The Undergraduate Summer School provides opportunities for talented undergraduate students to enhance their interest in mathematics. This program is open to undergraduates at all levels, from first-year students to those who have just completed their undergraduate education. There will be many organized activities, with some specifically targeted at students at the introductory level and others at more advanced students. There will also be time for study groups and individual projects guided by advisors, as well as other activities.
The 27th Annual PCMI Summer Session will be held June 25 – July 15, 2017.
Click HERE to apply to the Undergraduate Summer School program.
2017 Course Descriptions:
Antonio Auffinger, Northwestern University
Random matrices beyond random matrix theory
This course will focus on surprising and quite inspiring appearances of random matrices in other fields of mathematics. These include combinatorics, number theory, probability theory, statistical physics and beyond. We will discuss problems related to longest increasing subsequences, random tilings, connections to the Riemann zeta-function and spin glasses, among others.
We will give an introduction on each of these topics (no previous knowledge required!), focusing on questions where the distribution of eigenvalues plays (or is conjectured to play) a big role. Homework assignments will be a major part of the course.
Prerequisites: Linear algebra, probability theory, and real analysis at undergraduate level.
Mihai Stoiciu, Williams College
Introduction to Random Matrix Theory
Course Description: Initiated by research in multivariate statistics (Wishart, 1928) and nuclear physics (Wigner, 1955), the study of random matrices is nowadays an active and exciting area of mathematics, with numerous applications to theoretical physics, number theory, functional analysis, optimal control, and finance. Random Matrix Theory provides understanding of various properties (most notably, statistics of eigenvalues) of matrices with random coefficients.
This course will provide an introduction to the basic theory of random matrices, starting with a quick review of Linear Algebra and Probability Theory. We will continue with the study of Wigner matrices and prove the celebrated Wigner’s Semicircle Law. After this, we will turn our attention to Gaussian ensembles and investigate the Gaussian Orthogonal Ensemble (GOE) and the Gaussian Unitary Ensemble (GUE). In particular, we will derive the joint distribution of eigenvalues for GOE and GUE and discuss the spacing distributions of the spectrum for these ensembles. The last lectures of the course will be dedicated to random Schrodinger operators and their spectral properties (in particular, the phenomenon called Anderson localization).
Prerequisites : Multivariable Calculus, Linear Algebra, Probability Theory.
There will be a lot of other mathematics going on at PCMI. Participants in the USS will have the chance to meet and interact with mathematicians and math teachers from around the world. The Undergraduate Summer School is for all undergraduates or recent graduates.