Courses
Abstracts
C1 : Gröbner basis theory is an important computational tool in commutative algebra and algebraic geometry. It yields a broad spectrum of computational methods in many areas of mathematics and not only. Our lectures provide an introduction to the theory of Gröbner bases exemplifying the computational part by using the computer algebra system SINGULAR.
C2 : Four lectures encompass two topics : 1. Fibrations of real analytic maps, and 2. Bifurcation of affine maps. Each of them represent active research branches, which we will try to make accessible to students. We will sketch the state of the art and a number of open problems, especially intended to those who want to start a PhD.
C3 : We start with several examples of optimization problems, discuss the notion of algebraic degree of optimization, and introduce several tools from Singularity theory which help us solve such problems. We define and compute the Euclidean distance degree and discuss applications, and the maximum likelihood degree with its variants, discussing their computation in relation to the Huh-Sturmfels involution conjecture.
C4 : We explain several new results in the following two topics 1. Local fibrations and their topology; bouquet theorems; Morse numbers, and 2. Polar degree of projective hypersurfaces. The lectures will introduce into some beautiful problems that started over 65 years ago with Milnor’s study of hypersurface singularities, in the complex and real setup, and that continue developing today in connexion with many problems (see e.g. course C3).
T2: We study the topology of the complements of curves in the complex projective plane through different methods, like the fundamental group and other invariants that can be obtained from it. In certain cases, we will gain information about the topology of the complement through the existence of algebraic morphisms from the plane curve complement to the complex projective line.
T1 : We present the geometry behind the following classical optimization problem: given a point in the ambient space and given an algebraic variety, find the closest point on the variety to the given point with respect to the squared Euclidean distance function. We will present a geometric invariant of the variety, namely the Euclidean Distance Degree, which describes the algebraic complexity of solving this constrained optimization problem. This topic is related to C3.
T3 : We discuss the interaction between algebraic properties (freeness and freeness-like properties) and various invariants belonging to the combinatorial type of reduced complex projective curves (for instance the set of irreducible components, their degrees, the set of singular points, the topological types). We focus on examples of simple curves, and of arrangements of lines in the complex projective plane.
Short talk by Gabriel Esteban Monsalve : Examples of splitting and of vanishing for polynomials of 2 variables
Short talk by Francisco Braun : Counter-examples to some Jacobian problems
Poster by Rodrigo Thomaz da Silva : Properties of polynomial submersions via their bifurcation set
Poster by György Tötös : A surgery formula for the topological Poincaré series
Poster by Zsolt Baja : Weighted homogeneous surface singularities and numerical semigroups