Events
Department of Mathematics and Statistics
Texas Tech University
The Boussinesq equations concerned here model buoyancy-driven fluids such
as various atmospheric and oceanographic flows, and Rayleigh-Benard convection.
This talk presents recent stability results on the Boussinesq equations with partial
dissipation near two physically significant steady-states: Couette flow and the
hydrostatic balance. In the case of perturbation near a Couette flow, it is the enhanced
dissipation created by a linear non-self-adjoint operator that makes the nonlinear stability
possible. This is a joint work with Wen Deng and Ping Zhang.
To join Dr. Wu's talk please click
here.
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Beginning with the advent of classical Maxwell theory (1865) and continuing with its generalization into Yang-Mills theory (1954), gauge theories have become a powerful set of tools, both in pure and applied mathematics (and mathematical physics, with their quantization forming the basis of the Standard Model of particle physics). Beginning in 1986, following their success with the other three fundamental forces, Ashtekar began applying these same techniques to general relativity, the classical theory of the gravitational field, with the introduction of certain 'new variables'. In collaboration with Rovelli and Smolin, et al., these techniques eventually became known as loop quantization. As general relativity is a theory of differential geometry, under loop quantization, it becomes a theory of (differential) quantum geometry. Similar techniques may also be applied to classical symmetry-reduced cosmologies (e.g., Friedmann-Lemaitre-Robertson-Walker and Bianchi Type I), resulting in quantum cosmologies. It is not clear, a priori, however, whether loop quantization and symmetry reduction commute. I will present recent progress in this area, including the first concrete realization of 'quantum isotropy'; this work has been published in three recent papers, in collaboration with colleagues at Florida Atlantic University.
Watch online via this Zoom link.The Koszul Algebra structure for codepth 3 commutative local
rings has been studied significantly in recent years. Avramov
compiled a comprehensive list of properties for these structures using
their respective Poincaré and Bass series. He showed that codepth 3
local rings could be described with 5 distinct classifications;
$\textrm{C}(c)$, $\textrm{T}$,
$\textrm{B}$, $\textrm{G}(r)$, and
$\textrm{H}(p,q)$. This led to many other studies
of these structures, with $\textrm{H}(p,q)$ being
the most diverse of the given classes. Christensen, Veliche, and
Weyman determined strong restrictions for the values $p$ and $q$ could
take on with respect to the rank of the first and last free modules in
the minimal free resolution of the given ring. For this talk we
further restrict the bounds for $p$ and $q$ when the ring $R/I$ has
Koszul algebra structure $\textrm{H}(p,q)$ and $I$
is an artinian monomial ideal in $R=\Bbbk[x,y,z]$ where $\Bbbk$ is a
field. These bounds will depend only on the number of minimal
generators for the given monomial ideal and fixed values of either $p$
or $q$.
Join the seminar via this Zoom link