|Linear algebra||(~ 220 pages)||These are lecture notes for the year-long linear algebra class at ETH in the Fall Semester 2015 and Spring Semester 2016 for incoming mathematics and physics students. There is nothing original in the presentation, which is essentially targeted at giving a written detailed exposition of the class as I present it.|
|Probabilistic number theory||(~ 180 pages)||
These are lecture notes originally based on a class at ETH in
the Fall Semester 2015. The goal is to present some aspects of
probabilistic number theory that fall under the general
framework of convergence in law of "arithmetically defined"
probability measures to interesting limit measures. The main
specific results presented are: (1) Erdös(-Wintner) and the
Erdös-Kac Theorems; (2) Bagchi's Theorem on the functional
limit of vertical shifts of the Riemann zeta function; (3)
Selberg's Theorem on the normal behavior of the normalized
logarithm of the modulus of the Riemann zeta function on the
critical line (following Radziwill and Soundararajan); (4) the
Chebychev bias and (5) my own results with W. Sawin on the
statistic behavior of "Kloosterman paths".
These notes are scheduled to appear in book form in the Cambridge Studies in Advanced Mathematics series of Cambridge University Press (probably in late 2020 or early 2021).
Missing parts: More discussion of generalizations and further topics.
|Expander graphs||(~ 240 pages)||
These started as lecture notes for a class at ETH in the Fall Semester 2011, and were expanded later after various mini-courses and another class at ETH in 2015. They contain a general introduction to graphs and expansion in graphs, and a full account of the theorems of Helfgott and Bourgain-Gamburd concerning expansion in SL2(Z/pZ).
These notes will appear in book form in the series Cours Spécialisés of the SMF).
|Introduction to representation theory||(~ 300 pages)||
These are lecture notes for a class at ETH in the Spring Semester 2011, updated more recently with material for a "Lie Groups II" class in the Spring Semester 2013. These cover the basic formalism of representation theory (without assumptions on the group or field involved), the elementary representation of finite groups (mostly in characteristic coprime to the order of the group), the Peter-Weyl theory of representations of compact groups, and a few additional topics and applications. The last additions are some survey sections on representations of (some) non-compact groups.
Missing parts: The notes are now complete. The completed version has been published by the AMS.
|Exponential sums over finite fields, I: elementary methods||(~ 100 pages)||
These are lecture notes for a class at ETH in the Spring Semester 2010. They contain an introduction to elementary methods in the study of exponential sums over finite fields, with a proof of the Weil bound (i.e., special cases of the Riemann Hypothesis over finite fields) for character sums in one variable, based on a custom version of Bombieri's adaptation of Stepanov's method.
Missing parts: a few algebra lemmas on irreducibility of certain polynomials; proof of the bound for character sums involving rational functions (e.g., Kloosterman sums).
|Measure and integral||(~ 150 pages)||
These are lecture notes for a class at ETH in the Spring Semester 2010, containing an essentially standard course of measure theory and integration theory (with a smattering of probability). Up to minor adaptations and corrections, it is a translation of an earlier French lecture note (for a course given in Bordeaux in 2001-2002). J. Teichmann has also used this script in his lecture in Spring Semester 2012 and made a few corrections and additions; the link above corresponds to his version, whereas my older version can be found here.
Missing parts: a few facts concerning Fourier analysis are mentioned but not proved.
|Spectral theory in Hilbert spaces||(~ 130 pages)||
These are lecture notes for a class at ETH in the Spring Semester 2009, with a full treatment of the elementary spectral theory of linear operators acting on Hilbert spaces (starting from the spectral theorem for compact operators, which is not proved), covering normal bounded operators and normal unbounded operators. Applications include a very short introduction to the formalism of quantum mechanics.
Missing parts: (1) proof of the spectral theorem for compact operators; (2) Weyl law for the Laplace operator in bounded planar domains.
Last update 23.12.2019 by E. Kowalski