## Math 290, Topics in Analysis: Fourier Series and Beyond! (Fall 2022)

**Instructors:**

**Meeting times:** Tuesdays/Thursdays 12:00-1:15pm

**Description:** This special-topics course will introduce students to important and fascinating topics in mathematical analysis that have played a fundamental role in many applications. This semester, the course will focus on Fourier analysis and its application to partial differential equations (PDEs) and to imaging. The course will be divided into three modules: (1) Fourier series and analysis (2) PDEs for diffusion and reaction (3) application to Computed Tomography (CT). Prior experience with analysis or differential equations or imaging is not required. The course is designed to build student literacy in mathematics and to expose students to salient ideas before they have taken more advanced analysis courses. Topics in each module are given below.

**Prerequisites:** MATH 221 or MATH 218 (linear algebra). Calculus is also required. Some ideas from multivariable calculus will be used, but it is ok if students are taking MATH 212, MATH 219 or MATH 222 concurrently. This is not a “proofs” course: developing mathematical proofs will not play a significant role, and prior experience with mathematical proofs is not expected. Prior experience programming computers is not expected. Some activities in Modules 2 and 3 will involve use of MATLAB and/or Python, but instructions and sample codes will be provided.

**Assignments and Evaluation:** There will be regular readings, and homework assignments. There will be one midterm exam. There will be no final exam.

**Module 1. Instructor: Tarek Elgindi**

- Linear algebra review
- Convolution, approximation of the identity
- Fourier series
- L2 convergence, pointwise convergence
- Fourier transform on the real line
- Applications

**Module 2. Instructor: James Nolen **

- Heat equation
- Solution by separation of variables and Fourier transform
- Smoothing effect
- Reaction-diffusion systems, stability and instability
- Turing instability and pattern formation

**Module 3. Instructor: Hongkai Zhao **

- X-ray and Radon transform
- Fast Fourier Transform (FFT)
- Fourier slice theorem for Radon transforms
- Filtered back projection for CT

**Textbook:** Fourier Analysis: An Introduction, by Elias Stein and Rami Shakarchi, Princeton University Press, 2003. ISBN-13: 978-9380663463.

The instructors will also provide some notes to supplement the textbook.

Do you have **questions** about the class? If so, please contact Prof. Nolen