# Course Descriptions

## 16:640:573 - Special Topics Number Theory

### Subtitle:

Topics in Automorphic Forms and L-Functions

### Course Description:

A survey of automorphic forms will be given, buy the main goal is to present a variety of applications to problems in modern number theory.

### Text:

Henryk Iwaniec, Topics in Classical Automorphic Forms, Graduate Studies in Mathematics, Volume 17, AMS

### Prerequisites:

Basic knowledge of algebra and harmonic analysis

Prime Numbers

### Course Description:

The main topic of this course is the distribution of prime numbers. First I will cover traditional results such as
--the asymptotic formula for the number of primes up to x (elementary and analytic proofs)
--primes in short intervals
--primes in arithmetic progressions
The highlight of the course will be a complete proof of bounded gaps between primes (après James Maynard)
Primes represented by polynomials in two variables will be considered with sketch of proofs.

### Text:

I will follow selected research papers in the literature

### Prerequisites:

good skill in complex function theory

### Subtitle:

The zeros of L-functions

### Course Description:

The main objective of this course is to present fundamental problems about the zeros of the Riemann zeta function, the Dirichlet L-functions and L-functions of automorphic forms. The classical and most recent ideas (the random matrix theory) will be discussed in considerable details. The distribution of zeros on the critical line constitutes the highlight of the course, such as that the positive percentage of zeros satisfy the Riemann Hypothesis, or the statistical behavior (with respect to special families of L-functions) of zeros near the central point.

### Text:

H.Iwaniec and E. Kowalski, Analytic Number Theory, AMS Call 53, 2004

### Prerequisites:

Good skill in complex function theory and harmonic analysis.

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### Subtitle:

Exponential and Character Sums

### Course Description:

These are special topics which have applications to the Riemann zeta function and the Dirichlet L-functions.

### Text:

H.Iwaniec and E. Kowalski, Analytic Number Theory, AMS Call 53, 2004

### Prerequisites:

Skill in classic Fourier analysis

### Subtitle:

Spectral Theory of Automorphic Forms

### Course Description:

This course will cover the spectral resolution of the space of real-analytic automorphic forms. The highlight of the theory is the trace formula. Harmonic analysis of Kloosterman sums for congruence groups will be developed in conjunction with applications to analytic number theory.

### Text:

H. Iwaniec, SPECTRAL METHODS OF AUTOMORPHIC FORMS, AMS-GSM, Vol.53

### Prerequisites:

Skill in classic Fourier analysis

Sieve Methods

### Course Description:

Sieve methods are used for selecting and counting elements of an arithmetic sequence, such as prime numbers or solutions to special diophantine equations. I will present current state of sieve theory (combinatorial sieve, Selberg sieve, large sieve). Recent progress was made by implementation of harmonic analysis (estimates for exponential sums and bilinear forms) which will be included in the course.

### Text:

Opera de Cribro, AMS Colloquium Publications, Vol 57, by Friedlander and Iwaniec

### Prerequisites:

Knowledge of classical harmonic analysis and skill in complex function theory

### Subtitle:

Equations over Finite Fields

### Course Description:

Congruences of prime modulus p for polynomials in several variables can be viewed as equations over the field of p elements. This view point is powerful allowing to work in the field extensions. I will present analytic methods, mostly for algebraic curves, using exponential sums and the theory of L-functions. The highlight of the course will be a proof of the Riemann hypothesis for hyperelliptic curves. Numerous advanced topics will be presented in a survey fashion, in particular the L-functions of algebraic varieties

### Text:

Equations over Finite Fields, An Elementary Approach, W.M Schmidt, LNM 536

### Prerequisites:

Basic facts of algebra and harmonic analysis

Henryk Iwaniec

### Subtitle:

Diophantine Approximations and Transcendental Numbers

### Course Description:

This course concerns two subjects which are closely related: approximations of special numbers by algebraic numbers and theory of transcendental numbers. The main results will be covered in details, in particular :

-Roth theorem

-Baker theory oflinear forms of logarithms

Among several applications I will give a solution of the Gauss Class Number One Problem for imaginary quadratic fields.

No advanced knowledge of number theory is required, but participant's curiosity in special numbers will make the course enjoyable.

### Text:

J.W.S. Cassels, An Introduction to Diophantine Approximations, Alan Baker, Transcendental Number Theory

### Prerequisites:

None

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Henryk Iwaniec

### Subtitle:

Spectral theory of automorphic forms

### Course Description:

This will be a one semester course on automorphic forms from analytic point of view. The main topics are:

-spectral decomposition

- trace formula

-sums of Kloosterman sums

-distribution of eigenvalues of the Laplace operator (Weyl’s law, exceptional eigenvalues)

-distribution of Hecke eigenvalues

-hyperbolic lattice point problems

-application to equidistribution of roots of congruences

### Text:

Henryk Iwaniec, Spectral Methods of Automorphic Forms, AMS Grad.Stud. Vol.53, 2002

### Prerequisites:

Good knowledge of functional analysis and complex function theory will be helpful

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