Events for 11/26/2024 from all calendars

Nonlinear Partial Differential Equations

Time: 3:00PM - 4:00PM

Location: BLOC302

Speaker: Alex Vasseur, University of Texas at Austin

Title: From Navier-Stokes to discontinuous solutions of compressible Euler

Abstract: The compressible Euler equation can lead to the emergence of shock discontinuities in finite time, notably observed behind supersonic planes. A very natural way to justify these singularities involves studying solutions as inviscid limits of Navier-Stokes solutions with evanescent viscosities.The mathematical study of this problem is however very difficult because of the destabilization effect of the viscosities. Bianchini and Bressan proved the inviscid limit to small BV solutions using the so-called artificial viscosities in 2004. However, until very recently, achieving this limit with physical viscosities remained an open question. In this presentation, we will provide the basic ideas of classical mathematical theories to compressible fluid mechanics and introduce the recent method of a-contraction with shifts. This method is employed to describe the physical inviscid limit in the context of the barotropic Euler equation, and to solve the Bianchini and Bressan conjecture in this special case. This is a joint work with Geng Chen and Moon-Jin Kang.

Topology Seminar

Time: 4:00PM - 5:00PM

Location: Online

Speaker: Louisa Liles, University of Virginia

Title: Infinite Families of Quantum Modular 3-Manifold Invariants

Abstract: This talk will begin with an introduction to Witten-Reshetikhin-Turaev (WRT) invariants and a related q-series which first appeared in the work of Lawrence and Zagier and unified the WRT invariants of the Poincaré homology sphere via radial limits. Remarkably it was also a key first example of a quantum modular form, a term later coined by Zagier with this series in mind and an object of interest in number theory. The q-series was later expanded to an invariant of negative definite plumbed 3-manifolds by Gukov, Pei, Putrov, and Vafa, and more recently extended by Akhmechet, Johnson, and Krushkal (AJK) to an infinite collection of two-variable series providing a common refinement with Némethi’s theory of lattice homology. After introducing the AJK invariants this talk will present results based on joint work with Eleanor McSpirit, in which we establish modularity properties and radial limits for infinite families of manifolds. Zoom Link: https://tamu.zoom.us/j/7474850426