Events
Department of Mathematics and Statistics
Texas Tech University
The first symmetrization transformation was introduced by Jacob Steiner in 1838 in his attempt to find a geometric proof of the classical Isoperimetric Problem. It is amazing that for almost two centuries after its creation the method of symmetrization remains an incredibly powerful tool in many areas of mathematics and mathematical physics. Several ramifications and generalizations of this method were suggested by George Pólya, Gabor Szegö, Walter Hayman, Igor P. Mityuk, Al Baernstein II, Vladimir Dubinin, Friedemann Brock and some other outstanding mathematicians.
I became acquanted with symmetrization in the late 70's, when my thesis adviser, Prof. Igor P. Mityuk, suggested several research problems in Geometric Function Theory. One of those was the Pólya-Szegö problem on continuous Steiner symmetrization, which construction was suggested in one of my papers. I also used some other symmetrization methods and in this talk, will discuss symmetrization-type transformations which I used in my work on problems suggested by Prof. Mityuk and other prominent mathematicians.
At the end of my talk I present some open questions on transformations of symmetrization type.
In his Habilitationsschrift Riemann mentioned the geometric tangent 2D-indicatrix as a relevant extension of the Euclidean unit circle, thus conceiving the simplest Riemann-Finsler metrics. However, in 1818, Gabriel Lamé (1795-1860) had already provided a systematic treatment of supercircles and superellipses, with the aim of describing crystals geometrically.
The generalization of Lamé curves to the Superformula, a generic geometric transformation generalizing conic sections, turns out to be an excellent scientific tool for the study of natural shapes and phenomena, from the very small to the very large. These developments have found their way in numerous applications in technology and science. It also connects many fields of science in a simple way, e.g. shape and form (constant anisotropic mean curvature surface), and inequality in economics and biology.
In a dynamic and kinematic setting, recent developments include a complete differential geometry apparatus, adapted to these shapes. This aligns with Lamé’s Unique Rational Science: find the best fitting coordinate systems and then solve the relevant boundary value problems.
Zoom link: texastech.zoom.us/j/93346469342
A recent AI assisted discovery of a counterexample to a conjecture in commutative algebra includes an elegant construction of one-dimensional Gorenstein domains. I will explain the construction and discuss its origins and consequences.
Abstract: In population dynamics, the diffusive McKendrick-von Foerster system plays a vital role in modeling the evolution of structured populations comprising both immature and mature individuals. In this work, we propose a numerical method to address a multiple-coefficient inverse problem for this system. In particular, our method is aimed to simultaneously reconstruct four coefficients: the natural death rates of both population groups, the recruitment rate, and the initial distribution of mature individuals. Our inverse approach is based on some special transformations and the use of the Fourier-Klibanov basis, which together yield a nonlinear bi-directional ODE-PDE system. This system is then approximated using the variational quasi-reversibility method, which is designed using a suitable pair of perturbing and stabilized operators, along with associated conditional energy estimates. In this work, we prove that the approximation of the immature individuals converges at a Lipschitz rate, whereas the approximation of the mature ones converges at a Hölder rate. The latter convergence rate represents a significant extension of earlier studies on the variational quasi-reversibility inversion. Finally, some numerical examples are presented to show how the proposed inversion works.
When: 4:00 pm (Lubbock's local time is GMT -5)
Where: room Math 011 (Math Basement)
ZOOM details:
- Choice #1: use this Direct Link that embeds meeting ID and passcode.
- Choice #2: With this link, Join Meeting, you will then need to input the ID and Passcode by hand:
* Meeting ID: 949 9288 2213
* Passcode: Applied
Outreach program for advanced middle and high school math studentsThe United States has a long-standing issue with measles, even with high levels of vaccination. In this study, we develop and analyze a deterministic mathematical model to investigate measles transmission dynamics in a heterogeneous population. To capture both the burden of disease and its transmission, we include some of the most important epidemiological and demographic processes, such as recruitment, transitions between age groups, transitions between risk classes, and disease-induced mortality. We define the basic analytical properties of the model, such as positivity and boundedness of solutions, and calculate the basic reproduction number to describe the potential of transmission. We examine both local and global stability of the disease-free equilibrium (DFE) and establish the conditions under which measles can be eliminated, which includes a clear herd immunity level. We supplement the analytical results with the numerical simulation and the global sensitivity analysis to study the impact of the important parameters, determine the effect of the vaccination coverage, and the impact of the intervention strategies. According to this study, to eliminate measles from the United States, at least 94 % individuals should be vaccinated (herd immunity threshold), consistent with existing literature. Our results give a quantitative understanding of the role of age and risk heterogeneity in determining the processes of measuring the transmission and control of measles, and offer an informative framework to guide the processes of vaccination planning and decision-making in health on the part of the public in a heterogeneous population. Simulations also show that incorporating a single supplementary immunization activity (SIA) achieving 80% coverage of the susceptible population, alongside routine vaccination, reduces cumulative measles cases by approximately 72% over the 48-week outbreak period. We further analyze an optimal-control extension incorporating two intervention strategies: an awareness-based transmission-reduction program and a continuous vaccination program. Numerical simulations show that both strategies reduce infection burden, while the cost-effectiveness analysis identifies vaccination as the most cost-effective intervention under the assumed cost structure.
Since the seminal work of Gatheral, Jaisson, and Rosenbaum (2014), it has become widely accepted that volatility in derivatives markets exhibits rough behavior. Numerical methods for rough volatility models are often difficult to stabilize and scale to large volatility smile surfaces. In addition, calibration to volatility surfaces, rather than to historical data, remains a significant challenge, as does the reliable valuation of exotic derivatives. In this work, we clarify how machine learning tools can be used most effectively to address these problems.
Zoom link: texastech.zoom.us/j/3067000354
Meeting ID: 306 700 0354
Passcode: TTUMF