Advanced topics in mathematics 4

lmat2940  2023-2024  Louvain-la-Neuve

Advanced topics in mathematics 4
5.00 credits
0 h + 22.5 h
Q2

This biannual learning unit is being organized in 2023-2024
Teacher(s)
Ramos Gonzalez Julia;
Language
English
Prerequisites
Depending on the subject, mathematics skills at the level of the end of the Bachelor in Mathematics or first year Master in Mathematics.
Main themes
The topic considered varies from year to year depending on the research interests of the course instructor.
Learning outcomes

At the end of this learning unit, the student is able to :

1 Contribution of the course to learning outcomes in the Master in Mathematics programme. By the end of this activity, students will have made progress in:
  • Show evidence of independent learning.
  • Analyse a mathematical problem and suggest appropriate tools for studying it in depth.
  • Begin a research project thanks to a deeper knowledge of one or more fields and their problematic issues in current mathematics. He will have made progress in:
    • Develop in an independent way his mathematical intuition by anticipating the expected results (formulating conjectures) and by verifying their consistency with already existing results.
    • Ask relevant and lucid questions on an advanced mathematical topic in an independent manner.
Learning outcomes specific to the course.
The course aims to initiate research in the field under consideration. Specific learning outcomes vary depending on the field.
 
Content
1. Broad introduction to Random Matrix Theory.
2. Random matrix models with unitary symmetry and their relation to Orthogonal Polynomials.
3. Brief introduction to Determinantal Point Processes.
4. Steepest descent method for contour integrals in the complex plane, applications to Airy and Hermite functions.
5. The Gaussian Unitary Ensemble: Wigner's semicircle law, local statistics in the bulk and at the edge of the spectrum.
6. Riemann-Hilbert approach to Orthogonal Polynomials and Universality for random matrix models with unitary symmetry.
Teaching methods
Lectures with active participation from the students. Some exercises will be suggested during the course and discussed in class if the students wish.
Evaluation methods
Written + oral examination to assess the practical and theoretical skills acquired by the students.
Online resources
Moodle page
Faculty or entity
MATH


Programmes / formations proposant cette unité d'enseignement (UE)

Title of the programme
Sigle
Credits
Prerequisites
Learning outcomes
Master [120] in Mathematics
MATH2M
5