Events
Department of Mathematics and Statistics
Texas Tech University
 | Tuesday
Sep. 15
3:30 PM
MATH 017
|
| Real-Algebraic Geometry
Etale and Stalk
David Weinberg
Department of Mathematics and Statistics, Texas Tech University
|
This talk is the first in a series which reviews the seminal work of Soren Galatius, Ib Madsen, Ulrike Tillmann, and Michael Weiss. The main result is a refinement of the Pontryagin-Thom equivalence to a space level equivalence, going between the classifying space of the cobordism category and a certain spectrum. The authors use this equivalence to prove a conjecture by David Mumford about the cohomology of the mapping class group of Riemann surfaces. In this first talk, I will begin motivation and review the classical Pontryagin-Thom construction. I will then describe a categorification of the collapse map, which is claimed to induce an equivalence at the level of classifying spaces. Various bordism categories will be introduced, all of which will be equivalent. If there is time, I will introduce the Madsen-Tillmann spectrum.The discrete fracture model (DFM) has been widely used in the simulation of fluid flow in fractured porous media. Traditional DFM use the so-called hybrid-dimensional approach to treat fractures explicitly as low-dimensional entries (e.g. line entries in 2D media and face entries in 3D media) on the interfaces of matrix cells to avoid local grid refinements in fractured region and then couple the matrix and fracture flow systems together based on the principle of superposition with the fracture thickness used as the dimensional homogeneity factor. Because of this methodology, DFM is considered to be limited on conforming meshes and thus may raise difficulties in generating high qualified unstructured meshes due to the complexity of fracture’s geometrical morphology. In this talk, we clarify that the discrete fracture model actually can be extended to non-conforming meshes without any essential changes. To show it clearly, we provide another perspective for DFM based on hybrid-dimensional representation of permeability tensor modified from the comb model to describe fractures as one-dimensional line Dirac delta functions contained in permeability tensors. A finite element DFM scheme for single-phase flow on non-conforming meshes is then derived by applying Galerkin finite element method to it. Analytical analysis and numerical experiments show that our DFM scheme automatically degenerates to the classical finite element DFM when the mesh is conforming with fractures. Moreover, the accuracy and efficiency of the model on non-conforming meshes is demonstrated by testing several benchmark problems. This model is also applicable to curved fracture with variable thickness.
In this talk I will discuss the basics of higher topos theory with an emphasis on the theory's applications to geometry. Particular emphasis will be placed on diffeological spaces and sheaf toposes.Continuing the Financial Math seminar series concentrating on alternative financial indices, the agenda for the second seminar is:
1. Thilini Mahanama: introduction of a new index on crime
2. Jiho Park: near riskless rate portfolios
3. Yuan Hu: crypto-currency portfolios and option valuation
4. Zari Rachev: Vol-of-vol indices
Watch online at 2 PM this Friday the 18th via this Zoom link.