Probability, Differential Geometry and Mathematical Physics
Department of Mathematics and Statistics
Texas Tech University
See the colloquium at this link.See the colloquium at this link.See the colloquium at this link.We establish a uniqueness result for the $[\varphi,\vec{e}_{3}]$-catenary cylinders by their asymptotic behaviour.
Well known examples of such cylinders are the grim reaper translating solitons for the mean curvature flow. For such solitons,
F. Martín, J. Pérez-García, A. Savas-Halilaj and K. Smoczyk proved that, if $\Sigma$ is a properly embedded translating soliton with
locally bounded genus, and $\mathcal{C}^{\infty}$-asymptotic to two
vertical planes outside a cylinder, then $\Sigma$ must coincide with
some grim reaper translating soliton. In this talk, applying a strong
maximum principle for elliptic operators together with a compactness
result, we increase the family of $[\varphi,\vec{e}_{3}]$-minimal
graphs
where these types of results hold under different assumption of
asymptotic behaviour.
Watch online via this Zoom link.In this talk, we will discuss recent results on the issue of well-posedness for the 2D generalized surface quasi geostrophic (gSQG) equation. This family of equations was introduced to the mathematical community by Chae-Constantin-Cordoba-Gancedo-Wu in 2012. Both the 2D incompressible Euler and 2D SQG equations can be realized as belonging to this family. In light of recent ill-posedness results, we positively discuss the issue of well-posedness for the gSQG family, their regularized variations, as well as the effect of instantaneous smoothing, when it is present.
Watch online via this Zoom link.
This is joint work with T. Miura and F. Rupp.
In this talk we study the qualitative behavior of elastic flows of curves -- Can embedded curves develop self-intersections along the flow?
In general they can, as revealed by [S. Blatt (2010)], even for a larger class of higher order geometric flows. In a recent work we have observed that this loss of embeddedness may only occur only above a certain energy treshold, which can be quantified optimally.
My goal for this talk is to explain this optimal treshold geometrically. This discussion will lead us into the fabulous world of Euler's elastic curves.
Watch online via this Zoom link.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.First, we state a conformal Gauss-Bonnet theorem for four-manifolds with corners. Then we review the definition of the renormalized volume of asymptotically hyperbolic Einstein metrics, which came from the AdS/CFT correspondence in physics; and review the Gauss-Bonnet theorem for such manifolds in the ordinary case. We then state a new Gauss-Bonnet formula for half of an asymptotically hyperbolic Einstein space that has been partitioned into two by a minimal surface and use this to derive a variation formula for its renormalized volume under variations of the minimal surface. The latter two results are joint work with Matthew J. Gursky and Aaron J. Tyrrell.
Attend in person in the Math building or watch online Wednesday the 10th at 3 PM via this Zoom link.
Motivated by the Canham-Helfrich model for lipid bilayers, the minimization of the Willmore energy subject to the constraint of fixed isoperimetric ratio has been extensively studied throughout the last decade. In this talk, we consider a dynamical approach by introducing a non-local $L^2$-gradient flow for the Willmore energy which preserves the isoperimetric ratio. For topological spheres with initial energy below an explicit threshold, we show global existence and convergence to a Helfrich immersion as $t\to\infty$. Our proof relies on a blow-up procedure and a constrained version of the Lojasiewicz-Simon gradient inequality.
Watch online via this Zoom link.