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Texas A&M University
Mathematics

Events for 02/23/2022 from all calendars

Mathematical Physics and Harmonic Analysis Seminar

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Time: 09:00AM - 10:00AM

Location: Zoom

Speaker: Gilad Sofer, Technion

Title: Differences between Robin and Neumann eigenvalues on metric graphs

Abstract: We consider the Laplacian on a metric graph with Neumann vertex conditions. One may introduce a perturbation by placing δ vertex conditions at a selected subset of vertices. This results in an increase of the Laplacian eigenvalues. The differences between the eigenvalues of the perturbed and original operators are known as the Robin--Neumann gaps. The sequence of Robin--Neumann gaps has been recently studied for planar domains (Riviere--Royen), the hemisphere (Rudnick--Wigman) and star graphs (Rudnick--Wigman--Yesha).

The questions of interest concern the mean value of the Robin--Neumann gap sequence, the limiting values of the sequence and its bounds. We answer those questions for general metric graphs and compare our results to those obtained or conjectured for planar domains. In particular, we make a connection to inverse spectral problems, by showing that the mean value of the aforementioned sequence is determined by some geometric properties of the graph. The talk is based on a joint work in progress with Ram Band, Holger Schanz and Uzy Smilansky.


Groups and Dynamics Seminar

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Time: 3:00PM - 4:00PM

Location: BLOC 302

Speaker: Christopher Bishop, SUNY Stony Brook

Title: Dessins and Dynamics

Abstract: After defining harmonic measure on a planar domain, I will discuss "true trees", i.e., trees drawn in the plane so that every edge has equal harmonic measure and so that these measures are symmetric on each edge. True trees on the 2-sphere are a special case in Grothendieck's theory of dessins d'enfant, where a graph on a topological surface induces a conformal structure on that surface. I will recall the connection between dessins, equilateral triangulations and branched coverings (Belyi's theorem). I will also describe some recent applications of these ideas to holomorphic dynamics: approximating continuua by polynomial Julia sets, finding meromorphic functions with prescribed postcritical orbits, constructing new dynamical systems on hyperbolic Riemann surfaces, building wandering domains for entire functions, and estimating the fractal dimensions of transcendental Julia sets. There will be many pictures, but few proofs.