Events
Department of Mathematics and Statistics
Texas Tech University
Infection with severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) results in varied clinical outcomes including asymptomatic, mild, severe, or fatal disease. While the mechanisms responsible for the pathogenesis of COVID-19 are not fully understood, studies suggest that virus-induced chronic inflammation and tissue injury are associated with severe outcomes. To determine the role of tissue damage on immune populations recruitment and function, we develop a mathematical model of innate immunity following SARS-CoV-2 infection. The model was fit to published longitudinal immune marker data from patients with mild and severe COVID-19 disease and key parameters were estimated for each clinical outcome. Analytical, bifurcation, and numerical investigations were conducted to determine the effect of parameters and initial conditions on long-term dynamics. The results were used to suggest changes needed to achieve immune resolution. Such results can guide interventions.
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The Jones polynomial is an invariant of knots in R^3. Following a proposal of Witten, it was extended to knots in 3-manifolds by Reshetikhin–Turaev using quantum groups. Khovanov homology is a categorification of the Jones polynomial of a knot in R^3, analogously to how ordinary homology is a categorification of the Euler characteristic of a space. It is a major open problem to extend Khovanov homology to knots in 3-manifolds. In this talk, I will explain forthcoming work towards solving this problem, joint with Leon Liu, David Reutter, Catharina Stroppel, and Paul Wedrich. Roughly speaking, our contribution amounts to the first instance of a braiding on 2-representations of a categorified quantum group. More precisely, we construct a braided (∞,2)-category that simultaneously incorporates all of Rouquier's braid group actions on Hecke categories in type A, articulating a novel compatibility among them.
Bring your own lunch and discuss (mostly) biology-related topics in math. Students, postdocs, and faculty welcome from any discipline.
This week's PDGMP seminar may be attended at 3:00 PM CDT (UT-5) via this Zoom link.
Meeting ID: 933 4646 9342
Let \(R\) be a commutative Noetherian local ring of prime characteristic
\(p\). For an nonnegative integer \(e\), let \(R^e\) be the ring \(R\) viewed as a
module over itself via the \(e\)th iteration of the Frobenius
endomorphism; i.e., for \(r\in R\) and \(s\in R^e\), \(r\cdot s := r^{p^e}s\). In
the early 1970s, Peskine and Szpiro proved that for a finitely
generated module \(M\) of finite projective dimension over \(R\),
\(\mathrm{Tor}_i^R(R^e,M)=0\) for all \(i,e>0\). Shortly afterward, Herzog proved the
converse. Subsequently, several authors established stronger converse
statements under certain hypotheses, the strongest being in the case \(R\)
is a complete intersection. In this situation, Avramov and Miller
(2001) proved that for a finitely generated module \(M\), if
\(\mathrm{Tor}_i^R(R^e,M)=0\) for some positive \(i\) and \(e\), then \(M\) has finite
projective dimension. Whether or not that statement holds over an
arbitrary local ring is (to the best of my knowledge) an open
question. In this talk, I will discuss the history of this problem as
well as some recent progress in the case \(R\) is Cohen-Macaulay and also
when the module \(M\) is not assumed to be finitely generated.
The construction of robust and accurate numerical methods is essential for simulating complex fluid dynamics. Most partial differential equations that arise in such areas often exhibit certain physical and thermodynamic properties that should be preserved at the discrete level. We call such numerical schemes structure-preserving. Structure-preserving approximation techniques provide theoretical guarantees of reliability for situations where ad-hoc stabilization techniques can fail. In this talk, we give an introduction of relevant PDEs in the field of computational fluid dynamics such as the Shallow Water Equations. We then describe a dispersive extension of the SWEs known as the Serre-Green-Naghdi Equations which are applicable in coastal hydrodynamics. We discuss the respective structure-preserving approximation technique for the two models. We then conclude with a short overview on the verification and validation process in scientific computing.
When: 4:00 pm (Lubbock's local time is GMT -6)
Where: room MATH 011 (basement)
ZOOM details:
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* Meeting ID: 940 7062 3025
* Passcode: applied