### Advanced Algebraic Geometry: Algebraic Surfaces - 8EC

Max Noether, one of the founding fathers of algebraic geometry, used to say that algebraic curves were created by God and algebraic surfaces by the Devil. This does reflect the complexity of the theory of algebraic surfaces compared to that of curves. However, surfaces are still much more accessible than higher dimensional algebraic varieties. This course will be an introduction to the theory of algebraic surfaces and a walk through their classification.

Topics include: intersection form, birational maps between surfaces, Castelnuovo's contractibility theorem, ruled surfaces, rational surfaces, plurigenera, Castelnuovo's Theorem, irregular surfaces, Kodaira dimension, K3+Enriques+abelian+bilelliptic surfaces, properly elliptic surfaces, general type surfaces, Castelnuovo's inequality.

Prerequisites

Algebraic Geometry I (or any course treating the equivalent of Hartshorne, Chapter I). Useful but not needed are basic knowledge of Hartshorne Chapters II, III, but a working knowledge of these chapters will be developed along the way. Literature: Complex algebraic surfaces, A. Beauville LMS student texts (1996).

Aim of the course

Discrete analogues of Fourier analysis have been critical to many important results in combinatorics, probability and additive number theory.

In recent years these techniques have led to many sophisticated and deep results in discrete mathematics, probability and related fields, ranging from "sharp thresholds" in random graphs and percolation theory, strong bounds for sum-sets in additive number theory, to the theory of computing and social choice.

In this course we will introduce the students to the beautiful theory of discrete harmonic analysis and we will cover selected applications in the above mentioned areas.

Prerequisites

Basic knowledge of discrete mathematics, probability theory and linear algebra; significant mathematical maturity consistent with a completed BSc degree in Mathematics. Having done probabilistic and extremal combinatorics or a similar course is desirable but by no means necessary.

There will be homework exercises that count for 40% of the final grade.

### Advanced Hamiltonian Dynamics - 8EC

Aim of the course

The theory of Hamiltonian systems is very rich and has given rise to many modern developments in mathematics. In this course we plan to present two of the main directions in the theory of Hamiltonian systems. The first one, related mostly to geometry, deals with symmetries in Hamiltonian systems and their consequences and focuses mostly on integrable Hamiltonian systems. The second one, related mostly to dynamics, deals with perturbations of integrable Hamiltonian systems and the methods, the perturbation methods that we have for understanding the dynamics, and the main results in this direction, that is, KAM and Nekhoroshev theory.

More specifically, the following topics will be treated:

Part A: Geometry

• Symplectic formalism
• Symmetries and group actions
• Symmetry reduction, Marsden-Weinstein theorem
• Integrable Hamiltonian systems (Arnold-Liouville theorem)
• Convexity theorems and Delzant classification of toric systems
• Hamiltonian monodromy

Part B: Dynamics

• Regularization of the Kepler problem
• Perturbation theory
• Birkhoff-Gustavson normal form
• Lie series
• KAM theory
• Nekhoroshev theory

Prerequisites

Bachelor level multivariable calculus, dynamical systems, topology and manifolds, including notions of vector fields, tangent spaces and differential forms. Basic knowledge of algebraic topology (homology and fundamental groups), Lie groups and algebras. In particular, familiarity with the material covered in Chapters 1-6 of “An introduction to manifolds”, 2nd edition by L. W. Tu, Springer, 2011 will be helpful, especially in Part A (see below), while basic concepts from dynamical systems theory (equilibria, stability, etc.) will be helpful in Part B.