Curso avanzado de análisis (máster) 2012/2013

Advanced course in Analysis

Notice:

Grades: grades_dni.pdf. They are displayed the grades in the last sheet, the average in the first sheets (S) and the average in the voluntary problems (V).
For any claim or clarification, please contact me.
A list by names is posted in Moodle.

Please, fill the survey through Moodle or SIGMA. It only takes few minutes and it is important for the Master Program.

The week 18th March - 22nd March professor J. Jiménez-Urroz will give some lectures (4 hours in total) belonging to this course. The schedule is as follows:
Tuesday 19th: 14:30-16:00, M17-320 (the room M17-520 is also reserved)
Wednesday 20th: 14:30-16:00, M17-320
Thursday 21st: 11:00-12:00, M17-520
In the first two lectures some arithmetic applications of the DFT will be shown, specially quadratic reciprocity. The last session is to introduce some basic analytic concepts regarding L-functions of elliptic curves.

In principle the language of the lectures will be Spanish (except if foreign students require otherwise) but I will try to post all the information and some material in English.

According to the web page of the Math Department, the course starts on Wednesday 30th January.

This a course on applications of Harmonic Analysis mainly to Analytic Number Theory. Occasionally we shall also treat other applications, specially to Computer Science and Physics.
The planned syllabus is the following

Introduction. Summation formulas. Discrete Harmonic Analysis. The distribution of prime numbers.
Oscillatory sums and integrals. The uncertainty principle. Oscillatory integrals. Oscillatory sums.
Some arithmetic applications. The circle method. Lattice point problems. Sieve methods (optional).
Spectral theory of automorphic forms. Spectral resolution of Laplacian. Trace formulas. Some applications.

There is a web page for general reference about master courses in the Math Department.

Notes

Complete first version of the lecture notes

Errata: In p.91 in the second displayed formula it should read e(bn/q).

The distribution the topics mentioned in the "guía docente" is:
1 (except prime numbers) -> §1.1, §1.3
2 -> §2.1, §2.2, §2.3.1
3 (without circle method) -> §2.3.2, §2.3.3
4 Not covered in the lecture notes. If time permits, I will cover part of these notes.

There will be posted here excerpts of the theory and some complementary material. Check the content of each lecture (frequently updated).

1.	Introduction

	1.1. Warming up
	1.1.1	The original Fourier series	Lecture notes (Spanish)
	1.1.2	Some theorems
	1.1.3	Some examples and graphics

	1.2. Learning to add
	1.2.1	Partial summation	Lecture notes (Spanish)
	1.2.2	Poisson summation formula
	1.2.3	Complex analytic techniques

	1.3. Discrete harmonic analysis
	1.3.1	Characters and transforms	Lecture notes (Spanish) (For JPEG check the dark room) (For Fileters check the dark room)
	1.3.2	Implementation of the DFA
	1.3.3	DFT in the real world

2.	Oscillatory sums and integrals

	2.1. The uncertainty principle
	2.1.1	Uncertainty for mathematicians and physicists	Lecture notes (Spanish)
	2.1.2	A glimpse on quantum mechanics	Lecture notes (Spanish)

	2.2. Oscillatory integrals
	2.2.1	The stationary phase principle	Lecture notes (Spanish)
	2.2.2	The mathematics of difraction	Lecture notes (Spanish)

	2.3. Oscillatory sums and integrals
	2.3.1	Van der Corput method	Lecture notes (Spanish)
	2.3.2	The large sieve
	2.3.3	Some applications to number theory

References and index (frequent updates)

Exercises

Home assignments of the course

Sheet 1	Sheet 2	Sheet 3	Voluntary Sheet
due March/6	due April/24		due May/22

Grading

Check moodle for the grades.

Check the Guía docente.

Say that H is the average in the sheets, V in the voluntary problems and E in the final examination. Then the grade is given by the formula

2+0.4H+0.4max(V,E)

Note that the final examination is not mandatory it will take place the 22nd of May at the room 420. This examination can also be employed to raise the grade: It will be taken as final grade if it more advantageous than the previous formula.