Modular forms are a beautiful and central topic in number theory which proved to be a powerful tool for a wide range of applications. The theory of modular forms itself is very broad and combines perspectives from analytic number theory, algebraic geometry as well as representation theory.
Of the wide variety of applications the most prominent is certainly the proof of Fermat's Last Theorem. Other applications include explicit constructions of families of Ramanujan graphs as well as an unconditional proof of Linnik's theorem about the equidistribution of lattice points on spheres. More precisely, Linnik's result investigates the equidistribution of points

LaTeX: \left\{\left( \frac{x}{\sqrt{n}},\frac{y}{\sqrt{n}},\frac{z}{\sqrt{n}} \right) :  x,y,z,\in \mathbb{Z}, x^2+y^2 +z^2 = n \right\},

on the unit sphere in LaTeX: \mathbb{R}^3 as  LaTeX: n\to\infty.

This course will start with an introduction corresponding to the material from Chapter VII in Serre's book, based on which we will study Linnik's problem following the exposition in Sarnak's book.


Literature

  • D. Bump, Automorphic Forms and Representations, Cambridge University Press, 1998
  • F. Diamond and J. Shurman, A first course in modular forms, Springer, 2005
  • H. Iwaniec, Spectral Methods of Automorphic Forms, American Mathematical Society, 2002
  • J.S. Milne, Modular Functions and Modular Forms, Course notes available at www.jmilne.org/math/CourseNotes/mf.html
  • P. Sarnak, Some Applications of Modular Forms, Cambridge University Press, 1990
  • J-P. Serre, A Course in Arithmetic, Springer-Verlag New York, 1973.

The main text for the course will be "Statistical Learning with Sparsity: The Lasso and Generelizations" by Hastie, Tibshirani and Wainwright ( https://web.stanford.edu/~hastie/StatLearnSparsity/ )

The course is planned to cover most of Chapters 1--8 of this book. Additional material covering basics on convex sets, functions and convex optimisation will also be included, as well as additional material on proximal algorithms for solving convex optimisation problems relavant to the course contents.

The course will have a hands-on perspective, solving exercises and doing computer assignments, rather than plunging deep into theory.

Prerequisites are multivariable calculus, linear algebra, basic knowledge of optimisation, statistics including regression and preferrably logistic regression.