Intro to Research at UCR seminar

Wednesday 2:30-3:30 in 284 Skye Hall


Schedule:


Title: Impacts of Mathematical Modeling of Blood Clots on Mathematics and Biology

Abstract: While blood clot formation has been relatively well studied, little is known about the mechanisms underlying the subsequent structural and mechanical clot remodeling called contraction or retraction. Impairment of the clot contraction process is associated with both life-threatening bleeding and thrombotic conditions, such as ischemic stroke, venous thromboembolism, and others. Recently, blood clot contraction was observed to be hindered in patients with COVID-19. First, a novel multi-phase PDE model of blood clot embolization, life threatening condition, will be described and discussed [1]. Then, a novel three-dimensional multiscale model consisting of thousands of Langevin differential equations, will be described, and applied to quantify biomechanical mechanisms of the kinetics of clot deformation and contraction driven by platelet filopodia-fibrin fiber pulling interactions. These results provide novel mathematical modeling approaches as well as important biological insights [2,3]. The biomechanical mechanisms and modeling approach to be described can potentially apply to studying other blood clotting processes as well as to other biological systems in which cells are embedded in a filamentous network and exert forces on the extracellular matrix.


Title: Introduction to ergodic Schrodinger operators

Abstract: Ergodic Schrodinger operators arise naturally in the study of Schrodinger equations, which models the motion of a quantum particle in a disordered medium. On the other hand, spectral analysis of such operators may eventually be turned into questions in dynamical systems. In this talk, I will briefly introduce how the three different areas--mathematical physics, spectral analysis, and dynamical systems-- interact in my research.


Title: An An invitation to algebraic geometry

Abstract: We will introduce the field of mathematics known as algebraic geometry and will provide an overview of some of the problems that have motivated my research in this area. The main objects of study in algebraic geometry are algebraic varieties, which are spaces locally defined by the solution sets of systems of polynomial equations. My research primarily focuses on the study of varieties that can be described using combinatorial formalisms, such as toric varieties; intersection theory, particularly algebraic cobordism; birational geometry, especially the finite generation of total coordinate rings; and the geometry of moduli spaces. I greatly enjoy using algebra, geometry, combinatorics, and topology to study these topics, and it will be my pleasure to share some of this story with you during the talk.


Title: Biological Experiments in a Math Lab

Abstract: In developmental biology, the aim is to understand the principles underlying the growth of tissues or organs as well as the pattern formation during the processes. Mathematical modeling offers remarkable benefits to the research community for studying morphogenesis and growth control, which are the two fundamental questions in developmental biology, due to its capability to simulate complex biological systems in sillico with high efficiency. Different modeling approaches have been widely used to understand a variety of biological systems, especially those requiring challenging experiments and involving complex structural change. In a math lab, one can perform simulations under both wild type and mutant conditions as biological experiments in wet labs, as well as data analysis for calibrating models and investigating mechanisms. In this talk, I will show several models developed or used in my group to study classic biological systems in developmental biology.


Ttile: Curvature, minimal submanifolds, and topology of a manifold.

Abstract: In this talk, we investigate the relation between the curvature of its global properties. In particular, we will see that the existence of a metric with positive curvature restricts the topology as well as minimal submanifolds.


Title: Thirteen ways to say ``lower curvature bound" 

Abstract: We will define what it means for an intrinsic metric space to have a lower curvature bound, by comparing the space's geometry to that of a space of constant curvature. We will start with a description of the three model, two dimensional, spaces with constant curvature, the hyperbolic plane, the euclidean plane, and the two sphere. We will then discuss the wide open problem: Which closed smooth n-manifolds admit Riemannian metrics with nonnegative sectional curvature.


Title: On Fluids

Abstract: What is a fluid? How tangible is air? How do planes fly? What is turbulence? I don't know the answer to any of these questions, but I can say a bit about their mathematical idealizations. And give a rough idea of the kinds of questions that mathematical fluid mechanics are currently actively pursuing.


Title:  Lorentz gases on quasicrystals 

Abstract: The Lorentz gas was originally introduced as a model for the movement of electrons in metals. It consists of a massless point particle (electron) moving through Euclidean space bouncing off a given set of scatterers S (atoms of the metal) with elastic collisions at the boundaries of S. If the set of scatterers is periodic in space, then the quotient system, which is compact, is known as the Sinai billiard. 

There is a great body of work devoted to Sinai billiards and in many ways their dynamics is well understood. In contrast, very little is known about the behavior of the Lorentz gases with aperiodic configurations of scatterers which model quasicrystals and other low-complexity aperiodic sets. This case is the focus of our joint work with Rodrigo Treviño.We establish some dynamical properties which are common for the periodic and quasiperiodic billiard. We also point out some significant differences between the two. The novelty of our approach is the use of tiling spaces to obtain a compact model of the aperiodic Lorentz gas on the plane. 


Title: Representation theory: a tale of impossible coincidences

Abstract: The original idea of representation theory is that we can try to understand properties of an inherently nonlinear and complicated object (for example, a group) by realizing it as linear operators on a vector space, that is, essentially, representing it by some collection of matrices. But the beauty of representation theory lies in the fact that it often connects objects which a priori should not have anything in common. 

The aim of my talk is to provide an overview of several areas of representation theory which are connected with each other via root systems of Lie algebras. We will start with the classical problem solved by Jordan (and Kronecker, who actually solved a different problem but slightly earlier than Jordan) of describing equivalence classes of linear operators on a finite dimensional complex vector space  and build our way into Hall algebras, cluster algebras and Lie algebras. 


Title: Arithmetic Hyperbolic Surfaces and Beyond

Abstract: Arithmetic hyperbolic surfaces are fascinating, highly symmetric geometric objects that exhibit all sorts of extremal behavior. Want a surface with a large systole?  Look at arithmetic hyperbolic surfaces.  Want a surface with a large Cheeger constant?  Look at arithmetic hyperbolic surfaces.  he goal of this talk is to give a gentle introduction to arithmetic hyperbolic surfaces.  I will highlight numerous and deep connections between topology, geometry, Lie theory, linear algebra, and number theory.   It is these connections that first drew me to this area.  There will be many pictures.   I will then discuss some of the major areas of research involving arithmetic hyperbolic surfaces and their generalizations.  In the last few minutes, I will discuss some of my research in the area and directions for future research.