Events
Department of Mathematics and Statistics
Texas Tech University
One of the great developments in the study of continuous and discrete time dynamics was the theory of Lyapunov exponents and non-uniform hyperbolicity. It is an amalgamation of ideas from Linear algebra, Analysis, Differential geometry as well as Measure theory. Lyapunov exponents quantify the asymptotic rate of growth of perturbations in the dynamics. Asymptotic properties usually have a discontinuous dependence on initial states. The key discovery is that although discontinuous, Lyapunov exponents take a finite number of values almost everywhere. This simple discovery revealed further universal properties such as local stable and unstable manifolds, dense occurrence of periodic orbits, and periodic approximation theorems. This makes the theory applicable to general dynamical systems lacking a solution or even an explicit formulation. The talk will present some key ideas in this line of work, and some open problems of great relevance in the present day.
Understanding the dynamics of Ricci flow is a difficult problem. On homogeneous spaces, however, the Ricci flow PDE reduces to a system of ODEs, making it more manageable to analyze long-term behavior. Furthermore, the fewer isotropy summands a space has (i.e. irreducible subrepresentations of the isotropy representation), the more constrained the space of invariant metrics is.
I will present the dynamics of the normalized Ricci flow on Spin(8)/Gâ‚‚, the only homogeneous space built from a simple Lie group that has exactly two equivalent isotropy summands. I will show that every homogeneous metric evolves under the flow to have positive Ricci curvature, and I will describe the asymptotics of the system. This is one of the few remaining examples for which the dynamics are easily visualizable (being a 2x2 system of ODEs), and I will discuss what is known about the dynamics for the others. This is joint work with Eric Cochran (Syracuse), Nazia Valiyakath (Syracuse), and Arseny Mingajev (Trinity University).
This Differential Geometry, PDE and Mathematical Physics seminar is available over zoom.
 | Wednesday
Nov. 19
|
| Algebra and Number Theory
No Seminar
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Abstract: This talk contributes a computational method to reconstruct both the birth and mortality coefficients in an age-structured population diffusive model. In mathematical oncology, solving this inverse problem is crucial for assessing the effectiveness of anti-cancer treatments and thus, gaining insights into the post-treatment dynamics of tumors. Through some linear and nonlinear transformations, the targeted inverse model is transformed into an auxiliary third-order nonlinear PDE. Subsequently, a coupled age-dependent quasi-linear parabolic PDE system is derived using the Fourier-Klibanov basis. The resulting PDE system is then approximated through the minimization of a cost functional, weighted by a suitable Carleman function. Ultimately, an analysis of the minimization problem is studied through a new Carleman estimate, and some computational results are presented to show how the proposed method works. This is a joint work with Vo Anh Khoa (Texas Tech University).
When: 4:00 pm (Lubbock's local time is GMT -6)
Where: room Math 011 (Math Basement)
ZOOM details:
- Choice #1: use this
Direct Link that embeds meeting ID and passcode
or
- Choice #2: Log into zoom, then join by manually inputting the meeting ID and passcode ...
* Meeting ID: 915 2866 2672
* Passcode: applied
 | Thursday
Nov. 20
6:30 PM
MA 108
|
| Mathematics Education
Math Circle
Hung Tran
Mathematics and Statistics, Texas Tech University
|
Math Circle Fall Flyer
abstract 2 PM CST (UTC-6)
Zoom link available from Dr. Brent Lindquist upon request.