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. On the Topology of Complex Map Germs

Lecturer: Juan José Nuño Ballesteros

Title: On the Topology of Complex Map Germs

Abstract:

In this minicourse, we will focus on some topological aspects of Singularity Theory. The starting point is Milnor’s fibration theorem, which concerns the topology of complex analytic maps near critical points. First, we will review the construction of the Milnor fibration of a holomorphic function germ f : (C^(n+1), 0) → (C, 0), as well as the definition of the Milnor number μ(f) in the case where f has an isolated singularity. Following Hamm, the second step is to generalize these constructions to holomorphic map germs f : (C^(n+k), 0) → (C^k, 0), such that the zero locus (X, 0) is an isolated complete intersection singularity (ICIS). As we shall see, the algebraic computation of the Milnor number of an ICIS is provided by the well-known Lê–Greuel formula.

These results can be extended in several directions. We will consider holomorphic map germs f : (X, 0) → (C^k, 0), where (X, 0) is an arbitrary complex analytic variety. We will say that f admits a Milnor–Lê fibration if there exists an induced locally trivial fibration X ∩ Bε ∩ f⁻¹(Bδ \ Δ) → Bδ \ Δ, for sufficiently small 0 < δ ≪ ε ≪ 1. Here, Bε and Bδ denote balls centered at the origin of radii ε and δ, respectively, and Δ ⊂ (C^k, 0) is a hypersurface. According to Lê, every function f : (X, 0) → (C, 0) admits a Milnor–Lê fibration. This is essentially due to the fact that f admits a stratification satisfying Thom’s condition. However, there are examples for which f does not admit a Milnor–Lê fibration when k > 1, even if X = C^N.

We will see that if f, g : (X, 0) → (C, 0) are two functions and g has an isolated stratified critical point with respect to f, then the map germ Φ = (g, f) : (X, 0) → (C², 0) admits a Milnor–Lê fibration. Moreover, it is possible to obtain a Lê–Greuel-type formula computing the Euler characteristic of the fibre χ(FΦ) in terms of χ(Ff) and the number of critical points of g restricted to Ff. This can be achieved using standard arguments from Stratified Morse Theory. Furthermore, when (X, 0) has an isolated singularity, we obtain an algebraic formula that naturally extends the classical Lê–Greuel formula.

Finally, we will review other Lê–Greuel-type formulas for space curves, isolated determinantal singularities, and image Milnor numbers of map germs with isolated instability (in the sense of Mond).

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.