Centro: Facultad de Ciencias



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Asignatura: Curso Avanzado de Análisis

Código: 30073

Centro: Facultad de Ciencias

Titulación: Máster en Matemáticas y Aplicaciones

Nivel: Master M2

Tipo: Elective

Nº de créditos: 8



ASIGNATURA / COURSE TITLE


Curso Avanzado de Análisis

1.1.Código / Course number


30073

1.2.Materia / Content area


Analysis applied to number theory

1.3.Tipo / Course type


Elective subject

1.4.Nivel / Course level


Master (second cycle)

1.5.Curso / Year


2012/2013

1.6.Semestre / Semester


2nd (Spring semester)

1.7.Número de créditos / Credit allotment


8 ECTS

1.8.Requisitos previos / Prerequisites


It is required to have followed courses of harmonic analysis, real analysis and complex analysis.

1.9.Requisitos mínimos de asistencia a las sesiones presenciales / Minimum attendance requirement


It is strongly recommended to attend class regularly.


1.10.Datos del equipo docente / Faculty data


Professor(s): Fernando Chamizo Lorente

Department of Mathematics

Facultad de Ciencias

Office - Módulo 307 - 17

Telephone: +34 91 497 7640

Email: fernando.chamizo@uam.es

Homepage: http://www.uam.es/fernando.chamizo

Office hours: Tuesdays and Thursdays 17:30, or by arrangement.


1.11.Objetivos del curso / Course objectives


At the end of the course, the student should:

  • have mastered the basic techniques of analysis as applied to number theory (E1, E4);

  • manage to relate different topics of the syllabus, realizing similarities and differences among them(E8, E11);

  • be able to elaborate and develop the course material, using visual and technical aids that will improve effective communication of mathematics results (E14).

1.12.Contenidos del programa / Course contents


CHAPTER I: INTRODUCTION .

  • Summation formulas.

  • Discrete harmonic analysis.

  • The distribution of prime numbers.


CHAPTER II: Oscillatory sums and integrals

  • The uncertainty principle.

  • Oscillatory integrals.

  • Oscillatory sums.


CHAPTER III: SOME ARITHMETIC APPLICATIONS.

  • The circle method.

  • Lattice point problems.

  • Sieve methods.


CHAPTER IV: SPECTRAL THEORY OF AUTOMORPHIC FORMS.

  • Spectral resolution of Laplacian.

  • Trace formulas.

  • Some applications.



Note: In Chapter III one of the points could be skipped or changed according to time requirements.

1.13.Referencias de consulta / Course bibliography


Montgomery, H.L. Ten lectures on the interface between analytic number theory and harmonic analysis. CBMS Regional Conference Series in Mathematics, 84. American Mathematical Society, Providence, RI, 1994.


H. Iwaniec, E. Kowalski. Analytic number theory. American Mathematical Society

Colloquium Publications, 53. American Mathematical Society, Providence, RI, 2004.


Graham, S. W.; Kolesnik, G. van der Corput's method of exponential sums. London Mathematical Society Lecture Note Series, 126. Cambridge University Press, Cambridge, 1991.
Körner, T. W. Fourier analysis. Second edition. Cambridge University Press, Cambridge, 1989.
Terras, A. Fourier Analysis on Finite Groups and Applications. Cambridge University Press, Cambridge, 1999.
F. Chamizo. Métodos analíticos en teoría de números. http://www.uam.es/fernando.chamizo/libreria/libreria.html
F. Chamizo. Temas de teoría de números. http://www.uam.es/fernando.chamizo/libreria/libreria.html
Dym, H.; McKean, H. P. Fourier series and integrals. Probability and Mathematical Statistics, No. 14. Academic Press, New York-London, 1972.
Ramakrishnan, D.; Valenza, R.J. Fourier analysis on number fields. Graduate Texts in Mathematics, 186. Springer-Verlag, New York, 1999.
J. Steuding. An introduction to the theory of L-functions. 2006.
Katznelson, Y. An introduction to harmonic analysis. Second corrected edition. Dover Publications, Inc., New York, 1976.

2.Métodos docentes / Teaching methodology


This is an advanced course and the students are expected to participate actively and originally. Many proofs and some topics will be just sketched and the details will be left to the interested students.
Problem set assignments: Regularly given, with a predetermined deadline for their completion.
Seminars and essays given by the students.

3.Tiempo de trabajo del estudiante / Student workload







de horas

Porcentaje

Attendance activities

Class lectures

40h (20%)

70 h (35%)

Office hours

18h (4%)

Seminars and essays

12h (5%)

Others

-

Final exam

2h (1%)

Non attendance activities

Problems preparation

78h (39%)

130 h (65%)

Weekly study

46h (23%)

Exam preparation

6h (3%)

Total workload: 25 horas x 8 ECTS

200








4.Métodos de evaluación y porcentaje en la calificación final / Evaluation procedures and weight of components in the final grade

1) Home assignments: 40%.

2) Final exam or extra activities: 40%.

3) In-class exercises, participation: 20%


The extra activities can encompass student seminars, essays, higher difficulty exercises, etc. They can substitute to the final exam.


5.Cronograma* / Course calendar


Semana

Week


Contenido

Contents



Horas presenciales

Contact hours


Horas no presenciales

Independent study time

1

Summation formulas

4

10

2

Discrete harmonic analysis

5

10

3

The distribution of prime numbers.

5

10 Problem collection

4

The distribution of prime numbers

5

10

5

The uncertainty principle

5

10

6

Oscillatory integrals

5

10 Problem collection

7

Oscillatory sums

5

10

8

The circle method

5

10

9

Lattice point problems

5

10

10

Sieve methods

5

10 Problem collection

11

Spectral resolution of Laplacian

5

10

12

Trace formulas

5

10

13

Some applications

5

16 Problem collection

*This course calendar is tentative.





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