Events for 12/08/2022 from all calendars

Working Seminar on Banach and Metric Spaces

Time: 09:30AM - 11:00AM

Location: BLOC 302

Speaker: Pavlos Motakis, York University

Title: Uniqueness of c₀ asymptotic structures (after Freeman, Odell, Sari, Zheng)

Mathematical Physics and Harmonic Analysis Seminar

Time: 11:00AM - 11:50AM

Location: BLOC 302

Speaker: Alexander Kiselev

Title: Convergence of Neumann Laplacians on thin structures: an alternative approach

Abstract: Neumann Laplacians $A_\epsilon$ on thin manifolds, converging to metric graphs $G$ as $\epsilon\to0$, have been intensively studied by many authors, including Kuchment, Post, Pavlov, Exner, Zeng, among many others. The present-day state-of-the-art in this area is described in the monograph by O. Post.

It was proved that the spectra of $A_\epsilon$ converge within any compact $K\in \mathbb{C}$ in the sense of Hausdorff to the spectrum of a graph Laplacian $A_G$. In the book of Post, the claimed convergence was enhanced to the norm-resolvent type, with an explicit control of the error as $O(\epsilon^\gamma),$ with $\gamma>0$ explicitly given. The matching conditions at the vertices of the limiting graph turn out to be either:
(i) Kirchhoff (i.e. standard), if the vertex volumes are decaying, as $\epsilon\to0,$ faster than the edge volumes;
(ii) Resonant, which can be equivalently described in terms of $\delta$-type matching conditions with coupling constants proportional to the spectral parameter $z$, if the vertex and edge volumes are of the same order;
(iii) ``Dirichlet-decoupled" conditions (i.e., the graph Laplacian becomes completely decoupled), if the vertex volumes vanish slower than the edge ones.

In the talk, I will be primarily interested in the most non-trivial resonant case (ii). I will provide a straightforward, alternative to that of Post, proof of the fact that the Neumann Laplacians $A_\epsilon$ in this case converge in norm-resolvent sense to an ODE acting in the Hilbert space $L^2(G)\oplus \mathbb{C}^N$, where $N$ is the number of vertices. The operator to which it converges is in fact the one first pointed out by Kuchment as the self-adjoint operator whose spectrum coincides with the Hausdorff limit of spectra for the family $A_\e$.

I will show how a better error bound than that of Post is attained, namely, our estimate in the planar case is logarithmically worse than $O(\epsilon)$ and in the case of $\mathbb{R}^3$ is $O(\epsilo

Number Theory Seminar

Time: 1:00PM - 2:00PM

Location: BLOC 302

Speaker: Yen-Tsung Chen, Penn State University

Title: On Thakur's basis conjecture for multiple zeta values in positive characteristic

Abstract: Classical multiple zeta values (MZV's) are generalizations of special values of the Riemann zeta function at positive integers. The machinery of regularized double shuffle relations produces many linear relations among MZV's. There is a famous conjecture of Zagier predicting the dimension of the vector space spanned by MZV's of the same weight. Furthermore, due to the numerical computation, Hoffman proposed a conjectural basis for the space in question. In the parallel but quite different world, namely the global function fields, Thakur introduced an analogue of classical MZV's in the function field setting. In this talk, we give affirmative answers to the conjectures of Todd and Thakur which are analogues of the conjectures of Zagier and Hoffman respectively for MZV's in positive characteristic. This is joint work with Chieh-Yu Chang and Yoshinori Mishiba.