Events
Department of Mathematics and Statistics
Texas Tech University
When are two geometric shapes identical or similar? This question has intrigued mathematicians from ancient times with its roots going back to Euclid's Elements. In particular, in Elements Euclid provided several answers to the question: When are two triangles congruent and when they are similar? The answer to this innocent-looking question depends on what tools for measurement are available and which characteristics of a triangle we measure. In a classical Euclidean setting, we measure lengths of sides and angles of triangles. But often these measurements are not available. For example, if we hear a triangular drum playing on a neighboring street can we recognize the shape from its tones? Or, staying away from a triangular oven, we feel temperature when one of its sides is heated. If we know the temperature for each of the sides, can we recognize the shape of a triangle? These and some other questions, as well as some open problems, will be discussed in this talk. This topic belongs to the intersection of areas of Elementary Geometry, Complex Analysis, Mathematical Physics, and PDE's. Furthermore, Transcendental Special Functions appear in this study rather naturally. So, all people who are interested in these matters are invited to attend. Because triangles as a generic geometric configuration are simplest possible, most problems are understandable for graduate and undergraduate students (assuming they are familiar with elements of Euclidean geometry) and can be good topics for student research projects.
In recent decades, a series of eigenvalue estimates have been established for minimal-type hypersurfaces under geometric constrains. A foundational result was given by Choi and Wang (1983), who proved a lower bound for the first eigenvalue of closed minimal surfaces in complete Riemannian manifolds with positive Ricci curvature. Their wok was later extended by Cheng-Mejia-Zhou and Ding-Xin to closed f-minimal surfaces and closed self-shrinkers, respectively.
Beyond the closed case, Brendle and Tsiamis considered complete non-compact self-shrinkers, proving that the first eigenvalue admits a universal lower bound 1/4. More recently, Conrado and Zhou generalized this direction to f-minimal hypersurfaces in gradient shrinking Ricci solitons, establishing not only lower bounds but also the discreteness of the spectrum of the drifted Laplacian.
In this talk, I will present a further generalization to λ-hypersurfaces in R^{n+1}. Under the condition |λ|≤1/2-1/(2n), we show that the first eigenvalue is bounded below by 1/4- λ^2/2. This estimate recovers the result of Brendle-Tsiamis in the special case λ=0. Since self-shrinkers and λ-hypersurfaces can be regarded as minimal and constant mean curvature(CMC) hypersurfaces in a smooth metric measure space with weight function |x|^2/4, our result provides a natural extension of eigenvalue estimates from the minimal to the CMC setting.
This Differential Geometry, PDE and Mathematical Physics seminar is available over zoom.
Let \(R\) be a commutative noetherian ring. The G-level of an
\(R\)-complex with bounded and degreewise finitely generated homology
counts the number of mapping cone constructions it takes to build the
complex from the collection of finitely generated Gorenstein
projective \(R\)-modules. We prove that if \(d+1\) is an upper bound for
the $G$-level of perfect \(R\)-complexes, then \(R\) is Gorenstein of
Krull dimension at most $d$. Further, for a Gorenstein ring of Krull
dimension \(d\), we show that the G-level of an \(R\)-complex with bounded
and degreewise finitely generated homology is at most
\(\max\{2,d + 1\}\). This improves the bound of \(2(d + 1)\)
obtained by Awadalla and Marley a few years ago and aligns with the
bound on \(R\)-levels in case \(R\) is regular. The talk is based on joint
work with Kekkou, Lyle, and Soto Levins.
 | Wednesday
Oct. 22
4 PM
Math 011
|
| Applied Mathematics and Machine Learning
TBA
Andreas Mang
Department of Mathematics, University of Houston
|
Abstract. TBA.
When: 4:00 pm (Lubbock's local time is GMT -5)
Where: room Math 011 (Math Basement)
ZOOM details:
- Choice #1: use this link
Direct Link that embeds meeting and ID and passcode.
- Choice #2: join meeting using this link
Join Meeting, then you will have to input the ID and Passcode by hand:
* Meeting ID: 915 2866 2672
* Passcode: applied
abstract noon CDT (UT-5)
Zoom link available from Dr. Brent Lindquist upon request.