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.
The process of identifying and quantifying metabolites in complex mixtures plays a critical role in metabolomics studies to obtain an informative interpretation of underlying biological processes. Manual approaches are time-consuming and heavily reliant on the knowledge and assessment of nuclear magnetic resonance (NMR) experts. We propose a shifting-corrected regularized regression method, which identifies and quantifies metabolites in a mixture automatically. A detailed algorithm is also proposed to implement the proposed method. Using a novel weight function, the proposed method is able to detect and correct peak shifting errors caused by fluctuations in experimental procedures. Simulation studies show that the proposed method performs better with regard to the identification and quantification of metabolites in a complex mixture. We also demonstrate real data applications of our method using experimental and biological NMR mixtures.
The Statistics seminar may be attended online at 4:00 PM CDT (UTC-5) via this Zoom link.
Meeting ID: 904 545 1744
There are two oftentimes unspoken truths in measure theory. 1) Practically all useful measures in practice are given by Radon measures. 2) One does not really care so much about the sigma-algebra of measurable sets, but rather about its quotient by the ideal of null sets.
The quotient of measurable sets by null sets is, in the case of a given Radon measure, an example of what is called a measurable locale, and can be treated like a (usually point-free) space. We argue that this measurable locale can be constructed directly from a Grothendieck topology on the poset of compact sets. This opens the door to a purely sheaf-theoretic perspective on measure theory. As an application, we show that the locale of sublocales of a given Hausdorff space X equipped with a Radon measure can be equipped with a natural extension of the measure, invariant under measure preserving homeomorphisms.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.
Abstract. We develop and analyze numerical methods for solving optimal control problems governed by flows of diffeomorphisms. Our focus is on designing efficient algorithms and fast computational kernels to address large-scale inverse transport problems.
The goal is to compute a diffeomorphic transport map that transforms image intensities or densities from a reference dataset to match those of a target. The resulting optimization problem is highly non-linear and non-convex, leading to large-scale, ill-conditioned optimality systems. To address these challenges, we employ adjoint-based first- and second-order optimization methods enhanced by advanced preconditioning and acceleration strategies to ensure rapid convergence.
We demonstrate the performance of our approach using both synthetic and clinical datasets, evaluating convergence behavior, accuracy, scalability, and time to solution. Our GPU-accelerated software package CLAIRE is specifically optimized for clinically relevant image registration problems and exhibits excellent strong and weak scaling on modern architectures, scaling efficiently to hundreds of GPUs.
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.