Cancelled Random Matrices
Course content
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.
MSc Mathematics
MSc Mathematics with a minor subject
Knowledge:
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Concepts and techniques of high dimensional analysis
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Classical random matrix models
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Limit theorems for eigenvalues
Skills
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Ability to identify relevant observables of matrix spectra.
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Use of advanced mathematical tools (resolvent techniques, moment method, Dyson Brownian motion) to access such observables.
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To rigorously prove limit theorems in high dimensional interacting systems.
Competencies
- To understand and analyze spectral properties (eigenvalues and eigenvectors) of high dimensional random matrices.
5 hours of lectures and 3 hours of exercises per week for 7 weeks.
Standard knowledge in Analysis and Linear Algebra. Basic knowledge of Probability Theory is recommended.
- accessible also to students of physics
- ECTS
- 7,5 ECTS
- Type of assessment
-
Oral examination, 30 min under invigilationAn oral examination without preparation time.
- Aid
- Without aids
- Marking scale
- 7-point grading scale
- Censorship form
- No external censorship
Several internal examiners
Criteria for exam assessment
See learning outcome.
Single subject courses (day)
- Category
- Hours
- Lectures
- 35
- Preparation
- 149
- Exercises
- 21
- Exam
- 1
- English
- 206
Kursusinformation
- Language
- English
- Course number
- NMAK21007U
- ECTS
- 7,5 ECTS
- Programme level
- Full Degree Master
- Duration
-
1 block
- Placement
- Block 1
- Schedulegroup
-
A
- Capacity
- No limits
- Studyboard
- Study Board of Mathematics and Computer Science
Contracting department
- Department of Mathematical Sciences
Contracting faculty
- Faculty of Science
Course Coordinator
- Torben Heinrich Krüger (2-837a4f7c7083773d7a843d737a)
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