DIMACS REU 2024

Student:	Abbas Dohadwala
Office:	CoRE 448
School:	Department of Mathematics, Purdue University
Contact:	ad2129@rutgers.edu
Project:	When Fourier Analysis Meets Ergodic Theory and Combinatorics
Mentor:	Mariusz Mirek, Leonidas Daskalakis

Project Description

Research Log

Week 1 (6/5-6/9)

I arrived at the program this week, attended the orientation and got settled. I met with my mentors, as well as Ish and Erik. This week, I reviewed measure theory, working from Mariusz's slides and working on exercises given to us by Leonidas. I learned a large amount of the prerequisite ergodic theory and worked several exercises from various books. Our mentors discussed how we will use tools from harmonic analysis and analytic number theory to prove the desired result concerning pointwise convergence. In addition we were introduced to Tolev's paper on exponential sum estimates. There is an incredible amount to cover before we can begin the problem, so the first half of the program will be dedicated to acquiring the necessary background. Together with Ish, I prepared a slidedeck for the upcoming presentation.

Outside of program activities, I am adapting to the surrounding areas. I went climbing with a couple students, went on a meandering run arround campus (was disappointed by the lack of sidewalks), and went to New York City.

Week 2 (6/12-6/16)

This week, I learned much of the necessary material on Fourier series. I learned some necessary Hilbert space theory and some more important results in ergodic theory. I also worked several exercises in Fourier analysis, ergodic theory, and measure theory. Ish and I met with Leonidas most days so we could better understand the plan for our project. We also did our initial presentation week, and it was very interesting to see what others were doing in their projects.

Week 3 (6/19-6/23)

This week, we made more progress in terms of the necessary material, and we began going through the Tolev-Laporta paper to understand better the exponential sum estimates. I learned the basic theory of maximal operators and learned additional material in Fourier series.

This week, I also went to several talks in the "Beyond the Freshman Horizons Workshop". Many were close to what my project was concerned with and aligned with my general interests, and I showed up for the additive combinatorics, analytic number theory, random matrix theory, integrable systems, and fourier analysis talks.

Week 4 (6/26-6/30)

I made more progress this week in the Tolev Paper, and I began looking through a paper of Laporta-Tolev which many of the estimates were drawn from. I am starting to become more comfortable with exponential sum estimates and the philosophy of Vaughan's identity for the von Mangoldt function. I also learned some more theory of maximal operators and maximal inequalities, finishing the notes by Mariusz on the topic. I also began the notes on Ergodic theory and Oscillations. The theory of oscillation seminorms will be crucial for demonstrating our result holds on square-integrable functions. We met with Leonidas most days to discuss our progress and to learn more about general analysis principles to prove results of this nature, namely maximal inequalities and pointwise convergence results.

Week 5 (7/3-7/7)

This week we made significant progress towards understanding the exponential sum estimates of the minor arc. There is some uneasiness that Ish, Leonidas, and I have about some of the large steps the authors have skipped, and we intend to fill in the gaps. Otherwise, I made significant progress in the oscillations notes.

Week 6 (7/10-7/14)

Week 7 (7/17-7/21)

Project Summary

References

Acknowledgements

This work was carried out while the author Abbas Dohadwala was a participant in the 2024 DIMACS REU program at Rutgers University, CNS-2150186, supported by the Rutgers Department of Mathematics and NSF grant DMS-2154712.