The 5th Edition of the JAE School of Mathematics will take place from 2nd July 2012 until 27th July 2012 in Aula Roja (2nd3rd & 5th6th July), Aula Gris 2 (4th July) and Aula Gris 1 (9th27th July), at ICMAT.
The 2012 JAEIntro students and another degree students who wish to participate, are invited to do so. They should apply for an invitation to the email: esther.fuentes(a)icmat.es before June 27th, apart from a refference letter of a University lecturer or researcher, who will send personally to the same email address, at the end of June.
In order to receive the JAE School certificate, every student has to attend at least 3 courses. This is the minimum required, but they can register as many courses as they want.
Courses:
 José Maria Arrieta, Rosa Pardo y Aníbal Rodríguez Bernal: Qualitative and quantitative study on dynamical systems (10 hours) (SD)

Tomás L. Gómez y Marina Logares: Algebraic Geometry (10 hours) (GA)

Francisco Presas Mata: Periodic orbits of Hamiltonian systems (10 hours) (SH)

Carlos Palazuelos y Nacho Villanueva: Quantum Information Theory (10 hours) (IC)
 Antonio Córdoba, Javier Cilleruelo y Florian Luca: Number Theory (18 hours) (TN)
Other activities:

Apertura e inscripción en la Escuela JAE de Matemáticas. (Opening)

Rafael Orive (Vicedirector of ICMAT): Presentación “Qué es el ICMAT”. Líneas de investigación. Tesis doctorales en el ICMAT. (Intro ICMAT)

Presentación de másters de matemáticas de universidades (UAM, UC3M y UCM). (Máster Univ)

Curso rápido sobre el uso de recursos bibliográficos de la biblioteca. (Biblio)
Schedule:
First Week  Monday  Tuesday  Wednesday  Thursday  Friday 
2nd July  3rd July  4th July  5th July  6th July  
10:1510:30  Opening  
10:3012:00  TN 
TN  TN  TN 
GA (11:001200) 
12:3013:30  SH 
SH 
GA  SH  SH 
13:3015:00 
Lunch  Lunch  Lunch  Lunch  
15:0017:00  SD 
SD 
SD 
GA (15:0016:00)  
Second Week  Monday  Tuesday  Wednesday  Thursday  Friday 
9th July  10th July  11th July  12th July  13th July  
10:3011:30  SH  SH  GA  GA 
GA (10:3012:30) 
11:3013:30  SD  Intro ICMAT  Máster Univ  Biblio  
13:3015:00 
Lunch  Lunch  Lunch  Lunch  
15:0016:00  GA 
GA  SD (15:0017:00)  GA  
Third Week  Monday  Tuesday  Wednesday  Thursday  Friday 
16th July  17th July  18th July  19th July  20th July  
11:0013:00  IC  IC  IC  IC  IC 
13:0015:00 
Lunch  Lunch  Lunch  Lunch  
15:0016:30  TN  TN  TN  TN  
Fourth Week  Monday  Tuesday  Wednesday  Thursday  Friday 
23rd July  24th July  25th July  26th July  27th July  
10:3012:00  TN  TN  TN  TN  
12:3013:30 
SH  SH  SH  SH 
Programme:

José Maria Arrieta, Rosa Pardo y Aníbal Rodríguez Bernal: Qualitative and quantitative study on dynamical systems (10 hours) (SD)
Programme:
1. Introduction
2. Local Dynamics
2.1. Equilibria. Linearization. HartmanGrobman Theorem . Local invariant manifolds.
2.2 Periodic solutions. Non autonomous systems: Poincare map. Autonomous systems: Poincare sections.
3. Global dynamics
3.1. Absorbent sets. Omega limit sets. Unstable sets of invariant sets. Attractors.
4. Robustness
4.1. Persistence of equilibira, periodic solutions and attractors under erturbations

Tomás L. Gómez y Marina Logares: Algebraic Geometry (10 hours) (GA)
Programme:
1. Hilbert Nullstellensatz.
2. Calculations: the genus of a plane curve
3. Moduli of curves: the FrenchelNielsen coordinates.
4. Introduction to "geometric invariant theory"
5. Introduction to the moduli space of fiber.
Bibliography:
Fulton: Algebraic Curves

Francisco Presas Mata: Periodic orbits of Hamiltonian systems (10 hours) (SH)
Abstract:The Seifert conjecture states that there are nowhere vanishing vector fields in the $3$sphere that do not posses a periodic orbit. The course explains the history of the conjecture and its relations with symplectic geometry.
Programme:
1. Dynamic sytems in manifolds: fundamental concepts.
2. The Seifert conjecture. Smooth, differenctiable and null divergence counterexamples.
3. Hamiltonian dynamics. Basics properties of a hamiltonian flow. Basis of symplectic geometry : convex hypersurfaces.
4. The Seifert conjecture in convex hamiltonian systems. Case of the Euclidean space. General case: a dense set of level hypersurfaces admits periodic orbits.
5. Hamiltonian counterexamples to the Seifert conjecture.

Carlos Palazuelos y Nacho Villanueva: Quantum Information Theory (10 hours) (IC)
Programme:
1. Introduction. Comparation between classic and quantum resources.
2. Postulates for Quantum information . Heisenberg uncertainty principle.
3. CHSH inequality.
4. Tensor rules and Bells inequalities.
5. Shor's algorithm.
6. BB84 Protocol.
7. Cryptography based on violation of Bell inequalities.

Antonio Córdoba, Javier Cilleruelo y Florian Luca: Number Theory (18 hours) (TN)
Description: This course will consist of three sixhour courses given by three different researchers. There will be a variety of topics with problems and tecniques related to Analytic Theory and Enumarative combinatory.
Antonio Córdoba (ICMATUAM): Analytic number theory.
Programme:
1.Riemann zeta function
2.Prime numbers in arithmetic progression theorem
3.Reticle points in curves and circles
4.Trigonometric series of Van der Corput and Vinogradov 's method
5.Poisson's summation formula applications
6.Goldbach and Waring problem
Javier Cilleruelo (ICMATUAM): Combinatorial number theory.
Programme:
1.Roth's Theorem
2.Sidon sets
Florian Luca (UNAM, México): Multiplicative number theory.
Programme:
1.Brun's Sieve
2.Prime counting chains
3.Davenport's Constant
4.Infinity test of Carmichael numbers
5.phi(n)=sigma(m) equation
Organizers:
Luis Álvarez Cónsul (ICMATCSIC, School Director)
Daniel Faraco (ICMATUAM)
Javier Parcet (ICMATCSIC)
Aníbal Rodríguez Bernal (ICMATUCM)