Courses
Viviana Ene & Ayesha Qureshi C1: An Introduction to Gröbner Bases and their ApplicationsCezar Joița C2: Fibrations of Real Maps and Applications to BifurcationsLaurențiu Maxim C3: Applications of Singularity Theory to OptimizationMihai Tibăr C4: Topics on Singular Complex Hypersurfaces Invited LecturesAlex Dimca* Free curves in Algebraic GeometryEva Elduque Hodge theory of smooth algebraic varietiesAnca Măcinic Line arrangements and their combinatorics
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 be accessible to students. We will give 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).