Course Descriptions
Uhlenbeck Lecture Course: What KdV Teaches us About Waves
Lecturer: Monica Visan, UCLA
Teaching Assistant: Katie Marsden, UCLA
Abstract: In this course, we will use the Korteweg-de Vries equation as a model to understand the behavior of dispersive partial differential equations. Through this lens, we will discuss dispersion,
well-posedness, and solitons. We will also discuss the existence of a Lax pair for this equation and how this informs our understanding of the complete integrability of this model.
Terng Lecture Course: From Fourier restriction to number theory, combinatorics, and fractal geometry
Lecturer: Dominique Maldague, Cambridge University and UCLA
Teaching Assistant: Olivine Silier, UC Berkeley
Fourier series are classically used to construct solutions to partial differential equations such as the wave and Schrödinger equations. In Fourier restriction theory, additional conditions are imposed on the frequencies of these series, giving rise to a more geometric view of the underlying functions. We will discuss how to use this perspective to study diverse problems, like the distribution of prime numbers, size of arithmetic progressions in sets, and the Kakeya problem in fractal geometry.
Prerequisites for the courses:
Uhlenbeck Course: Familiarity with measure spaces, convergence theorems, function spaces, and the Fourier transform at the level of a first-year graduate course. A first course on harmonic analysis is recommended.
Terng Course: A first course in proof-based analysis is expected. Exposure to Fourier series, Holder's inequality, and $L^p$ spaces will be helpful.