Events
Department of Mathematics and Statistics
Texas Tech University
Newton’s iterative root finding method and the dynamics of the Newton maps of rational functions have been studied in recent years, resulting in the classification of rational functions whose Newton maps are Möbius conjugate to polynomials of degree 1, 2, and 3. In this dissertation, we describe a new approach for obtaining the same results, then classify the sets of rational functions whose Newton maps are Möbius conjugate to polynomials of degree 4, polynomials of degree 5, and polynomials of any finite degree.
If you cannot attend in person, join the talk on Zoom.
The shape of the Red Blood Cell (RBC) exhibits a unique toric-like shape and has posed challenges in mathematical modelling. In the 1970’s, Peter Canham and Wolfgang Helfrich pioneered techniques from elastic energy theory, reframing the study of cell membranes via the integral of the mean curvature squared over a surface called the Helfrich functional, $$\int\int_M \beta(H-c_o)^2 + K dS +\int \lambda dS + \int \Delta P dV$$ where $c_o$ is the spontaneous curvature, $\Delta P$ the partial pressure, $\lambda $ the surface tension, and $\beta$ the bending rigidity respectively. Varying this functional yields a fourth-order elliptic PDE for equilibrium membrane shapes.
Later, Canham introduced the now-standard “axisymmetric model” which reframes the Euler-Lagrange equation into a nonlinear second order ODE known as the shape equation. This model describes rotationally symmetric surfaces generated from a profile curve, parametrized in terms of a rotational radius $r$, and the angle $\psi(r)$ between the profile’s tangent and a horizontal line.
Cassini ovals, a family of toric-like curves, have been proposed as apt RBC models. We show that these curves fail to satisfy the shape equation, and therefore the Euler-Lagrange equation, ruling them out as energy minimizing solutions to the Helfrich energy functional.
The infinitesimal counterpart of a Lie group(oid) is its Lie algebra(oid). I will show that the differentiation procedure works in any category with an abstract tangent structure in the sense of Rosicky, which was later rediscovered by Cockett and Cruttwell. Mainly, I will construct the abstract Lie algebroid of a differentiable groupoid in a cartesian tangent category $C$ with a scalar $R$-multiplication, where $R$ is a ring object of $C$. Examples include differentiation of infinite-dimensional Lie groups, elastic diffeological groupoids, etc. This is joint work with Christian Blohmann.We discuss a new notion of lifting modules over a Noetherian local ring to a regular local ring. We define a module to be Serre liftable if it lifts to a module of correct dimension over the regular ring. We will discuss an interesting property of such Serre liftable modules: a finite length Serre liftable module must have length at least the Hilbert-Samuel multiplicity of the local ring. This project is joint with Nawaj KC.
 | Thursday
Apr. 3
6:30 PM
MA 108
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| Mathematics Education
Math Circle
Stone Fields
Mathematics and Statistics, Texas Tech University
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Math Circle Spring Poster
abstract noon CDT (UT-5)
Zoom link available from Dr. Brent Lindquist upon request.