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Ish Shah's 2024 REU Webpage

About

I am an undergraduate student at Rutgers University–New Brunswick studying mathematics (my main major) as well as computer science (as either a second major or a minor). As someone who has only just completed their second year of undergrad, I'm still exploring different areas, but my current interests mainly lie in mathematical analysis.

For pictures of animals I have seen during the REU, go to the bottom of the page.

My pronouns are they/them.

Project Description

Title: When Fourier analysis meets ergodic theory and combinatorics
Abstract: Pointwise convergence in ergodic theory is the most natural (as well as the most challenging to establish) mode of convergence. This line of inquiry began in the early 1930's with von Neumann's mean ergodic theorem and Birkhoff's pointwise ergodic theorem and admitted profound extensions, particularly following Furstenberg's ergodic proof of a celebrated Szemeredi's theorem (on the existence of arbitrarily long arithmetic progressions in subsets of integers with positive density) and the work of Bourgain in the late 1980s.

After Bourgain's groundbreaking papers, which were acknowledged in his Fields Medal announcement, Elias Stein began to call this area Discrete Analogues in Harmonic Analysis due to its discrete flavor and interactions with number theory, additive combinatorics, probability, and Fourier analysis, among other fields. Our focus will be on studying various intriguing open problems at the intersection of ergodic theory, harmonic analysis, additive combinatorics, and analytic number theory.
(Abstract credit goes to Prof. Mirek.)

Weekly Logs

Week 1 (May 28-31) Much of this week was spent reviewing the basics of measure theory. This includes \(\sigma\)-algebras, measure spaces, Lebesgue measure, measurable functions, and \(L^p\) spaces. Leonidas provided exercises to work on to check my understanding of the material; I decided to write up my solutions to some of them, which may be found below (see Presentations and other work). And of course, I also participated in other first week activities: meeting other students here (including my collaborator, Abbas), attending some programs put on by the staff to make the transition to the REU smoother, and setting up this page!
Week 2 (June 03-07) This week started with finishing my measure theory review (for now). With Abbas, I prepared some slides for the introductory presentation (which we gave on Tuesday). I revisisted my knowledge of Fourier series and general theory in the setting of Hilbert spaces, working out a few exercises to ensure I still have a grasp over the material. I also did a bit of reading on functions of bounded variation and how these functions interact with Fourier series. To wrap up this week, I helped reprove some lemmas and worked through some exercises with Leonidas and Abbas to prepare for when we start looking at the Laporta–Tolev papers and start our own work.
Week 3 (June 10-14) This week, I started reading through the Laporta–Tolev paper. Together, we began rederiving some of the bounds presented in this paper, making sense of some details omitted from the paper. We also now recognize some parts in which we may want to try improving these bounds to produce useful results for our purposes.
Week 4 (June 17-20) This week, I spent more time going through the Laporta–Tolev paper. I've been working on rederiving more bounds from the paper using Vaughan's identity for the von Mangoldt function \(\Lambda(n)\) and the Möbius function \(\mu(n)\): \[\Lambda(n)=\sum_{\substack{b\mid n\\b\leq y}}\mu(b)\log\left(\frac{n}{b}\right)-\sum_{\substack{b\mid n\\b\leq y}}\sum_{\substack{d\mid n/b\\d\leq z}}\mu(b)\Lambda(d)+\sum_{\substack{b\mid n\\b>y}}\sum_{\substack{d\mid n/b\\d>z}}\mu(b)\Lambda(d),\] where \(y\) and \(z\) are constants chosen such that \(y,z\geq1\), and \(n>z\). I also gained more exposure to important concepts/tools of harmonic analysis, such as the Hardy–Littlewood maximal function and the maximal inequalities. Having seen this, I got a peek into how the current thing we are working on will interact with the whole project.
Week 5 (June 24-28) This week, we spent more time going through the Laporta–Tolev paper. To help with the process of stregthening bounds, we began consulting another paper of Tolev cited in the Laporta–Tolev paper. This paper features some manipulations that were unclear to me, and thus I ended up getting stuck somewhere in this paper. However, I was able to follow a good portion of the paper, and using slightly different estimation methods, it seems I may have improved one of the bounds in the Laporta–Tolev paper (although in a smaller range). In addition, we continued to take a little bit of a look at what lies ahead once we finish these estimates.
Week 6 (July 01-05) This week, more progress was made. After receiving some help from Abbas, my confusion was resolved and I was able to continue working towards obtaining new estimates. It seems like I have finally been able to match the bound reported for one sum in the Laporta–Tolev paper. I may have even improved on it slightly, and not in a smaller range as I was thinking last week! In Hardy–Littlewood circle method terms, this means I have derived a minor arc estimate, which may be useful for our result. Now, I plan on turning towards obtaining better major arc estimates, which may end up being more important for us. In addition, we continued to see where these estimates are situated in the big picture of things.
Week 7 (July 08-12) We continued to look at the major arc. Again, we started following the same Tolev paper cited in Laporta–Tolev to see if we can match or improve the estimates presented. Unfortunately, once again there are some things which appear to not be so clear at a first glance, so we unfortunately were not able to finish the estimates at the time of writing. Once again, we also continued to look ahead as well, keeping in mind that we do want to eventually develop enough to prove our desired ergodic theorem.
Week 8 (July 15-19) (will add when the time comes)
Week 9 (July 22-26) (will add when the time comes)

Presentations and other work

Presentations:

Other original work (writeups, etc.):

Acknowledgements

Thanks to my advisors, Prof. Mirek and Leonidas, for their support.

This work was conducted while the author Ish Shah 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.

References and reading

References for our work:

Reading for background:

As of now, these lists are by no means complete, and may be updated as the summer progresses.

Some animal pictures

Here are some pictures of animals I've seen around campus over the course of the REU.


A fawn spotted napping outside the Hill Center on Busch (May 28)
A cat that was staring me down by the Silvers Apartments on Busch (May 28)

A different cat caught running away in the dark by the Buell Apartments on Busch (June 1)
A squirrel inspecting some dumpsters outside the Buell Apartments (June 2)

A snail sticking to a step outside BEST Hall on Busch (June 2)
A turtle relaxing atop a rock in Passion Puddle on Douglass (June 3)

A hawk with a squirrel it caught outside the Buell Apartments (June 10)
Eight turtles sharing a single rock in Passion Puddle (June 16)

A cat outside the CBIM building, south of the dining hall on Busch (June 17)
A deer having some grass behind the Dr. Samuel Dewitt Proctor Hall on Busch (June 17)

Some deer grazing in the woods opposite CoRE across Lot 64 on Busch (June 25)
A somewhat camoflauged squirrel outside the Physics Lecture Hall on Busch (June 25)

Two turtles and a duck sharing a rock in Passion Puddle (July 1)
Numerous deer opposite the Busch Student Center (July 1)

A squirrel eating an apple atop a trash bin outside CoRE (July 11)