Theses & Dissertations

Ph.D Students

Leonardo Gonçalves de Oliveira

Advisor: Simon Richard Griffiths
Thesis Title: Moderate Deviations of Triangle Counts in Sparse Random Graphs
Defense: 28/09/2022 | Abstract | Thesis
Depto de Matemática - Prédio Cardeal Leme - SL 856

In the first part of this thesis, we study the deviation of the number of triangles with respect to its mean in both the random graph models G(n,m) and G(n,p). We focus on the case where the random graph is sparse, in which the edge density goes to zero as the number of vertices increases to infinity. Also, our focus is in the case of moderate deviations, i.e., those of order in between the standard deviation and the mean. In addition, we derive the same kind of results for cherries (paths of length two). In the second part of this thesis, we study Freedman's inequality. This inequality gives bounds on the probability of the deviation of a bounded martingale using its conditional variance. In our work, we obtain a strengthening of Freedman's inequality, under additional symmetry conditions on the increments of the martingale process.

Hugo de Souza Oliveira

Advisor: Sinésio Pesco
Thesis Title: A RBF approach to the control of PDEs using Dynamic Programming equations
Defense: 27/09/2022 | Abstract | Thesis
Defesa realizada por meios de comunicação remota

Semi-Lagrangian schemes for discretization of the dynamic programming principle are based on a time discretization projected on a state-space grid.
The use of a structured grid makes this approach not feasible for high-dimensional problems due to the curse of dimensionality.
Here, we present a new approach for infinite horizon optimal control problems where the value function is computed using Radial Basis Functions (RBF) by the Shepard’s moving least squares approximation method on scattered grids.
We propose a new method to generate a scattered mesh driven by the dynamics and an optimal routine to select the shape parameter in the RBF. This mesh will help to localize the problem and approximate the dynamic programming principle in high dimension. Error estimates for the value function are also provided. Numerical tests for high dimensional problems will show the effectiveness of the proposed method. In addition to the optimal control of classical PDEs, we show how the method can also be applied to the control of non-local equations.

Edhin Franklin Mamani Castillo

Advisor: Rafael O. Ruggiero
Thesis Title: About the measure of maximal entropy and horospherical foliations of geodesic flows of compact manifolds without conjugate points
Defense: 16/09/2022 | Abstract | Thesis
Depto de Matemática - Prédio Cardeal Leme - SL 856

In this thesis, we study some dynamical and geometrical properties of the geodesic flow of certain compact manifolds without conjugate points. The thesis has two main parts. In the first one, we extend the work of Gelfert and Ruggiero about the existence of a expansive factor for the geodesic flow of compact surfaces without focal points to compact surfaces without conjugate points and genus greater than one. The main idea of their work is to define an equivalence relation that collapses bi-asymptotic orbits of the geodesic flow. The expansive factor is time-preserving semi-conjugate to the geodesic flow through the quotient map. We verify that this expansive model is topologically mixing, and has a local product. These properties imply that the model has the specification property and a unique measure of maximal entropy. We lift this measure to the unit tangent bundle and verify that it is the unique measure of maximal entropy for the geodesic flow. This provides an alternative proof ot the uniqueness of the measure of maximal entropy in the same context by Climegnaga-Knieper-War. In the second part of the thesis, we extend some results of Gelfert and Ruggiero from higher genus compact surfaces without conjugate points having continuous Green bundles to compact n-manifolds without conjugate points with Gromov hyperbolic fundamental group and having continuous Green bundles. In this setting, we see that if there exists a hyperbolic closed orbit for the geodesic flow, then hyperbolic periodic points are dense and the Pesin set agrees with a dense open set a. e. with respect to the Liouville measure. We deduce that Green bundles are the unique n-1 dimensional bundles invariant by the geodesic flow derivative. Moreover, these bundles are tangent to the horospherical foliations. We also show that horospherical foliations are the unique n-1 dimensional foliations of the unit tangent bundle, invariant by the geodesic flow. This fact was only known for compact surfaces without conjugate points by the work of Barbosa-Ruggiero, and in higher dimensions assuming the stronger condition of bounded asymptote by the work of Eschenburg. With respect to the expansive model for this case, we prove that the quotient space is a 2n-1 topological manifold.

Fiorella Maria Rendón Garcia

Advisor: Boyan Sirakov
Thesis Title: Global boundary weak Harnack inequality for general uniformly elliptic equations in divergence form and applications
Defense: 02/05/2022 | Abstract | Thesis
Depto de Matemática PUC Rio - Prédio Cardeal Leme - Sala 856

In this Ph.D. thesis we give a global extension of the interior weak Harnack inequality for a general class of divergence-type elliptic equations, under very weak regularity assumptions on the differential operator. In this way we generalize and unify all previous results of this type. As an application, we prove a priori estimates for a class of quasilinear elliptic problems with quadratic growth on the gradient and we investigate, under various assumptions, the multiplicity of the solutions obtained for this problem.

Master’s Students

Adailton José do Nascimento Sousa

Advisor: Sinésio Pesco
Thesis Title: Não divulgado
Defense: 30/09/2022 | Abstract | Thesis

Não divulgado

Daniel Byron Souza P. de Andrade

Advisor: Simon Richard Griffiths
Thesis Title: Matrizes aleatórias e a lei do semicírculo
Defense: 28/04/2022 | Abstract | Thesis
Depto de Matemática PUC Rio - Prédio Cardeal Leme - Sala 856

Nesta dissertação vamos abordar o famoso teorema de Wigner, “Lei de semicírculo”, que dá uma descrição do comportamento do espectro de autovalores de matrizes aleatórias simétricas. A demonstração combina ideias e técnicas de Combinatória e Probabilidade, incluindo uma análise cautelosa dos momentos da distribuição de autovalores.
A lei do semicírculo é muito usada em diferentes contextos. Atualmente tem aplicações em áreas como teoria quântica de campos.

Gabriel Dias do Couto

Advisor: Simon Griffiths
Thesis Title: em andamento
Defense: 19/04/2022 | Abstract | Thesis
Departamento de Matemática Sala 856

Aguardando título da tese.