Algebra is one of the main areas of mathematics, focused on studying the rules for manipulating mathematical objects. From solving elementary equations to studying more abstract structures like groups, rings, and other general algebraic structures, algebra is fundamental to the advancement of mathematical knowledge.

Areas of Study in Algebra

Algebraic Structures: Involves the study of groups, rings, fields, modules, and other structures that form the basis for many mathematical concepts.

Algebraic Geometry: Explores geometric aspects, intersection properties, and singularities of sets defined by the zeros of polynomials in several variables, such as curves and algebraic varieties.

Commutative Algebra: Studies commutative rings and their ideals, fundamental to algebraic geometry and number theory.

Number Theory: Investigates properties and relationships between integers, often using algebraic tools.

Intersection with Other Areas

Algebra is closely related to various areas of mathematics and applied sciences:

Combinatorics: Uses algebraic techniques to solve counting and structural problems.

Analytical Geometry: Employs algebra to represent and study geometric properties.

Modern Mathematical Physics: Applies algebraic concepts in modeling physical phenomena.

Mathematical Biology and Chemistry: Uses algebra to model complex biological and chemical processes.

Modern Applications Algebra has practical applications in various modern areas, including:

Cryptography: Uses algebra to develop secure algorithms for data protection.

Error-Correcting Code Theory: Applies algebra to create codes that detect and correct errors in data transmissions.

Recent Contributions

Recently, algebra has made significant contributions to technological and scientific areas, especially in digital security and communication. Its role in cryptography and error-correcting code theory highlights its growing importance in the information age.

Our program encourages research in algebra, providing a collaborative environment to explore both the fundamental aspects and practical applications of this fascinating area of mathematics.

Members

Ivan Kaygorodov

Renato Fehlberg

Thiago Filipe da Silva

Research Lines:

Free structures in division rings

Non-associative algebras

Algebraic theory of singularities

Lipschitz saturations of algebras

N-ary algebras

Poisson algebras

Derivations and general derivations

Deformations

Degenerations

Superalgebras

The area of Mathematical Analysis is dedicated to the study of mathematical objects, formalizing concepts such as asymptotic proximity, infinitesimals, and regularity. Its methods are widely applied in various mathematical disciplines, including differential geometry, probability, and dynamical systems.

Subareas and Applications of Analysis

Real and Complex Analysis: Studies functions of real and complex variables, investigating properties such as continuity, differentiability, and integrability.

Functional Analysis: Focuses on function spaces and linear operators, fundamental in operator theory and optimization problems.

Differential Geometry: Applies analytical techniques to the study of curves, surfaces, and manifolds, exploring their geometric and topological properties.

Probability: Uses analytical methods to model and analyze random phenomena, with applications in statistics, physics, and finance.

Dynamical Systems: Examines the long-term behavior of evolving systems using differential equations and chaos theory.

Differential Equations

Differential equations are equations involving derivatives of spatial and temporal variables. They are crucial for modeling most physical phenomena, such as:

Wave Evolution: Describes how waves propagate in different media.

Heat Propagation: Models the diffusion of heat in materials.

Fluid Dynamics: Investigates the movement of liquids and gases.

Study Objectives

In the analysis of differential equations, the main objectives include:

Existence: Determining if there are solutions to the equations.

Uniqueness: Establishing if the solutions are unique.

Regularity: Examining the smoothness and other analytical properties of the solutions.

Contributions and Impacts

Analysis has a profound and broad impact on various areas of knowledge. Its techniques and results are essential for developing theories and applications in physics, engineering, biology, and economics. The ability to formalize and solve complex problems, as well as a deep understanding of natural phenomena, highlights the importance of this area in modern mathematics.

Our program offers a rich environment for research in Analysis, encouraging the exploration of its various subareas and their interconnections with other mathematical disciplines. We invite interested individuals to delve deeper and contribute to this fundamental area of mathematics.

Members

Daniela Demuner

Fabiano Petronetto

Fabio Julio Valentim

Fernando Pereira Paulucio Reis

Ginnara Mexia Souto

Renato Fehlberg Júnior

Research Lines:

Theory of differential operators

Discretization of differential operators

Fractional calculus

Dynamical systems, impulses, and differential equations

Dynamics of diffeomorphism germs in (C^2,0).

Analysis of differential equations

Differential geometry is a mathematical theory dedicated to the study of curves, surfaces, and differentiable manifolds. This area is a bridge between various disciplines, including differential equations, topology, and physics, and is currently in a period of great development.

Intersections with Other Areas

Differential Equations: Differential geometry uses differential equation techniques to analyze curvature and other geometric properties of surfaces and manifolds.

Topology: Topological theory provides the fundamental structure for studying differentiable spaces, allowing deep analysis of their global properties.

Physics: Many physical theories, such as general relativity, are formulated in the context of differential geometry.

Recent Contributions

Differential geometry has been instrumental in solving old and important mathematical problems. A notable example is the resolution of the Poincaré Conjecture, which was a significant milestone in mathematics. Moreover, new problems of great relevance are constantly emerging, driving the advancement of theory and its applications.

Topology

Topology studies the properties of topological spaces, structures where notions of limit and continuity can be defined. This area is fundamental to understanding how spaces can be transformed and classified without altering their essential properties.

Structures and Tools Topological Spaces: Structures that allow the definition of concepts like continuity and limits.

Homology/Homotopy: Algebraic tools that associate a sequence of groups or rings with a topological space, helping to distinguish and classify these spaces.

Intersections with Other Areas Differential Geometry: Topology and differential geometry complement each other, enabling the analysis of global and local properties of differentiable manifolds.

Dynamical Systems: Topology provides the foundation for understanding the long-term behavior of dynamical systems.

Algebra: The association of algebraic objects such as homology or the fundamental group allows a deeper and more rigorous analysis of topological spaces.

Advances and Relevance

Topology continues to be a dynamic and expanding area, with new concepts and techniques regularly emerging. The combination of ideas from differential geometry, dynamical systems, and other areas has allowed significant advances, providing new ways to understand and explore the structure of topological spaces.

Our program values and promotes research in differential geometry and topology, offering a stimulating environment for the development of new ideas and the resolution of complex problems. We invite all interested individuals to engage with these fascinating areas of mathematics.

Members

Brayan Cuzzuol Ferreira

Carolina de Miranda e Pereiro

Fernando Pereira Paulucio Reis

José Victor Goulart Nascimento

Leonardo Câmara

Maico Felipe Silva Ribeiro

Marta Batoréo

Renan Mezabarba

Thiago Filipe da Silva

Wescley Bonomo

Research Lines:

Bi-harmonicity, biconservativity, and constant angle surfaces

Existence of periodic orbits in symplectic and contact dynamics

Embeddings and symplectic invariants

Milnor fibrations and singularity topology

Geometry of holomorphic foliations

Lipschitz geometry of singularities

Topological invariance of the algebraic multiplicity of foliation germs

Categorical notion of integral closure for modules and Thom regularity of analytic applications

Residues of structures transverse to holomorphic distributions

Symmetries of dynamical systems

Rotation theory and topological dynamics techniques

General topology and set theory

Surface braid groups

Coincidence and fixed point theory

Applied Mathematics is dedicated to applying mathematical methods in various areas of knowledge, with the aim of solving practical problems through the formulation and study of mathematical models.

Subareas and Applications

Numerical Analysis: Focused on the construction, analysis, and implementation of algorithms for approximating solutions to mathematical problems. This area is crucial for solving complex equations that arise in physics, engineering, and other sciences.

Computer Graphics: Develops tools and techniques to convert data into visual representations through graphic devices. It is essential for creating graphics, animations, and visual simulations in various disciplines.

Mathematical Modeling: Involves creating mathematical models to simulate real systems, allowing the prediction and analysis of their behaviors. It is widely used in natural sciences, engineering, and economics.

Probability Theory

Probability Theory studies mathematical models that formalize the idea of uncertainty. This area is increasingly relevant in various disciplines due to its ability to model random and uncertain phenomena.

Applications and Relevance Statistical Mechanics: Uses probability to understand and predict the behavior of physical systems composed of many particles.

Partial Differential Equations: Probability helps analyze and solve equations that model continuous and dynamic phenomena.

Statistics: Based on probability theory, statistics allows the analysis and interpretation of data for informed decision-making.

Financial Derivatives Theory: Uses probabilistic models to evaluate and manage risk in complex financial instruments.

Contributions and Impacts Applied Mathematics and Probability Theory are essential areas that drive technological and scientific advancement. The ability to model, simulate, and predict the behavior of complex systems has vast applications, from engineering and physics to economics and social sciences.

Our program offers a robust environment for research and development in these areas, encouraging interdisciplinary collaboration and innovation. We invite all interested individuals to explore and contribute to the advancement of Applied Mathematics and Probability Theory, areas at the forefront of science and technology.

Members

Fabiano Petronetto do Carmo

Fabio Julio da Silva Valentim

Gabriel Luchini

João Paulo Costalonga

Valdério Reisen

Research Lines:

Visualization and analysis of multidimensional data

Hypergraphs: signal processing and applications

Computer graphics: fluid flow

Mathematical modeling for the simulation of real systems and prediction of their behavior

Interacting particle systems

Mathematical statistics and stochastic processes

Mathematical physics

Combinatorics: matroid and graph theory

Time series