Events
Department of Mathematics and Statistics
Texas Tech University
Recall that a commutative noetherian ring $R$ is Gorenstein if
and only if every finitely generated $R$-module has finite Gorenstein
projective dimension. In papers from 2011 (Murfet and Salarian), 2017
(Estrada, Fu, and Iacob), and 2018 (Christensen and Kato) Gorenstein
rings are characterized in terms of qualitative properties of their
complexes---but always under some additional assumption on \(R\) such as
finite Krull dimension or finite projective dimension of flat
\(R\)-modules. These assumptions are, we now know, superfluous, and I
will sketch a proof of the equivalence of the following conditions:
\(R\) is Gorenstein.
Every acyclic complex of injective \(R\)-modules is totally
acyclic.
Every acyclic complex of flat \(R\)-modules is totally acyclic.
Every acyclic complex of projective \(R\)-modules is totally
acyclic.
For every acyclic complex \(P\) of projective \(R\)-modules the complex
\(\mathrm{Hom}_R(P,R)\) is acyclic.
For every acyclic complex \(P\) of projective \(R\)-modules and every
injective \(R\)-module \(E\) the complex \(E \otimes_R P\) is acyclic.
This talk focuses on the construction, analysis, and derivation of a new class of exponential methods — specifically, two-derivative exponential Runge--Kutta (TDEXPRK) methods — designed for stiff PDEs. Interestingly, while TDEXPRK methods are constructed based on a fixed linearization of the vector field similar to exponential Runge--Kutta (EXPRK) methods, they require a significantly smaller number of order conditions than EXPRK methods. This enables the derivation of high-order and efficient schemes with only a few stages. Numerical examples for both one- and two-dimensional problems are provided to validate the accuracy and efficiency of TDEXPRK schemes compared to state-of-the-art exponential methods such as EXPRK and exponential Rosenbrock methods. This work is a collaboration with my PhD student, Hoang V. Nguyen.
Bring your own lunch and discuss (mostly) biology-related topics in math. Students, postdocs and faculty welcome from any discipline.
See the pdf flyer for this semester's schedule.
abstract Noon CST (UT-6)
Zoom link available from Dr. Brent Lindquist upon request.
It is well known that four-dimensional Riemannian manifolds carry many peculiar properties, which give rise to the existence of unique canonical metrics (e.g. half conformally flat metrics). In their study of self-dual solutions of Yang-Mills equations, Atiyah, Hitchin and Singer adapted the celebrated Penrose’s construction of twistor spaces to the Riemannian context, showing that a Riemannian four-manifold is half conformally flat if and only if its twistor space is a complex manifold: this paved the way for the study of many other characterizations of curvature properties for Riemannian four-manifolds. After giving an overview of the Riemannian and Hermitian structures of twistor spaces in the four-dimensional case, we present some new rigidity results for Riemannian four-manifolds whose twistor spaces satisfy specific vanishing curvature conditions. We also address the problem of classifying Einstein four-manifolds with positive sectional curvature, establishing a partial result obtained via twistor methods. This is based on joint work with Giovanni Catino and Paolo Mastrolia.
Watch online Tuesday at 4 PM CST (UT-6) via this Zoom link.
Partition Lie algebras are sophisticated algebraic objects introduced by Brantner–Mathew to control infinitesimal deformations in positive characteristics. This talk will present a Koszul duality between partition Lie algebras and specific complete filtered derived rings. This duality helps to understand the homotopy operations on partition Lie algebras. Additionally, a “many-object” version of this duality connects partition Lie algebroids with infinitesimal derived foliations in the sense of Toën–Vezzosi.Moving boundary (or often called “free boundary”) problems are ubiquitous in nature and technology. A computational perspective of moving boundary problems can provide insight into the “invisible” properties of complex dynamics systems, advance the design of novel technologies, and improve the understanding of biological and chemical phenomena. However, challenges lie in the numerical study of moving boundary problems. Examples include difficulties in solving PDEs in irregular domains, handling moving boundaries efficiently and accurately, as well as computing efficiency difficulties. In this talk, I will discuss three specific topics of moving boundary problems, with applications to ecology (population dynamics), plasma physics (ITER tokamak machine design), and cell biology (cell movement). In addition, some techniques of scientific computing will be discussed.
Short Bio: Dr. Shuang Liu currently is an assistant professor in the Department of Mathematics at the University of North Texas. During January 2021 and June 2023, she was a SEW Assistant Professor in the Department of Mathematics at University of California, San Diego. Before that, she was a Postdoc Research associate in applied mathematics and plasma physics group at Los Alamos National Laboratory. Dr. Liu received her PhD in 2019 from the Department of Mathematics from the University of South Carolina.
When: 4:00 pm (Lubbock's local time is GMT -6)
Where: room MATH 011 (basement)
ZOOM details:
- Choice #1: use this direct link that embeds meeting and ID and passcode.
- Choice #2: join meeting with this link, then you will have to input the ID and Passcode by hand:
* Meeting ID: 979 1333 6658
* Passcode: Applied (Note the capital letter "A")
 | Thursday Nov. 7 6:30 PM MA 108
| | Mathematics Education Math Circle Álvaro Pámpano Department of Mathematics and Statistics, Texas Tech University
|
Math Circle Fall Poster
We study an ecosystem of three keystone species: salmon, bears, and vegetation. Bears consume salmon and vegetation for energy and nutrient intake but the food quality differs significantly due to the nutritional level difference between salmon and vegetation. We propose a stoichiometric predator-prey model that not only tracks the energy flow from one trophic level to another but also nutrient recycling in the system. Analytical results show that bears may coexist with salmon and vegetation at a steady state but the abundance of salmon may differ under different regimes. Numerical simulations reveal that a smaller vegetation growth rate may drive the vegetation population to extinction whereas a large vegetation growth rate may drive the salmon population to extinction. Moreover, a large vegetation growth rate may stabilize the system where the bear, salmon, and vegetation populations oscillate periodically.
Zoom link
Meeting ID: 564 796 5924
Passcode: Wenjing