Events
Department of Mathematics and Statistics
Texas Tech University
Variational methods have formed the foundation of classical mechanics for several hundred years. In this lecture, I will show how the applications of variational principles, coupled with some ideas from geometry, can solve a wide variety of seemingly disconnected problems using the same mathematical approach. After a general and gentle introduction, I will illustrate this method on the examples of modeling figure skating (a system with nonholonomic constraint), and fluid-structure interactions applied to porous media containing incompressible fluid (two media coupled through the incompressibility constraint). I will also discuss the limitations of these methods, i.e., what progress can be achieved by algorithmic thinking alone and at what point ingenuity and creativity must take over. Finally, I will also discuss the importance of teaching such universal, fundamental principles and methods for the successful employment of students and postdocs in the industry.
This is part of the W. Dayawansa Colloquium Series: Foundational and Applied Mathematics: Bridges to Industry, and is held in conjunction with the Applied Math seminar group. This event is hybrid, and available virtually via this Zoom link.
Higher Teichmuller theory studies representations of surface groups into Lie groups of higher rank, in contrast with the classical Teichmuller theory that concerns PSL(2,R). In this talk we will describe a scheme to find the analog of Thurston compactification for generalizations of Teichmuller space in the case where the Lie group is real split of rank 2 (SL(3,R), Sp(4,R), G2). For concreteness, we will mostly focus on SL(3,R), where the theory involves the study of convex real projective structures on surfaces, which in some sense extend the notion of hyperbolic structures.
This Job Colloquium is held in tandom with the PDGMP seminar group. Please watch online via this Zoom link.A "geometric Cauchy problem" means a geometric formulation of the Cauchy problem for some geometric PDE. For instance the Björling problem from the 19th Century is to construct the (unique) minimal surface that contains a given space curve with the surface normal prescribed along the curve. In that case the Schwarz formula gives the solution from the Cauchy data via analytically extension and taking the real part. This is based on the Weierstrass representation of minimal surfaces in terms of holomorphic data.
More generally, harmonic maps into symmetric spaces also have solution methods based on holomorphic maps into infinite dimensional spaces. It turns out that an infinite dimensional analogue of Schwarz's formula can be applied to solve Cauchy problems for surfaces with harmonic Gauss maps. The geometric Cauchy problem for Willmore surfaces can be stated in different ways. One way is to prescribe both the surface and its dual surface along a curve, as well as the conformal Gauss map. From this data, we obtain a unique solution. This can be used, for example, to classify equivariant Willmore sufaces.
Please virtually attend Dr. Brander's online Colloquium at this Zoom link. Passcode: Brander
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See the colloquium at this link.This week's Applied Math seminar details available at this pdf and will be available via this Zoom link
Foxby defined the (Krull) dimension of a complex of modules over a commutative Noetherian ring in terms of the dimension of its homology modules. In joint work with Iyengar we prove that the dimension of a bounded complex of free modules of finite rank can be computed directly from the matrices representing the differentials of the complex.