Events
Department of Mathematics and Statistics
Texas Tech University
Consider any Leray-Hopf weak solution of the three-dimensional Navier-Stokes equations for incompressible, viscous fluid flows. We prove that any Lagrangian trajectory associated with such a velocity field has an asymptotic expansion, as time tends to infinity, which describes its long-time behavior very precisely.  | Tuesday
Sep. 22
3:30 PM
MATH 017
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| Real-Algebraic Geometry
Sheafification
David Weinberg
Department of Mathematics and Statistics, Texas Tech University
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In the second installment of this series, I will introduce the topological bordism category and discuss its classifying space. Then I will review some basics of sheaf theory and discuss the geometric realization (or shape) of sheaves on manifolds. A sheaf variant of the bordism category will be introduced and it will be shown (next time) that its geometric realization is precisely the classifying space of the bordism category.In this talk, I will introduce the Madsen–Tillmann spectrum and discuss its connection with the Thom spectrum. I will prove the first of the two main theorems that imply Mumford's conjecture.The Hochschild cohomology is a tool for studying associative algebras that has a lot of structure: it is a Gerstenhaber algebra. This structure is useful because of its applications in deformation and representation theory, and recently in quantum symmetries. Unfortunately, computing it remains a notoriously difficult task. In this talk we will present techniques that give explicit formulas of the Gerstenhaber algebra structure for general twisted tensor product algebras. This will include an unpretentious introduction to this cohomology and to our objects of interest, as well as the unexpected generality of the techniques. This is joint work with Tekin Karadag, Dustin McPhate, Tolulope Oke, and Sarah Witherspoon.
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In this talk I will begin by finishing the discussion of a theorem relating infinity toposes and infinity stacks which started last week. After this, I will give basic results on over-infinity-toposes and bundles over fixed elements. I will end with a discussion on truncated objects and a correspondence theorem between groupoids internal to an infinity topos and infinity stacks. Advances in VLSI (Very Large Scale Integration) design and fabrication have resulted in the availability of low-cost, low-power, small-sized devices that have significant computational power and are able to communicate wirelessly. In addition, advances in MEMS (Micro Electric Mechanical Systems) technology have resulted in wide availability of solid-state sensors and actuators. The net result is ubiquitous sensing, communication, and computation that can be incorporated into small low-power devices.
In this talk, I will discuss how the above-mentioned technological advances present important opportunities and interesting challenges for control system designers. To this effect, I will discuss how the introduction of digital communication in control loops gives rise to a need for new tools for the design and analysis of feedback control systems. I will also describe recent work demonstrating that optimization-based approaches to path planning – which have been enabled by fast computation – can lead to solutions that significantly outperform previously proposed heuristics.
See pdfLow-cost, low-power embedded computation enables the use of online optimization to solve nonlinear control problems with hard state and input constraints, leading to the popularity of Model Predictive Control (MPC) in numerous industrial applications. More recently, online optimization also became popular to solve estimation problems that can take advantage of known constraints on the state, measurement noise, and disturbances. In particular, Moving Horizon Estimation (MHE) computes states estimates that are maximally compatible with measurements observed over a finite window of time.
In this talk, we discuss an optimization-based approach to solve output feedback control problems that combines state estimation and control into a single min-max optimization. We discuss the challenges involved in guaranteeing the convergence of the closed loop systems, as well as the computational techniques that are needed to solve the resulting optimizations in real-time control systems with sampling times on the order of just a few milliseconds.
See pdfComputer-based sensors are heavily used in monitoring and controlling complex large-scale physical system, such as the power grid, transportation systems, chemical processes, and manufacturing plants. While these sensors can yield great benefits in terms of improved efficiency, lower costs, and increased safety, they are often prone to attacks and can introduce significant security risks.
In this talk we explore how the formulation of classical estimation problems needs to be revisited to address scenarios where sensors are prone to attacks. By considering the joint design of estimators and attack policies, we obtain “resilient” estimators that use redundancy in an optimal fashion. While the design and construction of these optimal resilient estimators may be computationally expensive, we shall see that is often possible to find quasi-optimal solutions that are computationally attractive. For concreteness, we illustrate these ideas in a case study involving the estimation of power system oscillations using Phase Measurements Units.
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