Seminários

Differential equations and Strong minimality

Resumo: Firstly, I would like to give an introductory talk about the model theory of differential fields, Kolchin's constrained (or differentially closed) extensions of differential fields, and their reduction modulo p by the author and Leon Sanchez. Later, I will explain the model-theoretic contribution to the study of certain differential equations in characteristic zero, such as the Painlev\'{e} equation studied by Nagloo and Pillay and the order three differential equations satisfied by modular j-function by Freitag and Scanlon. Finally, I want to share my current research interest.

Curvature homogeneous hypersurfaces

Resumo: We provide a classification of all curvature homogeneous hypersurfaces in space forms. Besides the obvious cases and a single example in $H^5$ discovered in the 80's, that we recover, the only case left was completely open: the 3-dimensional one in both $S^4$ and $H^4$. We show that, in each of these two space forms, there is a one parameter family of hypersurfaces without symmetries, and a single algebraic example with a circle of symmetries. Moreover, these are essentially all of them.

This is a joint work with Robert Bryant and Wolfgang Ziller.

Neural Conjugate Flows: Physics-Informed Architectures with Differentiable Flow Structure

Resumo: We present Neural Conjugate Flows, a novel design for physics-informed neural networks with flow structure. We prove that this architecture is an universal approximator for solutions of differential equations and demonstrate how its group and topological lead to computational and theoretical gains when simulating dynamical systems with neural networks.

Data Science and Innovation

Resumo: Data science and innovation have become overloaded terms, leading to some confusion. To be successful, the innovation process involves not only inventions (e.g., new methods) but also context, e.g., user behavior and timing, e.g., market readiness. In this talk, I discuss the impact of data science on innovation, using selected success stories (some of which I was involved in). I also give hints to promote innovation within companies, in particular, using open innovation. Finally, I describe some innovations in the context of the Inria-Brasil partnership.

Category Theory applied to Data Visualization

Resumo: Category Theory (CT) is a branch of mathematics that studies general abstract structures through their relationships, and it is unmatched in its ability to organize and relate abstractions. In recent years, CT has found applications in a wide range of disciplines, such as chemistry, biology, natural language processing, and database theory. We present a novel application by formalizing Data Visualization within Category Theory. This formalization creates a bridge between Mathematics and Data Visualizations. Moreover, it provides a framework to express complex visualizations, which can be implemented computationally by leveraging the well-established connection between CT and Functional Programming.