This year's program concerns Geometry, Topology and Dynamical Systems. There are important links between these fields, as an example one can cite the importance of gemetrical and topological methods in the study of dynamical systems. The courses will cover real and complex differential geometry, their link to partial differential equations, topology and dynamical systems. In particular we propose introductory courses to complex analysis in several variables, complex geometry and metric geometry (for instance metric hyperbolicity in the sense of Gromov), (pseudo-)Riemannian geometry (and its link to mathematical general relativity) and algebraic topology. The more advanced lectures deal with knot theory, topology in low dimensions and dynamical systems in dimension 3.
Introductive courses
Manifolds, Differential forms and Sheaves (Philippe Eyssidieux, Catriona Maclean and Pierre Py)
The purpose of this preliminary course is to give a lightning quick introduction to differentiable manifolds and their most accessible topological invariant, the De Rham cohomology. The construction will then be interpreted in terms of sheaf cohomology, the central tool of the Cartan-Serre approach to complex algebraic and analytic geometries.
Tentative program:
1. Differentiable Manifolds and differentiable mappings. Examples and construction methods.
2. Vector fields and differential forms. Exterior differential Calculus. De Rham Cohomology.
3. Sheaves. Abelian Sheaves. Basic Homological Algebra. Cohomology of sheaves.
References:
[1] S. Gallot, D. Hulin, J. Lafontaine, Riemannian Geometry Universitext, Springer (2004).
[2] R. Godement Topologie algébrique et théorie des faisceaux, Actualités scientifiques et industrielles 1252, Hermann (1997).
[3] M. Spivak A comprehensive introduction to Differential Geometry Publish or Perish (1999).
Complex Algebraic Geometry (Catriona Maclean)
This course will define and study affine and projective varieties over a base field K, by which we mean subsets of the affine space or the projective space of dimension n defined by algebraic equations. Various tools will be developped to study these objects, such as their structure sheaves, local rings, dimension theory and tangent spaces. The course will culminate in the study of sheaf cohomology of projective varieties.
Time permitting, some aspects of toric geometry may be introduced in the latter part of the course.
Tentative program:
1. Affine algebraic sets, their ideals and the Nullstellensatz
2. Projective algebraic sets and projectivisation of an affine algebraic set. Projective ideals.
3. Structure sheaves of affine and projective varieties, sheaves of modules.
4. Morphisms of algebraic varieties
5. Dimension, tangent spaces and singular points.
6. Sheaf cohomology of varieties.
7. Toric varieties.
References:
[1] D. Perrin, Algebraic Geometry, an Introduction, Universitext, Springer (2008).
Complex Differential Geometry (Philippe Eyssidieux)
Complex differential geometry arose in the 1940s in the work of Hodge and Kodaira as a way to use the methods of global differential geometry to study complex algebraic manifolds and more general complex manifolds. Its characteristic feature is to combine the ideas of Riemannian Geometry with the theory of holomorphic functions of several variables. The course will introduce the basic metric concepts of complex differential geometry and give their main application to the topology of complex projective manifolds (the Hodge-Lefschetz package). Then it will focus on their application to the study of Dolbeault cohomology and on the numerous applications of the L2 estimates for the Cauchy- Riemann equations.
Tentative program:
1. Complex Manifolds and Vector bundles. Hermitian metrics. Connections. Chern connection, Levi-Civita connection, curvature tensors. Kähler Manifolds.
2. Hodge theorem. Hodge-Lefschetz package.
3. Dolbeault cohomology. Curvature and Vanishing theorems.
4. The Kodaira embedding Theorem.
References:
[1] J.-P. Demailly, Complex Analytic and Differential geometry, https ://www-fourier.ujf-grenoble.fr/ de- mailly/manuscripts/agbook.pdf.
[2] P. Griffiths, J. Harris, Principles of algebraic geometry, Wiley & sons (2014).
[3] N. Mok, Metric Rigidity Theorems on Hermitian Locally Symmetric Manifolds, World Scientific (1989).
[4] C. Voisin, Hodge Theory and Complex Algebraic Geometry I, Cambridge Studies in advanced Mathematics 76, Cambridge University Press (2010).
Lie groups, Lattices (Pierre Py)
The aim of this course is to give an introduction to the theory of Lie groups and their lattices.
We will start by defining Lie groups as well as the basic objects of the theory (Lie algebras, adjoint representation, exponential map), and will then study many examples. We will then learn some basics about the structure of Lie algebras, the covering maps between Lie groups, and the correspondence between Lie groups and Lie algebras.
In the second half of the course, we will focus on locally compact groups, on their Haar measures, and on the existence of invariant measures on their homogeneous spaces. We will then define the notion of a lattice in a locally compact group, and will present many examples (e.g. in semisimple groups and in nilpotent groups). We will conclude by a complete study of lattices in nilpotent Lie groups (Malcev theory).
It is recommended to follow this course as a prerequisite for M. Deraux’s advanced course.
Tentative program:
1. Lie groups, closed subgroups of linear groups
2. (A tiny bit of) Structure of Lie groups and Lie algebras 3. Locally compact groups, Haar measures, lattices
4. Generalities on lattices, examples
5. Lattices in nilpotent Lie groups
6. Examples of lattices in semisimple Lie groups
References:
[1] Y. Benoist, Five lectures on lattices in semisimple Lie groups, in Géométries à courbure négative ou nulle, groupes discrets et rigidités, Séminaires et Congrès 18 (2009).
[2] A. Borel, Introduction aux groupes arithmétiques, Actualités Scientifiques et Industrielles 1341, Hermann (1969).
[3] A. W. Knapp, Lie groups beyond an introduction, second edition, Progress in Mathematics 140, Birkhäuser, Boston (2002).
[4] M. S. Raghunathan, Discrete subgroups of Lie groups, Ergebnisse der Mathematik und ihrer Grenzgebiete 68, Springer-Verlag, New York-Heidelberg (1972).
Advanced courses
Random Complex Geometry (Damien Gayet)
The complex smooth hypersurfaces of the complex n-dimensional projective space of given degree d are the zero sets of homogeneous polynomials of degree d in n + 1 complex variables. For a fixed d, these hypersurfaces have all the same topology. For instance, when n = 2, they are closed connected Riemann surfaces of genus half of (d − 1)(d − 2). However, their geometry depends a lot on the defining polynomial. For large d, they can be concentrated near a hyperplane, or tend to fill out the ambient space. When we choose the polynomial at random (there exists a natural measure for any d), one would like to understand the average behaviour of the random hypersurface. Here, "behaviour" can describe various analytical, topological or geometric observables : the current defined by the hypersurface, the number of connected components of it in a fixed ball, the sign of the Ricci curvature for the restriction of the ambient metric, etc. One surprising fact is that these probabilistic methods can provide deterministic results. The goal of this course is to introduce this subject, at the intersection of complex algebraic geometry, differential and Riemannian geometry and probability. If time allows, we will generalize this to random divisors in projective manifolds.
References:
[1] P. Griffiths, J. Harris, Principles of algebraic geometry, Wiley & sons (2014).
[2] X. Ma, G. Marinescu, Holomorphic Morse inequalities and Bergman kernels, Progress in Mathematics 254, Birkhäuser (2007).
[3] B. Shiffmann, S. Zelditch, Distribution of zeros of random and quantum chaotic sections of positive line bundles, Communications in Mathematical Physics 200 (1999), 661–683.
Complex geometry and discrete groups (Martin Deraux)
The goal of this course will be to investigate various constructions of ball quotients of finite volume : arithmetic constructions, explicit generating sets, and uniformization.
The complex unit ball, equipped with the Bergman metric, is the Hermitian symmetric space of negative constant holomorphic sectional curvature. The fundamental group of any n-dimensional finite volume ball quotients embeds as a finitely presented (but infinite!) discrete subgroup of its group of biholomorphisms, which is isomorphic to the Lie group PU(n,1). However, it is however very difficult to determine when a giving set of matrices generate a discrete subgroup.
The arithmetic construction gives a general way to construct such discrete subgroups (finiteness of the volume then follows from a theorem of Borel-Harish Chandra), but it is usually very difficult to study detailed properties of the corresponding ball quotients, to find explicit generating sets, or to work out group presentations for their fundamental group.
In the cocompact (torsion-free) case, one can also think of the quotient as a Kähler manifold of general type satisfying the equality case in the Miyaoka-Yau inequality, which allows to use tools from (complex) algebraic geometry to characterize and construct ball quotients, but it is often difficult to check whether the corresponding examples come from an arithmetic construction.
We will explain these three classes of constructions, and present some techniques that allow us to build bridges between the three classes of constructions. Among others, we will present the state of the art for building non-arithmetic ball quotients.
The uniformization techniques require knowledge of basic algebraic geometry, while most others require some basic geometry and topology and a bit of elementary number theory (which are not considered as a prerequisite for this course).
References:
[1] P. Tretkoff, Complex Ball Quotients and Line Arrangements in the Projective Plane, Mathematical Notes, Princeton University Press (2016).
[2] R. P. Holzapfel, Ball and Surface Arithmetics, Aspects of Mathematics 29, Springer (1998).
[3] M. Deraux, J. Paupert, J. Parker, New non-arithmetic complex hyperbolic lattices (I&II) Inventiones Mathematicae 203 (2016), 681-–771, Michigan Math. J. 70 (2021), 133-205.