Events
Department of Mathematics and Statistics
Texas Tech University
Social hierarchies are ubiquitous in social groups such as human societies and social insect colonies, however, the factors that maintain these hierarchies are less clear. Motivated by the shared reproductive hierarchy of the ant species Harpegnathos Saltator, we have developed simple compartmental nonlinear differential equations to explore how key life-history and metabolic rate parameters may impact and determine its colony size and the length of its shared hierarchy. Our modeling approach incorporates nonlinear social interactions and metabolic theory. The results from the proposed model, which were linked with limited data, show that: (1) the proportion of reproductive individuals decreases over colony growth; (2) an increase in mortality rates can diminish colony size but may also increase the proportion of reproductive individuals; and (3) the metabolic rates have a major impact in the colony size and structure of a shared hierarchy.
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We will discuss some foundational aspects of three-dimensional incompressible fluid turbulence, including guiding experimental observations, Kolmogorov's 1941 theory on the structure of a turbulent flow, Onsager's 1949 conjecture on anomalous dissipation and weak Euler solutions, and Landau's Kazan remark concerning intermittency. Mathematical examples and constructions that exhibit features of turbulent behavior will be discussed.
This week's Analysis seminar may be attended at 4:00 PM CST (UT-6) via this Zoom link.
Meeting ID: 976 4978 7908
Passcode: 973073
Abstract: We prove that the category of diffeological spaces does not admit a model structure transferred via the smooth singular complex functor from simplicial sets, resolving in the negative a conjecture of Christensen and Wu. Embedding diffeological spaces into sheaves of sets (not necessarily concrete) on the site of smooth manifolds, we then prove the existence of a proper combinatorial model structure on such sheaves transferred via the smooth singular complex functor from simplicial sets. We show the resulting model category to be Quillen equivalent to the model category of simplicial sets. We then show that this model structure is cartesian, all smooth manifolds are cofibrant, and establish the existence of model structures on categories of algebras over operads. We use these results to establish analogous model structures on simplicial presheaves on smooth manifolds, as well as presheaves valued in left proper combinatorial model categories, and prove a generalization of the smooth Oka principle established in arXiv:1912.10544. We finish by establishing classification theorems for differential-geometric objects like closed differential forms, principal bundles with connection, and higher bundle gerbes with connection on arbitrary cofibrant diffeological spaces. arXiv:2210.12845.
 | Wednesday
Feb. 1
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| Algebra and Number Theory
No Seminar
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Surface quasi-geostrophic equations is a special example of an active scalar (other examples include Burgers' and porous media equations) with many applications in geophysics and fluid mechanics. It has caught much attention from mathematicians due to its similarity to the Navier-Stokes/Euler equations in terms of the structures of the equations, as well as the behavior of its solutions according to numerical simulations. Despite the difficulty created by its non-linear term (the Fourier symbol of the velocity is odd in frequency due to Riesz transform), convex integration technique has been successfully applied to the surface quasi-geostrophic equations in the past several years, and we now know that there exist infinitely many weak solutions (e.g., one can construct solutions with prescribed energy). We review these results and remaining open problems in both deterministic and stochastic cases..
When: 4:00 pm (Lubbock's local time is GMT -6)
Where: room MATH 011 (basement)
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* Meeting ID: 940 7062 3025
* Passcode: applied