Mini courses

Minicourse Lecturers

Each minicourse will consist of 6 hours of lectures. 

Mini course 1. Introduction to singular fiber topology and its application to data analysis

Lecturer: Osamu Saeki

Abstract: In this mini course, we consider smooth generic maps f : M → N between smooth manifolds M and N with dim M ≥ dim N ≥ 1 and explore various topological properties of their singular fibers. A singular fiber is, roughly speaking, the inverse image f−1(y) for a singular value y ∈ N of f . We first explain how the notion of a singular fiber is mathematically formulated, and then we explore how the singular fibers are classified under various settings. Such classification results lead to various topological consequences, such as local topological structures of Reeb spaces, cobordism invariants for singular maps, etc. If time permits, we also explain how the notion of singular fibers can be used for data analysis, especially for developing new technologies for visualizing large scale data.

Mini course 2. TBA

Lecturer: Juan José Nuño Ballesteros

Abstract: TBA

Mini course 3. Geometry Meets Dynamics: Newton Polyhedra in the Study of Functions and Vector Fields

Lecturers: Regilene Oliveira, Thaís Dalbelo and Otavio Perez

Abstract: Newton polyhedra are objects of convex geometry that naturally generate subdivisions of the plane into convex polyhedral cones. These structures serve as a unifying framework connecting toric varieties, analytic functions, and planar vector fields. In toric geometry, they guide the resolution of singularities by producing smooth toric varieties through combinatorial and geometrically controlled methods. In the study of functions, Newton polyhedra encodes critical information about singularities and asymptotic behaviour, while in vector fields, they help classify local dynamics and topological structures.

The goal of this minicourse is to provide an accessible introduction to the use of the Newton polyhedron in planar vector fields and analytic functions, discussing both classical and recent results. A central theme will be the interplay between the study of functions and planar vector fields, highlighting how Newton polyhedra serve as a bridge between these domains. The course will focus on theoretical background as well as applications, with several examples provided throughout to help students grasp the profound concepts involved. Applications include the resolution of singularities of planar analytic vector fields, as well as the determination of topological normal forms near isolated singularities and near the infinity of the Poincaré–Lyapunov disk.