Events
Department of Mathematics and Statistics
Texas Tech University
A history of genetic pest management is provided, culminating with the explosive development of gene drives in the past few decades. Gene drives are any natural or synthetic mechanism of propagating a gene into a target population, even if the gene imposes a fitness cost. This technology offers a promising solution to the burden posed by crop pests and vectors of important human diseases. However, gene drive dynamics in the wild are presently unknown, so scientists must leverage mathematical and computational models to understand how gene drives behave in a natural population. One of the most critical factors affecting gene drive performance is insect dispersal. The role of dispersal is explored in the case of the yellow fever mosquito, Aedes aegypti, using a spatially explicit patch model. Numerical results illustrate the complex relationship between gene drive function and dispersal behavior.
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I will introduce a class of diffeological spaces, called elastic, on which the left Kan extension of the tangent functor of smooth manifolds defines an abstract tangent functor in the sense of Rosický. On elastic spaces there is a natural Cartan calculus, consisting of vector fields and differential forms, together with the Lie bracket, de Rham differential, inner derivative, and Lie derivative, satisfying the usual graded commutation relations. Elastic spaces are closed under arbitrary coproducts, finite products, and retracts. Examples include manifolds with corners and cusps, diffeological groups and diffeological vector spaces with a mild extra condition, mapping spaces between smooth manifolds, and spaces of sections of smooth fiber bundles. arXiv:2301.02583.
 | Wednesday
Feb. 8
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| Algebra and Number Theory
No Seminar
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Bring your own lunch and discuss (mostly) biology-related topics in math. Students, postdocs, and faculty welcome from any discipline.
This week's PDGMP seminar may be attended at 3:00 PM CDT (UT-5) via this Zoom link.
Meeting ID: 933 4646 9342
The Navier-Stokes equations at high Reynolds are well known to have very chaotic dynamics. One way to measure this chaos is via the top Lyapunov exponent, a number which measures the infintesimal exponential rate of separation of nearby trajectories. A positive Lyapunov exponent is seen as a hallmark of chaos. Despite this conceptually simple characterization, proving positivity of the top Lyapunov exponent is notoriously challenging, especially in infinite or high dimensional dynamical systems. In this talk, I will present some recent progress in this direction for arbitrary Galerkin truncations of the stochastic 2d Navier-Stokes equations on the Torus. Using tools from the theory of random dynamical systems and Hormander's hypoelliptic theory, I will present a new estimate that gives a quantitative lower bound on the top Lyapunov exponent in terms of fractional regularity of a certain invariant probability measure that tracks the statistics of unstable tangent directions. Using computer assisted techniques from algebraic geometry to veryify a certain Lie algebra condition, I will explain how this estimate gives rise to the first rigorous proof of a positive Lyapunov exponent for the Galerkin-Navier-Stokes system for arbitrary large frequency cutoff. The above technique is quite general and can be applied to many other high-dimensional dynamical systems with stochastic forcing. This work is joint with Jacob Bedrossian and Alex Blumenthal.
When: 4:00 pm (Lubbock's local time is GMT -6)
Where: room MATH 011 (basement)
ZOOM details:
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* Meeting ID: 940 7062 3025
* Passcode: applied