Undergraduate Summer School 2018

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 graduated and completed their undergraduate studies. There will be many organized activities, with some specifically targeted at students at the introductory level and others at more advanced students.

There will be a lot of mathematics happening throughout the PCMI Summer Session. The research topic, harmonic analysis, is a central field of mathematics with a number of applications to geometry, partial differential equations, probability, and number theory, as well as physics, biology, and engineering. Participants in the USS will have the opportunity to meet and interact with mathematicians and math teachers from around the world. PCMI’s Undergraduate Summer School is an excellent way to learn about how to become a professional mathematician.

The 28th Annual PCMI Summer Session will be held July 1 – 21, 2018.

2018 Course Descriptions:

Ricardo Sáenz, University of Colima

Introduction to Harmonic Analysis

Harmonic functions have a key role in mathematics, physics, engineering, biology, and many other areas. They are the solutions to the Laplace equation, which describes, for example, the thermal equilibrium reached by the solution to the heat equation, as well as the behavior of, for instance, electric or fluid potentials.

In this course we study the basic properties of harmonic functions, such as the mean value property, the maximum principle, or Liouville's Theorem. We discuss explicit solutions to the Laplace equation in special cases, such as the ball or the upper half space. We also study the behavior in the boundary of domains, and introduce the Hilbert transform.

We will also discuss Dirichlet's principle, which states that the harmonic functions are the minimizers of energy in domains. We will finish this course with a study of the construction of harmonic functions in fractals.

Prerequisites: Linear Algebra, Multivariable Calculus, Introduction to Real Analysis

Eyvindur Palsson, Virginia Tech

Oscillations in Harmonic Analysis

The origin of harmonic analysis can be traced back to the fundamental work of Fourier in the early 19th century who asked whether arbitrary functions could be written as infinite sums of the most basic trigonometric functions. Today harmonic analysis is an active and exciting area of mathematics with a vast range of applications to fields such as compression algorithms, inverse problems, partial differential equations, and number theory.

In this course we will follow the story of oscillations in harmonic analysis beginning with Fourier series and their properties. Next we will continue with their natural continuous analog, the Fourier transform, and highlight applications to classical partial differential equations. Finally we will expand our discussion to more singular objects with the calculation of the Fourier transform of the natural measure on the unit sphere being a particular highlight. This will then motivate discussion of open problems in Harmonic Analysis such as the Falconer distance problem and the restriction problem.

Prerequisites: Linear Algebra, Multivariable Calculus, Introduction to Real Analysis

The Organizers of PCMI’s Undergraduate Summer School are Thomas Garrity, Williams College, and Irena Swanson, Reed College.