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