SEPTEMBER 23, Wednesday, 4:00 P.M. with reception to follow
250 Mathematics Building

LECTURE 1. 
The Beauty and Usefulness of the Complex Plane
An attempt will be made to demystify complex analysis. This naturally leads to the introduction of the Riemann-Hilbert and d-bar formalisms. For many years, the employment of the Wiener-Hopf technique to acoustics and other physical problems, was the only manifestation in applications of the Riemann-Hilbert formalism. However, in the last 50 years this formalism has appeared in many problems in mathematics and mathematical physics. In addition, in the early 1980s the d-bar formalism also began to emerge in applications. Several of these applications will be mentioned, including the solution of integrable nonlinear evolution PDEs in one, two, and three dimensions, as well as medical imaging.

SEPTEMBER 24, Thursday, 4:00 P.M.
250 Mathematics Building
LECTURE 2.
The Unified Transform Method for Linear and Integrable Nonlinear PDEs
The Unified Transform, also known as the Fokas method, provides a new and powerful method for analysing boundary value problems for linear and integrable nonlinear PDEs. For linear PDEs, it provides an unexpected extension of classical works of D'Alembert, Fourier, Laplace, Green and Kelvin. For nonlinear PDEs, perhaps the most striking application of the new method is the solution of the x-periodic problem for integrable PDEs, which was first considered in the early 1970s, and which attracted the involvement of many distinguished mathematicians. This lecture will review the above novel method with emphasis on linear evolution PDEs.

SEPTEMBER 25, Friday, 4:00 P.M.
250 Mathematics Building
LECTURE 3.
Deformations to the Complex Plane, Novel Asymptotic Techniques, and the large t-Asymptotics of the Riemann zeta function
The combination of one of the main ideas related to the Unified Transform, namely, deformation to the complex plane, and the use of novel asymptotic techniques jointly obtained with Jonatan Lenells, has led to unexpected and exciting results regarding the large t-asymptotic analysis of the celebrated Riemann zeta function. The following two results will be discussed. First, a simple formula will be presented for the difference of the functions defining the error terms in two historic problems: in Atkinson’s formula and in the formula for the Dirichlet divisor problem; it will be shown that this difference equals π/ 2 plus a function which is simply related to the square of the Riemann zeta function. Second, a remarkable integral identity satisfied by the Riemann zeta function will be presented; this identity is obtained from an earlier identity derived by the speaker via contour deformation in the complex plane. The asymptotic analysis of this integral equation gives rise to an interesting identity satisfied, for large t, by a sum generalizing the Dirichlet divisor sum. Also, and more importantly, it gives rise to a specific integral transform suitable for the large t-asymptotic analysis of the Riemann zeta function.

Also note: Tea-time at 3:30pm each day.

Bio: Athanassios Fokas, PhD, MD, has published in a remarkably broad range of topics in Mathematics, Physics, Engineering, Biology, Medicine, Philosophy, and Arts. He is listed among the most highly cited researchers in the ISI Web of Sciences. He introduced the ‘Fokas method’ and plays a pivotal role in the solution of several mathematical problems arising in medical imaging. 

He was awarded the European Academy of Sciences Blaise Pascal Medal/Lecture (2023) “for the Fokas method, considered the most important development in the solution of partial differential equations since Fourier, Laplace and Cauchy”.

The Society for Industrial and Applied Mathematics (SIAM) honored Fokas with the Kruskal Award (2024) “for his contributions to the development of the inverse scattering transform, for his new method for boundary-value problems, and for his work on the asymptotics of the Riemann Zeta function."

Fields of Scholarship

• Applications in medicine and biology
• Electro-magneto-enchephalography
• Integrable systems
• Boundary value problems of linear and integrable nonlinear PDEs
• Asymptotic analysis
• Functional medical imaging
• Painleve equations, orthogonal polynomials and random matrices

Related Links

•  Faculty Profile
•  Google Scholar
• The Academy of Europe
• Seven Secrets of a Beautiful Mind (USC Viterbi)