportional to the relative changes of the prey’s density, which is consumed but may move away from the predator. This allows us to generalize a Keller-Segel type of system of equations with cross-diffusion.
In case when prey is not mobile we derive two Keller-Segel types of the models which manifest two essentially different types of traveling phenomena both of which have a clear biological interpretation. Research is based on recent results obtained jointly with Rahnuma Islam, graduate student of the Department of Mathematics and Statistics, Texas Tech University
Please virtually attend Dr. Ibraguimov's talk on Wednesday at 3 PM via this zoom link.The talk is accessible to beginning graduate students. Consider a geometric functional such as the area or the volume. In order to find local extrema, we calculate the second variation which is normally a symmetric bilinear form in a function space. The Morse index, which counts the maximal dimension of a subspace on which the form is negative definite, is the generalization of the 2nd derivative test in elementary calculus. Now consider the natural extension of finding extrema subject to certain constraints. How would we examine the index with constraints? In this talk, we’ll answer that question in a general abstract framework and then apply it to study capillary surfaces. This is joint work with Detang Zhou.
We discuss equilibrium surfaces for an energy which is a linear combination of the classical bending energy for curves and a surface energy containing the squared L2 norm of the difference of the mean curvature and the spontaneous curvature, i.e. the Helfrich energy.
Please virtually attend this week's PDGMP seminar via this zoom link Wednesday the 17th at 4 PM.This week's PDGMP seminar details available at this pdf
Please virtually attend Ms. Atampalage's presentation at 3 PM on Wednesday the 24th at this zoom link.
Blaschke showed that a surface with one family of spherical curvature lines can be parametrised via a certain flow of an initial curve on a sphere. In this talk we characterise when this surface is additionally a Lie applicable surface, by restricting the flow and the initial curve. It turns out that the initial curve must project to a constrained elastic curve in some space form, which leads us to a Lie geometric characterisation of such curves.
Please virtually attend this week's PDGMP seminar via this zoom link Wednesday the 31st at 10 AM.We address optimal quasiconformal immersions of surfaces in Euclidean space, with the goal of computing folding-free deformations with prescribed boundary. It is shown that extremal quasiconformal maps between Riemann surfaces can be characterized as critical points of the Dirichlet energy with respect to a certain metric, which leads to an iterative algorithm for the computation of maps which minimize the maximum conformality distortion. These so-called Teichmuller maps are shown to be robust and computable from extrinsic data, with applications to discrete surface deformation as well as remeshing.
Please virtually attend the PDGMP seminar on Wednesday, April 7th at 3 PM via this zoom link.This is an expository talk about a subject about which the speaker is currently learning. The theory of regularity structures was invented by Martin Hairer in 2014 (for which he won the Fields Medal). In short, many equations in physics such as Kardar-Parisi-Zhang equation, parabolic Anderson model, Phi4 model from quantum field theory, and even Navier-Stokes equations, were studied under the forcing by white noise (typically white in both space and time) which was so singular that the most fundamental question about the existence of its solution was not proven rigorously. In fact, solutions in the classical sense (or even weak sense) actually do not exist. The theory of regularity structures was a first (along with the theory of paracontrolled distributions due to Gubinelli et al.) systematic way to actually prove the existence of a limiting solution (after much work of renormalization, computing Wick products, and writing tree diagrams akin to Feynman diagrams). Its impact has been immense and has generated waves of new results in many related fields (even stochastic quantization of Yang-Mills).
Please virtually attend this seminar via this zoom link Wednesday the 14th at 3 PM.Optimal control problems constrained by PDEs are certainly among the most interesting and challenging topics in mathematics. Not only is that the case for the practical applications, but also for the underlying theoretical and computational questions that arise in their study. We will address some of these questions for a certain class of optimal control problems where boundary data act as the steering force, and where the constraint equations are typical models in continuum mechanics.
Please virtually attend this week's PDGMP seminar at 3 PM Wednesday the 21st at this zoom link.
In this talk we will construct via Daniel's sister correspondence in H2xR a 2-parameter family of Alexandrov-embedded constant mean curvature 02xR with 2 ends and genus 0. They are symmetric with respect to a horizontal slice and k vertical planes disposed symmetrically. We will discuss the embeddedness of the constant mean curvature surfaces of this family, and we will show that the Krust property does not hold for 02xR.
Please virtually attend this week's PDGMP seminar via this zoom link, Wednesday the 28th at 1 PM (UTC-5).