Second Chicago Summer School In Analysis
June 16 - July 3, 2015
This is the second series of NSF funded summer schools in analysis
at the university of Chicago. It intends to introduce advanced
undergraduates as well as beginning graduate students to a broad range of topics
which are important to modern analysis. This includes Partial Differential Equations, Solitons, Probability and Stochastic
Analysis, Numerical Analysis, Harmonic Analysis and Pseudodifferential
Operators and Geometric Measure Theory. The program focuses on foundational material,
and should be accessible to undergraduate and graduate students with a solid background in multivariable calculus,
complex variables, measure theory, and basic functional analysis (such as Hilbert spaces).
Organizers: M. Csornyei, C. Kenig, R. Fefferman,
W. Schlag, L. Silvestre, P. Souganidis.
Check the poster. For questions, write to chicagoanalysis@math.uchicago.edu
The registration to this summer school is now
closed. Per NSF regulations RTG funding is restricted to US citizens and permanent residents. Housing will be available in the university dormitories only for those participants receiving financial aid. Participants who do not qualify for financial aid will be responsible for their own accommodations. The deadline to apply for financial support is on
March 31st.
Schedule of lectures 
New!
All lectures take place at room 112 in the Stevanovich center.
List of courses
Minicourse by Antonio Auffinger.
Introduction to percolation and growth models. description
This course is an introduction to percolation, first-passage percolation and some examples of growth processes. For the past 50 years, these probabilistic models have provided a large collection of problems, remarkable phenomena and open questions to the whole mathematical community. Our goal is to present some of the main tools used to tackle these problems. Some of the topics covered in class: Percolation, FKG and BK inequalities, Russo’s Formula, RSW theorem, critical probabilities, Sub-additve Ergodic Theorem, order of fluctuations of the limit shape, geodesics and connections to stochastic homogenization. The course is aimed at advanced undergraduate and beginning graduate students. A first course in probability theory is the only pre-requisite.
References:
- Grimmett - Percolation (book)
- Grimmett - The Random Cluster Model (book)
- Grimmett - Percolation since Saint Flour (short survey of recent progress)
- Kesten - Aspects of First Passage Percolation (article/survey)
- Course notes.
Minicourse by Marianna Csornyei.
Tangents of sets and differentiability of functions and measures. description
One of the classical theorems of Lebesgue tells us that Lipschitz functions on the real line are differentiable almost everywhere. We study possible generalisations of this theorem and some interesting geometric corollaries.
References: There is no canonical reference for the course. All needed material will be introduced in the course.
Minicourse by Robert Fefferman.
An Introduction to Fourier Series. description
This will be a relatively self-contained class on Fourier series. We shall investigate
the basic properties of these series, as well as their convergence and divergence
properties. In so doing, we shall discuss how the analysis of Fourier series relates to
central areas of mathematics such as functional analysis, complex analysis and
probability theory.
Prerequisites: A knowledge of Lebesgue measure and elementary complex analysis.
References: Trigonometric Series, A. Zygmund; An Introduction to Harmonic Analysis, Y. Katznelson
Minicourse by Carlos Kenig.
Pseudodifferential operators with applications to linear Schrodinger equations. description
We will introduce the classical theory of pseudodifferential
operators proving boundedness properties, composition and adjoint rules
and the Garding inequality. Applications of these results will be given,
in particular to smoothing effects and well-posedness of linear
Schrodinger equations.
References:
H.Kumano-Go, Pseudodifferential operators, MIT Press, 1981.
E.M.Stein, Harmonic Analysis:real variable methods, orthogonality, and
oscillatory integrals, Princeton University Press, 1993.
C.Kenig, The Cauchy problem for the quasilinear Schrodinger equation, with
an appendix by Justin Holmer, arXiv:1309.329.
Minicourse by Wilhelm Schlag.
Introduction to nonlinear wave equations. description
Using the book by Nakanishi and Schlag, this course will
develop some basic aspects of the theory, with emphasis on the long-time dynamics
of the solution. The method of concentration compactness will be discussed.
Lecture notes
Minicourse by Panagiotis Souganidis.
An introduction to the theory of homogenization. description
The course will discuss some homogenization models for elliptic/parabolic and hyperbolic (Hamilton-Jacobi) equations. All necessary material will be presented in the lectures.
Minicourse by Luis Silvestre.
Viscosity solutions for nonlinear equations. description
We overview the concept of viscosity solution for nonlinear elliptic and parabolic equations. We will state and prove existence and uniqueness theorems in a very general setting. Some applications will be discussed.
Notes on viscosity solutions.
L. Silvestre offers to buy a coffee to anyone who sends him correct solutions to at least 20 of the exercises in the notes before June 29th.
(offer valid only to students attending the summer school)
Minicourse by Jonathan Weare.
Computing with deterministic and stochastic differential equations. description
The lectures will introduce the numerical simulation of ordinary and stochastic differential equations along with several typical applications. The focus will be on the estimation of averages with respect to a given density by Markov chain Monte Carlo and Langevin simulation along will typical difficulties encountered in applications and methods to address them. The emphasis will be on introducing important computational techniques. Elements of stochastic calculus will be informally introduced as needed. No background in probability will be assumed though exposure to it will be useful.
These activities are financed by the University of Chicago RTG grant (DMS-1246999).
Any opinions, findings and conclusions or recomendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation (NSF).