Jiří Lebl

Singular Levi-flat hypersurfaces

Levi-flat hypersurfaces appear as natural objects at the intersection of several different problems in several complex variables. They are pseudoconvex from both sides, they are invariant sets of holomorphic foliations, and they are simply the flat manifolds from the point of view of CR geometry. Singularities appear naturally and are a current topic of active research. We will assume basic theory of several complex variables, and we will cover the more specialized background necessary, including CR geometry, the real-algebraic and analytic geometry, and basics of holomorphic foliations, to get to the current state of the art of Levi-flat hypersurfaces. The main technique we will consider in studying Levi-flat hypersurfaces is the Segre variety and the related techniques, which found many other uses in CR geometry.

Purvi Gupta

Attaching holomorphic discs to real submanifolds in complex spaces

Given a set K in ℂn, a holomorphic disc attached to K is a nontrivial holomorphic map from the complex one-dimensional unit disc into ℂn that maps the boundary of the disc into K. The presence of such discs (individually, or as leaves of a foliation) is responsible for several phenomena in  complex analysis of several variables, such as the (non)solvability of linear partial differential equations, simultaneous analytic extensions, lack of polynomial approximability, regularity of the homogeneuous complex Monge–Ampère equation, etc. Furthermore, holomorphic discs generalize to the notion of J-holomorphic curves, which have played a significant role in symplectic and (almost) complex geometry. 

In this course, we will elaborate on some of the techniques used in proving the existence, regularity, and abundance of holomorphic disks attached to certain special submanifolds in ℂn. Specifically, we will focus on the case of totally real submanifolds, where the question can be formulated as a nonlinear Riemann--Hilbert problem. While all the technical details will not be provided, we will emphasize the recurring tools in this subject. 

Sahil Gehlawat

An Introduction to foliation theory

The aim of this minicourse is to provide an introduction to the fascinating theory of foliations. A foliation can be understood as a decomposition of a manifold M into a collection of disjoint submanifolds, known as leaves, which share the same dimension. We will begin by exploring real foliation theory, discussing fundamental properties and presenting significant examples. Following this, we will shift our focus to holomorphic foliation theory, specifically examining holomorphic foliations of dimension 1 and codimension 1, including those that may have singularities. We will look at various equivalent ways of constructing such foliations. This part of the course will cover some rigidity properties of holomorphic foliations, such as the Identity Principle. At last, we will detail the classification of all possible singular holomorphic foliations on the complex projective plane ℂℙ2.