Orchidea Maria Lecian - Jan 26, 14h:00 (zoom)

The scarred wavefunctions of the desymmetrised PSL(2,Z) group  and those of its congruence subgroups are newly studied. The construction of irrationals after the (Farey)-Pell method is compared with the qualities of the quadratic fields. The action of automorphisms on trees is recalled to explain the action of the Hecke operaors on the Maass waveforms. The Margulis measure (which acquires a multiplicative constant under the action of certain U flows) is used. The opportune Kirkhoff reduced surfaces of section are chosen.

Closed geodesics are newly proven to scar the waveforms according to the quadratic field they are constructed after:

a) In the desymmetrised PSL(2,Z) group, the scarred waveforms are newly proven to be obtained under the action of the $U$ flow on the  Margulis measure which acts on the  quadratics fields which define the (also, classes) of closed geodesics.

b) In the congruence subgroups of the desymmetrised PSL(2,Z) domain, the scarred wavefunctions are newly proven to occur under the effect of the action of the Bogomolny transfer operators on the Margulis measure.

Dmitrii Korshunov - Jan 26, 16h:00 (Sala 863 - Departamento de Matemática PUC Rio)

In a joint work with Sasha Anan'in we prove a conjecture of Ian Agol: all isometric realizations of a polyhedral surface with boundary sweep out an isotropic subset in the Kapovich-Millson moduli space of polygons isomorphic to the boundary. For a generic polyhedral disk we show that boundaries of its isometric realizations make up a Lagrangian subset. As an application of this result, weconclude that a generic equilateral polygon cannot be domed (in the sense of a problem of Kenyon, Glazyrin and Pak).