This year, we will cover selected topics in discrete harmonic analysis and its applications to additive combinatorics, probabilistic combinatorics, number theory, percolation, random graphs and theoretical computer science.
Discrete analogues of Fourier analysis have been critical to many important results in graph theory and additive number theory. In recent years these techniques have led to many sophisticated and deep results in discrete mathematics, probability and related fields, ranging from "sharp thresholds" in random graphs and percolation theory, strong bounds for sum-sets in additive number theory, to the theory of computing and social choice.
Basic knowledge of discrete mathematics, probability theory and linear algebra; significant mathematical maturity consistent with a completed BSc degree in Mathematics.
The course is aiming at students in theoretical and applied mathematics with interest in probability, graphs and networks, and emerging Network Science. The purpose of the course is to introduce the students to network phenomenon and its applications and to provide a solid background in its mathematical foundations within the theory of random graphs.
Network science is an exciting multidisciplinary research area that studies the network phenomenon in real-life systems (such the World Wide Web, transportation networks, neural networks in the brains, protein interactions, and social connections), and processes on networks, for example, infection and information spreading.
The first lecture will be a short introduction in most essential properties of real-life networks. We will learn why your friends usually have more friends than you do and address the famous small-world phenomenon and its formal description.
The major part of the course will be devoted to the theory of random graphs. Mathematically, it is natural to model a network as a graph of nodes connected by edges. The connections in these networks, such a friendships in social networks, are often a result of random events, therefore, the theory of random graphs is most suitable for their formal analysis. The methodology leverages on two important classes of stochastic processes: branching processes and martingales. We will first review the necessary probabilistic background and then cover three essential random graph models: Erdös-Rényi random graph, Configuration model, and Preferential Attachment model.
In the last part of the course we will cover two essential topics in Network Science: centrality measures in networks, including Google PageRank, and dependencies between degrees of neighbouring nodes.
The course will finish with hands-on research project and presentations.
The course is aimed at master students in theoretical and applied mathematics and possibly physics and computer science. Solid knowledge of probability theory is necessary. Programming skills are optional. Students who like programming and applied problems can put it to use in a research assignment and homework.
- Docent: Nelly Litvak
The aim of this course is to give an introduction to the Langlands programme. This programme, first formulated by R. P. Langlands in the late 1960s and developed by many people, links two different areas of mathematics: number theory (in particular Galois theory) on the one hand, and representation theory (more precisely automorphic forms) on the other hand.
On the "number-theoretical" side, one takes algebraic varieties (defined by polynomial equations with rational coefficients) and associates Galois representations to them. Fundamental examples of these are Tate modules of elliptic curves. On the "automorphic" side, one studies certain highly symmetric functions on Lie groups known as automorphic forms. The classical examples of these are modular forms.
To objects on both sides, one can attach L-functions, a certain kind of analytic functions similar to the Riemann zeta function. The Langlands programme then predicts, roughly speaking, that there is a correspondence between Galois representation and automorphic forms, under which two objects on both sides correspond to each other if they give rise to the same L-function.
In this course, we will introduce both sides of this correspondence and give examples of the kind of objects that occur. We then explain Langlands's conjectures, as well as the cases of them that have been proved. We will strive to give many concrete examples to illustrate the general ideas and techniques.
In this course, we will assume some familiarity with the following topics:
- basic knowledge of complex analysis (holomorphic/meromorphic functions, Taylor/Laurent series)
- basic knowledge of group theory (groups, (normal) subgroups, cosets, quotients, group actions)
- algebraic number theory (including Galois theory)
- algebraic geometry
Furthermore, knowledge of one or more of the following topics will be an advantage:
- elliptic curves (e.g. the course by Bright and Streng, autumn 2015)
- modular forms (e.g. the course by Bruin and Dahmen, spring 2016)
- p-adic numbers (e.g. the parallel course by Beukers and Dahmen, autumn 2016)
- representation theory
The course considers various aspects of queueing systems with Levy input, special cases being the classical M/G/1 queue and reflected Brownian motion. Introducing this class of queues as the solution of a Skorokhod problem, we subsequently characterize its stationary and transient distributions. Emphasis is put on the situation that the driving Levy process is spectrally one-sided; for the two-sided Wiener-Hopf theory is treated (but not in full detail). Then various other performance metrics are considered, such as busy periods and the workload correlation function. Explicit tail asymptotics are provided, which are derived by e.g. inversion techniques and change-of-measure arguments. Then the results are extended to networks of queues, e.g., tandems and feedforward networks. The course concludes by discussing applications in communication networks and mathematical finance.
Basic knowledge of probability theory and stochastic processes.