Course Descriptions

16:640:573 - Special Topics Number Theory

Henryk Iwaniec

Subtitle:

Statistic of L-functions and their zeros

Course Description:

This will be about recent developments in the analytic theory of L-functions which is modeled on the Random MatrixTheory. Some basic results on the distribution of zeros and estimates for the moments of central values will be presented in detail. However a significant part of the material will be dealing with conjectures which only recently emerged from ideas of RMT.

Text:

Recent Perspectives in Random Matrix Theory and Number Theory, LMS Lecture Notes Series 322, Cambridge University Press

Prerequisites:

Fascination with the Riemann Hypothesis

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

Spring 2018

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

Fall 2017

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

Schedule of Sections: