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JAE School of mathematics 2012

 

The 5th Edition of the JAE School of Mathematics will take place from 2nd July 2012 until 27th July 2012 in Aula Roja (2nd-3rd & 5th-6th July), Aula Gris 2 (4th July) and Aula Gris 1 (9th-27th July), at ICMAT.

The 2012 JAE-Intro students and another degree students who wish to participate, are invited to do so. They should apply for an invitation to the e-mail:  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 e-mail 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 (Vice-director 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
10:15-10:30 Opening        
10:30-12:00 TN
TN TN TN

GA

(11:00-12-00)

12:30-13:30 SH
SH
GA SH SH
13:30-15:00
Lunch Lunch Lunch Lunch  
15:00-17:00 SD
SD
SD
GA (15:00-16:00)  
           
Second Week Monday Tuesday Wednesday Thursday Friday
9th July 10th July 11th July 12th July 13th July
10:30-11:30 SH SH GA GA

GA (10:30-12:30)

11:30-13:30 SD Intro ICMAT Máster Univ Biblio
13:30-15:00
Lunch Lunch Lunch Lunch  
15:00-16:00 GA
GA SD (15:00-17:00) GA  
           
Third Week Monday Tuesday Wednesday Thursday Friday
16th July 17th July 18th July 19th July 20th July
11:00-13:00 IC IC IC IC IC
13:00-15:00
Lunch Lunch Lunch Lunch  
15:00-16:30 TN TN TN TN  
           
Fourth Week Monday Tuesday Wednesday Thursday Friday
23rd July 24th July 25th July 26th July 27th July
10:30-12:00 TN TN TN TN  
12:30-13:30
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. Hartman--Grobman 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 Frenchel-Nielsen 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 no-where 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 six-hour 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 (ICMAT-UAM): 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 (ICMAT-UAM): 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 (ICMAT-CSIC, School Director)

Daniel Faraco (ICMAT-UAM)

Javier Parcet (ICMAT-CSIC)

Aníbal Rodríguez Bernal (ICMAT-UCM)