Events for 10/28/2022 from all calendars

Mathematical Physics and Harmonic Analysis Seminar

Time: 09:00AM - 10:00AM

Location: Zoom

Speaker: Francesco Tudisco, Gran Sasso Science Institute, Italy

Title: Nodal domain count for the generalized graph p-Laplacian

Abstract: In recent years, there has been a surge in interest towards nonlinear extensions of graph operators such as the p-Laplacian and the generalized p-Laplacian (or p-Schrödinger) operators. This interest is prompted by applications connected to data clustering and semisupervised learning, where the limiting cases p=1 and p=\infty are especially noteworthy. In particular, similarly to the linear case, an important relation connects the nodal domains of the p-Laplacian and the k-th order isoperimetric constant of the graph.

In this talk, we consider a set of variational eigenvalues of the generalized p-Laplacian operator and present new results that characterize several properties of these eigenvalues, with particular attention to the nodal domain count of their eigenfunctions. Just like the one-dimensional continuous p-Laplacian, we prove that the variational spectrum of the discrete generalized p-Laplacian on forests is the entire spectrum. Moreover, we show how to transfer Weyl’s inequalities for the Laplacian operator to the nonlinear case and thus we prove new upper and lower bounds on the number of nodal domains of every eigenfunction of the generalized p-Laplacian on graphs, including those corresponding to variational eigenvalues. When applied to the linear case p=2, the new results imply well-known properties of the linear Schrödinger operator as well as novel ones.

Based on a joint work with P.Deidda and M.Putti.

Mathematical Physics and Harmonic Analysis Seminar

Time: 1:50PM - 2:50PM

Location: BLOC 306

Speaker: Matthias Hofmann, TAMU

Title: Spectral minimal partitions of unbounded metric graphs

Abstract: We investigate the existence or non-existence of spectral minimal partitions of unbounded metric graphs, where the operator applied to each of the partition elements is a Schrödinger operator of the form $-\Delta + V$ with suitable (electric) potential $V$, which is taken as a fixed, underlying ``landscape'' on the whole graph. We show that there is a strong link between spectral minimal partitions and infimal partition energies on the one hand, and the infimum $\lambda_{\text{ess}}$ of the essential spectrum of the corresponding Schrödinger operator on the whole graph on the other, which recalls a similar principle for the eigenvalues of the latter: for any $k\in\mathbb N$, the infimal energy among all admissible $k$-partitions is bounded from above by $\lambda_{\text{ess}}$, and if it is strictly below $\lambda_{\text{ess}}$, then a spectral minimal $k$-partition exists. We illustrate our results with several examples of existence and nonexistence of minimal partitions of unbounded and infinite graphs, with and without potentials. The nature of the proofs, a key ingredient of which is a version of Persson's theorem for quantum graphs, strongly suggests that corresponding results should hold for Schrödinger operator-based partitions of unbounded domains in Euclidean space. Joint project with James Kennedy and Andrea Serio.

Algebra and Combinatorics Seminar

Time: 3:00PM - 4:00PM

Location: BLOC 302

Speaker: Daoji Huang, U of Minnesota

Title: Bumpess pipe dream RSK, growth diagrams, and Schubert structure constants

Abstract: The cohomology ring of the complete flag variety has a basis given by classes of the Schubert varieties. A central open question in Schubert calculus is to give a combinatorial interpretation of the multiplicative structural constants of the Schubert classes. While the general question remains open, in the Grassmannian case, the Schubert structure constants are known as Littlewood-Richardson coefficients and well-understood, and many of these classical rules are based on tableaux combinatorics. In this talk, we aim to generalize some of these results using bumpless pipe dreams. In particular, we introduce analogs of left and right RSK insertion for Schubert calculus of complete flag varieties. The objects being inserted are certain biwords, the insertion objects are bumpless pipe dreams, and the recording objects are decorated chains in Bruhat order. As an application, we adopt Lenart's growth diagrams of permutations to give a combinatorial rule for Schubert structure constants in the separated descent case.