# Course Descriptions

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

Henryk Iwaniec

### Subtitle:

Harmonic Analysis in Arithemtic

### Text:

I will use mostly my own notes which will be available for the students.

### Prerequisites:

No special knowledge of the subject matter is required, however some skill in applying classical analysis will be helpful.

### Course Description:

This course is about problems of arithmetical nature which can be treated by analytic methods, especially by harmonic analysis. These include the circle method (for counting solutions to diophantine equations), exponential sums methods (for counting integral points on varieties) and bilinear forms methods for counting prime numbers.

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Please note: the instructor and course information changes from semester to semester for this course number.  Specifics for each semester below.

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

None

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 (Weylâ€™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

## Contacts

Departmental Chair
Michael Saks