91´ó»ÆѼ

August 28— "Eulerian and Lagrangian transport in wall-bounded turbulent flows"

• Presenter: Andrew Grace, CU-APPM
• Abstract:ÌýThe dynamics of fluids in Earth's waters and atmosphere profoundly affect many aspects of our lives, including weather forecasting, disaster response, and our climate. In this talk, I will present some results surrounding fluid mixing, turbulence, and Lagrangian particle dispersion, which represent several of the important underlying physical processes in these applications. I will begin by providing an overview of how we model the motion of incompressible fluids (the incompressible Navier-Stokes equations under the Boussinesq approximation) and show examples of numerical solutions to these equations in different contexts. I will then highlight recent results focused on how we can apply these equations to model turbulent transport of solid particles in buoyancy driven and mechanically driven turbulent flows. Specifically, I will discuss how multi-scale fluid structures in wall-bounded turbulent flows mediate dispersion processes near solid surfaces, and how we can model these interactions. Finally, I will finish the talk by highlighting several exciting avenues for future research.

September 25 — "Oscillation of graph eigenfunctions and its applicationsÌý"

• Presenter: Gregory Berkoilaiko, Texas A&M

• Abstract:ÌýOscillation theory, originally due to Sturm, seeks to connect the number of sign changes of an eigenfunction of a self-adjoint operator to the label k of the corresponding eigenvalue. ÌýIts applications run in both directions: knowing k, one may wish to estimate the zero set, or the topology of its complement, useful in clustering and partitioning problems. ÌýConversely, knowing an eigenvector (and thus the number of its sign changes), one may want to determine if it is the ground state, useful in the linear stability analysis of solutions to nonlinear equations. Ìý

  Within the setting of generalized graph Laplacians, Fiedler’s theorem says that the k-th eigenvector of a tree (a graph without cycles) changes sign across exactly k-1 edges. ÌýWe present a formula for the number of sign changes on a general graph, which attributes the excess sign changes to the cycles in the graph and their intersections.

  This result has many interesting connections. ÌýFirst, it allows one to derive a simple formula for the effective mass tensor of a particular class of crystals (periodic lattices), namely the maximal abelian covers of finite graphs. ÌýSecond, it can be used to efficiently determine stability of a stationary solution on a coupled oscillator network, such as the non-uniform Kuramoto model for the synchronization of a network of electrical oscillators. ÌýFinally, the determinant of the matrix which determines the excess sign changes is closely related to the graph’s Kirchhoff polynomial (which counts the weighted spanning trees), hinting at connections to both Feynman amplitudes and matroids.

  Based on Joint work with Jared Bronski (UIUC ) and Mark Goresky (Princeton).

October 2 — Ìý"Applied Math for the Heart; Take a few PDEs and call me in the morning"

• Presenter: Flavio Fenton, Georgia Institute of Technology
• Abstract: TBD
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November 6— "TBH"

• Presenter: Christopher Wikle, University of Missouri
• Abstract:
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December 4— "TBH"

• Presenter: Maria D'Orsogna, California State University Northridge,
• Abstract:
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