Teacher: Khazgali Khozasov

Statistical concepts such as principal component analysis, empirical means, or covariance matrices are naturally defined for data and probability distributions in linear spaces. Geometric statistics develops tools for analyzing data that lie in (possibly) nonlinear spaces such as manifolds. Since the notion of a metric is essential for this purpose, Riemannian geometry provides a solid foundation for the theory.
 

Teacher: Michele Ancona

This course provides an introduction to the theory of Riemann surfaces, with an emphasis on their geometric, analytic, and cohomological properties. Students will explore fundamental constructions, classical theorems, and the tools needed to study complex curves from both analytic and algebraic viewpoints.
 

Programme

• Definition of Riemann surfaces; first examples
• Tangent bundle; holomorphic maps and meromorphic functions
• Riemann–Hurwitz formula
• Projective spaces and plane algebraic curves
• Plücker’s genus formula for plane curves
• Holomorphic line bundles and their sections
• Sheaf of sections and Čech cohomology
• Divisors and associated line bundles
• Statement of the Riemann–Roch theorem and Serre duality, with an outline of the proof

Teacher: Rémy Rodiac

Programme

Introduction to Distributions

• Definitions; differentiation in the sense of distributions
• Convolution of a distribution with a Cc1 function
• Fundamental solutions (example: the Laplacian)
• Tempered distributions; Fourier transform; application to solving the heat equation on Rn

Differential Forms and Hodge Theory on Rn

• Definition of differential forms
• Pullback of differential forms
• Integration of differential forms and Stokes’ theorem
• Hodge decomposition
   

Variational Problems

• Solving minimization problems and the link with Euler–Lagrange equations
• Example: Poisson’s equation on an open set
• First eigenvalue of the Laplacian
• Diagonalization theorem for compact self‑adjoint operators
• Applications to evolution PDEs on bounded domains (heat and wave equations)
  • Sobolev inequalities and embeddings
  • Poincaré and Poincaré–Wirtinger inequalities
  • Trace operators
  • Weak convergence in Sobolev spaces

Sobolev Spaces

• Definitions; density of smooth functions

Teacher: Emmanuel Militon

Topics Covered
 

• Topological dynamics: recurrence, minimality, periodic orbits; examples include circle rotations, torus translations, Anosov automorphisms, and symbolic systems.
• Conjugacy and semi‑conjugacy: Markov partitions, symbolic codings, introduction to KAM methods, Poincaré–Siegel theorem.
• Ergodic theory: invariant measures, Poincaré recurrence, Birkhoff’s ergodic theorem, ergodicity, unique ergodicity, mixing.
• Circle dynamics: classification of circle homeomorphisms, rotation numbers.
• Additional topics (time permitting).

Teacher: Ursula Ludwig

This course introduces the basic tools of analysis on manifolds. The focus is on analytic and geometric techniques used to study differential operators, heat kernels, and spectral properties on smooth manifolds. The material provides a foundation for the analytic aspects of the Atiyah–Singer index theorem for geometric Dirac operators, although the classical Dirac operator on a spin manifold is not defined in this course.
 

Techniques Covered

Analytic tools:

• Sobolev spaces
• Sobolev embedding
• Fredholm theory
• Heat operator
• Laplace‑type and Dirac‑type operators

Heat Equation and Index:

• Heat kernel
• Formal heat kernel
• Heat operator via functional calculus
• Weyl’s law (for the eigenvalue counting function)

Eigenvalues of the Laplacian:

• Simple lower bounds for λₖ
• Faber–Krahn inequality (for λ₁)
• Continuity and multiplicity of eigenvalues
• Isospectral manifolds

Analysis of Dirac and Laplace‑Type Operators

• Sobolev spaces on compact manifolds
• Dirac and Laplace‑type operators (definition, examples, Gauss–Bonnet and signature operator)
• Analysis of Dirac‑type operators:
  • Gårding inequality
  • Bochner formula
  • Elliptic estimates and elliptic regularity
  • Functional calculus
  • Friedrich mollifier
  • Smooth kernels
  • Resolvent via functional calculus
  • Recap: spectral theorem for compact self‑adjoint operators
  • Hodge theory

Programme

Differential Operators on Manifolds

• Differential operators on manifolds: definition and examples
• Vector bundles and connections (possibly restricted to the tangent bundle)
• Intrinsic definition of differential operators using connections
• Symbol of a differential operator; ellipticity
• Recap: de Rham operator and de Rham cohomology

 
Main Results

• Hodge Theorem
• Heat kernel and small‑time asymptotics of the heat trace (construction via formal kernel, spectral theory, and functional calculus)
• Spectrum of Laplace‑type operators and its properties
• Weyl’s law

Geometric tools:

• Connections and curvature
• (Possibly restricted to the Levi‑Civita connection on the tangent bundle)