# General Information

Welcome to POSTGRADUATE PROGRAM IN MATHEMATICS (PPGMAT) at Ufes

• Algebra is a main area of research in mathematics and, essentially, it concerns the study of the rules of manipulation of mathematical objects. This field has a wide range of topics from solving elementary equations up to the modern study of groups, rings and other more general algebraic structures. Algebra has ties with combinatorics, analytic geometry, modern mathematical physics, mathematical biology, chemistry, amongst others. In particular, Algebraic Geometry is a field of mathematics that studies geometric aspects, singularities and intersection properties of null sets of polynomials in several variables, such as algebraic curves and its generalizations (algebraic varieties). There exists a strong overlap with fields of mathematical study such as commutative algebra and number theory. Applications are broad and, recently, algebraic geometry has been supplying important contributions to cryptography and to coding theory.

**Lines of research**: non-associative algebras, superalgebras, n-ary algebras, Poisson algebras, derivations and general derivations, deformations, degenerations (Ivan Kaygorodov); algebraic curves, Weierstrass semigroups, moduli spaces of punctuated curves, maximal curves over finite fields (Gilvan de Oliveira); free algebras, division rings (Renato Fehlberg Júnior).

• Differential geometry is a mathematical theory that studies questions related to curves, surfaces and differential manifolds. This theory overlaps many other fields of mathematics (differential equations, topology, etc) and physics. Differential geometry is in a period of big development. Old and important mathematical problems have been solved using geometrical techniques (a recent example is the solution to the Poincaré conjecture). There is a constant flow of new and relevant problems to study.

In Topology it is carried the study of the so called topological spaces, a mathematical structure on which the notion of limit and continuity can be defined. In this field it is recurrent the association of certain algebraic structures (homology), as it allows recognizing different spaces. This association can be carried out through various viewpoints: be it from differential geometry(e.g., De Rham Cohomology), or from dynamical systems (e.g., Morse homology), amongst others.

**Lines of research**: bi-harmonicity, bi-conservativeness, constant angle surfaces (Apoenã Passamani); integrability vector fields germs in C3, integrability of rational differential equations in the complex plane, dynamics of germs of diffemorphisms in C^(2,0) (Leonardo Câmara); symplectic topology/dynamics (Marta Batoréo); algebraic and differential topology, equivariant cobordism (Patrícia Desideri); Symmetries of dynamical systems (Wescley Bonomo).

• Applied mathematics is the application of mathematical methods by different fields. The goal is work on practical problems by formulating and studying mathematical models. In particular, in Numerical Analysis the goal is to build, analyze and implement approximation algorithms to solutions in mathematical problems. The area of Computer Graphics develops and analyzes tools for converting data into graphical devices by using a computer.

Probability is a field of inquiry in pure and applied mathematics. It studies mathematical models which formalize the concepts of uncertainty and statistical pattern. This theory goes back to early models of games of chance and it has been gaining an important relevance in several areas of knowledge providing tools of analysis for several other areas of knowledge such as statistics, statistical mechanics, partial differential equations, quantum mechanics, financial derivatives theory, amongst many others.

**Lines of research**: mathematical modeling for simulation and prediction of real systems (Fabiano Petronetto); probability methods for the study of physical systems  hydrodynamic behavior of interacting particle systems (Fábio Júlio Valentim);
Combinatorics: Theory of Matroides and Graphs (João Paulo Costalonga); Times series (Valdério Reisen).

• The field of Analysis concerns the study of mathematical objects formalizing the concepts of asymptotic neighborhood, infinitesimal and smoothness. Its methods are employed in several other fields of mathematics such as differential geometry, dynamical systems, probability, amongst many others.
Differential equations are equations involving derivatives (say, temporal and/or spatial). Examples abound WHERE physical phenomena can be well modeled by differential equations, for instance: waves, heat propagation, and fluid dynamics in general. One tries to understand: existence, uniqueness and smoothness of the solutions and other analytical properties.
**Lines of research**: bi-harmonicity, bi-conservativeness (Apoenã Passamani); impulsive differential equations (Daniela Demuner); discretization of differential operators (Etereldes Goncalves); theory of differential operators (Fábio Valentim); discretization of differential operators (Fabiano Petronetto); germs integrability and dynamics (Leonardo Câmara); impulses and differential equations (Ginnara Mexia Souto); dynamical Systems with impulses (Jaqueline da Costa Ferreira); substitutions, Rauzy fractals, perturbations in interval exchanges (Milton Cobo); fractional Calculus (Renato Fehlberg Júnior); cicles and bifurcations (Tiane Marcarini).