## Curricular Proposal – Doctoral Program

The Doctoral program aims to train researchers in Pure or Applied Mathematics and is addressed to students who have a Master’s Degree in Mathematics or related fields, as well as to undergraduates in these areas who have an exceptional talent for Mathematics.

The doctoral program has a minimum duration of 24 months. The maximum duration is 48 months, extendable for up to two years. Upon completion of the program, the students receive the degree of:

*“Doctor in Mathematics”*

The degree is granted in one of the following areas of concentration:

Algebra

Analysis

Applied Mathematics

Geometry and Topology

**Regular courses, divided into four groups, as listed below.**

a) Mandatory “Teaching Practicum” (4 credits), mandatory for all students. The Teaching Practicum consists of didactic activities supervised by a professor (called a tutor), in which the student teaches classes in undergraduate courses.

b) “Mathematics Colloquium” (2 credits). The course consists of lectures given by local or invited researchers and is offered whenever possible.

c) Master’s thesis (6 credits).

**The Doctoral curriculum is structured as follows:**

- Regular courses, divided into two groups, as listed below.
- No mandatory courses.
- Mandatory “Teaching Practicum” (4 credits), consisting in didactic activities supervised by a professor (called a tutor), in which the student teaches classes in undergraduate courses.
- “Mathematics Colloquium” (2 credits), consisting in lectures given by local or invited researchers and offered whenever possible.
- Doctoral thesis (12 credits).
- Meeting the requirements specified in the program’s bylaws regarding the minimum workload.
- Two written qualifying examinations, according to Article 36 of the program’s bylaws, in addition to an oral qualifying examination

*About qualifying examinations:*

- The written qualification exams are offered at the beginning of each semester, if there are registered candidates.
- If a student is attending the second year of the doctoral program, they may request a qualifying examination at the end of the second semester of the current year
- The student must do the first written exam until the beginning of the fourth semester and pass the two written exams until the beginning of the fifth semester.
- In case of failure in a written exam, the student may repeat it once, mandatorily at the next opportunity, and in the same area of study of the original exam.

Regular courses in the Doctoral program, each of 06 credits:

**Jointly with Master program:**

MTM410019 Linear Algebra

MTM410024 Computational Linear Algebra

MTM410028 Numerical Analysis I

MTM410027 Measure and Integration

MTM410026 Topology

MTM410057 Dynamical Systems

MTM410071 Finite Groups and their Representations

MTM410073 Mathematical Methods for Statistics

**Doctoral program courses:**

MTM510011 Operator Algebras

MTM510009 Differentiable Manifolds

MTM510008 Algebraic Topology

MTM510014 Introduction to Hopf Algebras

MTM510015 Theory of Non-Commutative Rings

MTM510005 Corings and Comodules

MTM510012 Distribution Theory and Sobolev spaces

MTM510007 Probability and Markov Processes

MTM510016 Ergodic and Information Theory

MTM510006 Symbolic Dynamics

MTM510013 Numerical Analysis II

MTM510010 Convex Analysis

MTM510050 Fibers in Differentiable Manifolds

MTM510034 Commutative Algebra

MTM510020 Introduction to Regularization Theory

MTM510040 Introduction to Category Theory

MTM510049 Mathematical Modeling Biomathematics

MTM510001 C*-Algebras

MTM510051 Von Neumann Algebras

MTM510037 C*- Dynamical Systems and Crossed Products

MTM510032 Hilbert Modules and Fell bundles

MTM510017 K-Theory for C* -Algebras

MTM510052 Advanced Module theory

MTM510018 Nonlinear Partial Differential Equations

MTM510033 Partial Differential Equations

MTM510053 Introduction to the Mathematical Theory of Navier-Stokes Equation

MTM510013 Theory of Semigroups and Applications in PDE

MTM510026 Infinite-Dimensional Attractors

MTM510025 Computational Linear Algebra II

MTM510027 Spectral Methods

MTM510019 Computational Methods in Optimization

MTM510022 Introduction to Continuous Optimization

MTM410057 Applied Functional Analysis

MTM510054 Riemannian Geometry

MTM510055 Iterative Methods for Inverse Problems

MTM 510059 Non-Linear Analysis Methods

MTM510056 Elements of Spectral Theory

MTM510057 Introduction to Group Cohomology

MTM510056 Groupoids, Inverse Semigroups and their C*-algebras

MTM510044 Discontinuous Galerkin Finite Element Methods for Problems of Multiphase Flows in Porous Media

Specialized topics

**LANGUAGE PROFICIENCY**

The Program’s Bylaws establish that:

Art. 50. Throughout the first academic year, proof of foreign language proficiency (English, French, German or Spanish) will be required. The Master’s program requires proficiency in one foreign language, and the doctoral program requires proficiency in two foreign languages.

**INFRASTRUCTURE**

**Computer Resources**

Master’s students have access to a computer laboratory. The graduate program also has a few laptops that can be borrowed for certain periods.

Since 2016, there is a room dedicated to postdoctoral researchers and another to the visiting faculty. All faculty rooms, and the computer lab, are connected to the internet.

**Library**

All students enrolled, as well as the university staff, have access to the central and branch libraries. In spite of a relatively small bibliographical collection in the area of Mathematics, this collection is incremented annually with the suggestions of our faculty. Access to journals is also available through the Capes online publication website.