Events
Department of Mathematics and Statistics
Texas Tech University
 | Tuesday
Sep. 8
3:30 PM
MATH 017
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| Real-Algebraic Geometry
Presheaf
David Weinberg
Department of Mathematics and Statistics, Texas Tech University
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In this talk, we provide a general overview of a few concepts in computability theory which have motivated our study of restriction categories and related structures: partial combinatory algebras, their so-called functional completeness, and their ability to generate categories known as realizability toposes, with structures known as triposes as an intriguing intermediate step in one account of how realizability toposes may be constructed. We then introduce Turing categories as a certain categorification of partial combinatory algebras based on restriction categories, and illustrate how some of previous concepts translate into this new setting.The electrocardiogram (ECG) has long been used to monitor cardiac health. With each feature of the ECG corresponding to a part of the cardiac cycle, the automated identification of such features can play a role in developing decision-support tools that provide further insight regarding a patient’s clinical status. Although algorithms for carrying out such feature identification are well-studied in adult populations, suitable algorithms are lacking for the pediatric congenital heart disease population. This work presents a framework for identifying representative subsets of ECG beats from large data sets and subsequent steps for automated feature identification. ECG subset selection is performed using methods related to the discrete empirical interpolation method (DEIM), and the approach to feature identification relies on techniques more commonly seen in the informatics and data science communities. With methods that generalize beyond the pediatric congenital heart disease population, both subset selection and feature identification aspects of this work can be used toward real-time analyses for the clinical setting. Watch online on Tuesday the 8th at 3:30 PM via this Zoom link.
 | Wednesday
Sep. 9
3:00 PM
Online
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| Algebra and Number Theory
No Seminar
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Explicit time integrators for parabolic PDE are subject to a restrictive time-step limit, so A-stable integrators are essential. It is well known that although there are no A-stable explicit linear multistep methods and implicit multistep methods cannot be A-stable beyond order two, there exist A-stable and L-stable implicit Runge-Kutta (IRK) methods at all orders. IRK methods offer an appealing combination of stability and high order; however, these methods are not widely used for PDE because they lead to large, strongly coupled linear systems. An s-stage IRK system has s-times as many degrees of freedom as the systems resulting from backward Euler or implicit trapezoidal rule discretization applied to the same equation set. In this talk, I will introduce a new block preconditioner for IRK methods, based on a block LDU factorization with algebraic multigrid subsolves for scalability. I will demonstrate the effectiveness of this preconditioner on the heat equation as a simple test problem, and compare in condition number and eigenvalue distribution, and in numerical experiments with other preconditioners currently in the literature. Experiments are run with IRK stages up to s = 7, and it is found that the new preconditioner outperforms the others, with the improvement becoming more pronounced as spatial discretization is refined and as temporal order is increased.In this talk, I will survey three convenient categories for studying the homotopy theory of spaces equipped with the action of a group. I will present a theorem of Elmendorf, which shows that all three variants are equivalent.