NMAK21007U Random Matrices
MSc Mathematics with a minor subject
The theory of random matrices has its origin in applications in mathematical statistics and nuclear physics in the first half of the 20th century. It has become a paradigm for the study of high dimensional non-commutative disordered systems that connects numerous subfields of mathematics and physics, such as probability theory, high dimensional analysis, combinatorics, quantum and statistical physics with diverse applications e.g. to communication theory, condensed matter and high energy physics, number theory and neural networks.
We will provide an introduction to random matrices and learn basic concepts and techniques that are used to analyze them. In particular, we will introduce prominent models such as the Gaussian Unitary Ensemble, Invariant Ensembles and Wigner matrices. We will show that despite having random entries their spectral densities become approximately deterministic with increasing dimension and we will study fine details of their eigenvalue distributions. We will interpret the eigenvalues as an interacting particle system (Dyson Brownian motion) and show that its fast approach to local equilibrium implies universal spectral statistics across a wide range of random matrix models, one of the hallmarks of the theory. Finally we will discuss applications.
Concepts and techniques of high dimensional analysis
Classical random matrix models
Limit theorems for eigenvalues
Ability to identify relevant observables of matrix spectra.
Use of advanced mathematical tools (resolvent techniques, moment method, Dyson Brownian motion) to access such observables.
To rigorously prove limit theorems in high dimensional interacting systems.
- To understand and analyze spectral properties (eigenvalues and eigenvectors) of high dimensional random matrices.
- 7,5 ECTS
- Type of assessment
- Oral examination, 30 min under invigilationAn oral examination without preparation time.
- Exam registration requirements
To participate in the exam 5-7 compulsory assignments given during the course must be approved and valid.
- Without aids
- Marking scale
- 7-point grading scale
- Censorship form
- No external censorship
Several internal examiners
The same as the ordinary exam.
If the compulsory assignments were not approved during the course, they must be handed in and approved no later than three weeks before the beginning of the reexamination week.
Criteria for exam assesment
See learning outcome.
- Course code
- 7,5 ECTS
- Full Degree Master
- 1 block
- Block 1
- Course capacity
- No limits
- Course is also available as continuing and professional education
- Study board
- Study Board of Mathematics and Computer Science
- Department of Mathematical Sciences
- Faculty of Science
- Torben Heinrich Krüger (2-837a4f7c7083773d7a843d737a)