Combinatorial Number Theory, 7.5 Credits

Contents

Additive combinatorics, sometimes called combinatorial number theory, is a young and extremely active field of mathematical research. It has coalesced into an area in its own right about twenty years ago. It has since then developed tremendously, and continues to grow rapidly.
The aim of additive combinatorics is to understand additive structure in sets of integers. The questions studied are often deceptively easy to state, but quite difficult to prove. Here are four typical examples of such questions:

i.    A set A is called sum-free if it does not contain elements x,y, z with x+y=z. How many sum-free subsets of {1,2,... , n} are there?
ii.    What is the maximal size of a subset of {1,2,... n} containing no triple {a, a+d, a+2d} of equally spaced elements?
iii.    What is the largest subset of {1,2,..n} in which all pairs of elements have distinct sums?
iv.    What is the smallest subset A of {1,2,... n} such that every element of {1,2,.. n} not contained in A can be written as a sum of two elements from A?

Besides the simplicity and naturalness of the questions studied in additive combinatorics, the discipline has two extremely appealing features.

The first is that additive combinatorics uses a mixture of techniques from many different areas of mathematics, such as probability theory, graph theory, harmonic analysis, incidence geometry, algebraic geometry and ergodic theory. Through the study of additive combinatorics, one can learn many powerful tools that have important applications in other fields.

The second is that additive combinatorics has seen spectacular developments in recent years, with the ground-breaking research in the area recognized by the awards of two Fields medals, four European Mathematics Society prizes and an Abel prize. Some of the highlights of the field include Gowers's seminal proof of Szemerédi's theorem via higher order Fourier analysis and the celebrated Green-Tao theorem on arithmetic progressions in the primes. Even more recently, there have been exciting breakthroughs on the cap-set problem and the monochromatic sum-product problem, with short and elegant solutions.

In this course, we will introduce some of the fundamental methods, problems and techniques in additive combinatorics. High points will include at least three proofs of Roth’s theorem, discussion of the most recent breakthroughs in the area and proofs of several beautiful results in combinatorics with far-reaching applications.

Expected learning outcomes

For a passing grade, students must be able to:

Knowledge and understanding

• independently and in-depth account for the fundamental aspects of the Probabilistic Method
• independently and in-depth account for the fundamental theory of discrete harmonic analysis and its applications to number theory
• independently and in-depth account for central results and concepts in additive combinatorics such as Freiman homorphisms, Gowers norms, and statements of the theorems of Van der Waerden, Szemerédi and Green-Tao
• independently and in-depth account for the arguments involved in the Fourier-analytic, ergodic-theoretic and graph-theoretic proofs of Roth’s theorem

Skills and abilities

• apply the Probabilistic Method with minimal hints to problems in number theory and combinatorics
• state, prove and apply the Balogh-Szemerédi-Gowers theorem, Turán’s theorem, Sperner’s theorem, Ramsey’s theorem, the Szemerédi regularity lemma, and the Szemerédi-Trotter theorem.
• independently restitute at least one of the proofs of Roth's theorem

Required Knowledge

The course requires 90 ECTS in Mathematics. Proficiency in English equivalent to Swedish upper secondary course English 5/A. Where the language of instruction is Swedish, applicants must prove proficiency in Swedish to the level required for basic eligibility for higher studies.

Form of instruction

Teaching is mainly in the form of lectures.

Examination modes

The course is examined by a written exam. For the course, one of the following grades is assigned: Fail (U), Pass (G), Pass with distinction (VG).

A student who has received a passing grade on a test is not allowed to retake the test in order to receive a higher grade. A student who has not received a passing grade after participating in two tests has the right to be assigned another examiner, unless there are certain circumstances prohibiting this (see the Higher Education Ordinance, chapter 6, 22§). A request to be assigned another examiner should be addressed to the head of department for the department of mathematics and mathematical statistics. The possibility of being examined based on the current version of the syllabus is guaranteed for at least two years following the student's first participation in the course.

Credit transfer
All students have the right to have their previous education or equivalent, and  their working life experience evaluated for possible consideration in the corresponding education at Umeå university. Application forms should be adressed to Student services/Degree evaluation office. More information regarding credit transfer can be found on the student web pages of Umeå university, http://www.student.umu.se, and in the Higher Education Ordinance (chapter 6). If denied, the application can be appealed (as per the Higher Education Ordinance, chapter 12) to Överklagandenämnden för högskolan. This includes partially denied applications

Other regulations

In a degree, this course may not be included together with another course with a similar content. If unsure, students should ask the Director of Studies in Mathematics and Mathematical Statistics. The course can also be included in the subject area of computational science and engineering.