Seminário das Sextas - Prof. Rodrigo Matos(PUC-Rio) e Miguel dos Anjos (Aluno de graduação PUC-Rio)

 Mutually orthogonal latin squares, their relation to finite projective planes, and all that

09h:00-11h:00 | (02/06/2023)

Miguel dos Anjos Batista - Aluno de graduação |  PUC-Rio


I will introduce the concept of a finite projective plane (FPP), their basic structure, and show the existence of some of them. Beside that, I will talk about Mutually Orthogonal Latin Squares (MOLS), a "categorical" way to visualize it and how they interact with FPP. If possible I will talk about more complex FPP existence theorems.

introduction to Fermi surface (Fermi Varieties), Bloch Varieties, and an interface of algebraic geometry,
spectral theory and mathematical physics

14h:00-16h:00 | (02/06/2023)

Prof. Rodrigo Matos - Professor do Departamento de Matemática | PUC-Rio


The structure of the dispersion relation is one of the central aspects to the study of periodic Schroedinger operators. 

Besides the intrinsic interest from the viewpoint of several complex variables and algebraic geometry, the dispersion relation also carries relevant information
for the spectral theory of periodic media. In particular, for the structure of spectral boundaries, isospectrality, and existence of eigenvalues for locally perturbed operators. 

I will discuss some of these connections as well as recent irreducibility theorems for the Bloch and Fermi varieties,
focusing on two joint works with Jake Fillman and Wencai Liu (arXiv:2107.06447 & J. Funct. Anal. 2022, arXiv 2305.06471). 

These recent papers cover a wide class of lattice geometries in arbitrary dimension and verify the discrete version of certain conjectures of Kuchment for various discrete models.