Daniel K. Rui

not-yet-Professor of Mathematics, Pianist, and Teacher

"Finished" Works

June 6, 2024
	Notes for a (2hr) seminar talk I (and another graduate student!) gave on primes in arithmetic progressions (APs) and sieve theory, split into 3 parts: Dirichlet's 1837 theorem on the infinitude of primes in APs to introduce Dirichlet characters, especially the real/quadratic ones and the problem of "Siegel poles/zeroes" (i.e. conspiracy of a quadratic Dirichlet character with the function 1, or the Liouville function λ); an introduction to the main ideas/philosophy/"black-boxable tools" of sieve theory and its limitations (the parity problem); and Matomäki&Merikoski&Teräväinen's use of sieve theory in the Siegel pole/zero/neither cases to prove Linnik's theorem (the first prime in an AP appears in "polynomial time"). A preliminary recording is here. I will try to make a complete recording (might be ~3hrs because I want to give the material its due space to breathe, which we sadly didn't get the time to do during the actual seminar) of the material of the talk and link it here sometime soon.
March 6, 2024
	Slides for a talk giving an overview of elementary analytic number theory (introducing the Riemann zeta function and the relationship between its analytic properties/properties revealed by analytic methods, and number theoretic questions; (cheap and bona-fide) Mertens' theorems; Mobius inversion and cancellation; divisor sums and taking advantage of averaging/Fubini; and providing what I think to be a very well-motivated approach to the Selberg symmetry formula, Brun-Titchmarsh and its factor of 2 related to the parity problem, etc.). Also I had a "hidden agenda" of introducing some important "tricks/themes" from analysis, like dyadic decomposition; big-O/little-o notation and the idea that away from the main term we can actually be much looser than what one may be used to in one's standard "intro analysis training" (and the IMHO unexpected phenomenon of "discrete resolution jumps", e.g. a O(1) error term might resolve further to c+O(1/log x), where there's a discrete gap between constant-order resolution and 1/log(x)-order resolution); and the flexibility of flowing between sums/integrals/measures. Culminates in Terry Tao's proof of the prime number theorem (PNT) using the language/machinery of Banach algebras.
Febuary 12, 2024
	Notes on a talk about the lace expansion method in the analysis of self-avoiding random walks (SAW) I (and 2 other graduate students!) gave during a Winter2024 seminar on perturbative methods in probability.
August 26, 2021
	Brief overview of Dirichlet's theorem on primes in arithmetic progressions (truly baffles me to know that Dirichlet gave a completely rigorous proof of this in 1837, more than twenty years before Riemann's foundational work in extending the zeta function to the complex plane in 1859; compare this with Galois's discovery of the Galois group in 1830... just awe-inspiring that in one paper someone can lay the foundation of an entire branch of mathematics). Full of links to resources (notes/articles) to learn more, mixed with some commentary. (Feb. 7, 2022 uploaded newer version; old version found here)
June 11, 2021
	My senior thesis, i.e. the final masterpiece from my college career, about providing a motivation-rich exposition to topology. In my opinion, I cover an absurd amount of material considering the length of the document, but topology has a way of drawing in a huge wealth of ideas. We will start with some basic definitions, and quickly move into motivating the development of abstract topological spaces by thinking about ways of defining local and global continuity of functions without the notion of distance. We will follow a narrative that meanders around ideas like partitions of unity, some theorems of Urysohn, Tietze's extension theorem, the countability and separation axioms, metrization, compactness, ordinals (yes, some wacky set theory! ...but perhaps not as surpising considering the connection between compactness and AC via Tychonoff), topological dimension, and manifolds, ultimately culminating in a proof of a version of Whitney's embedding theorem. There's some fun bonus material at the end regarding Brouwer's fixed-point theorem (via a combinatorial result, Sperner's lemma!) and deep results that typically come from algebraic topology, like the invariance of dimension and the Jordan curve theorem. (2024: the section on JCT was presented by me in this video.)
Aug 10, 2020
	Should've posted this earlier, but oh well. This paper is an exposition of the Longest Increasing Subsequence (LIS) problem and the Baik-Deift-Johansson theorem, as well as important results about the Tracy-Widom (TW) distribution and the problems/contexts in which it appears. As an added bonus, I ran some computer programs to try to empirically determine the rate of convergence of the length of the LIS to the TW-distribution, which I found seemed to be on the order of n^-1/6. This result/conjecture seems to not have appeared before in the literature.
Jun 3, 2020
	An introduction to measure theory, building up to the Caratheodory extension theorem. I've long thought that the Caratheodory measurability criterion was very unintuitive, and very mysterious, so I spent 20 pages trying to demystify it in the context of the real line and Lebesgue measure. I believe that mathematics should be a journey of discovery, and should be communicated as such; one should play around and tinker before making definitions. I attempt to do this in this paper, where I go over outer measure, inner measure, Lebesgue's definition of measurability, the open-closed/inner-outer definition of measurability, Caratheodory's definition (now intuive in the context of the other definitions), sigma-algebras (and other set structures, like semi-rings), and finally the extension theorem itself. My magnum opus of my life so far.
	The entire collection of homeworks I did for MATH 522. This class continually gets me to produce some of my highest quality work, which I'm very happy about, but also kind of stressed about; a blessing and a curse. In fact, my 33X homeworks kind of decreased in quality because (a) there were too many problems to think deeply on and (b) they weren't particularly satisfying or interesting to me, so I will not post them :(
	The entire collection of homeworks I did for MATH 523. Only 4 homeworks and a midterm, but the rest of the work I did for this class lies in the Longest Increasing Subsequence paper I have yet to complete/upload.
Dec 30, 2019
	Every wonder where in the world did the formulas for dots and cross products come from? Why didn't we just define vector multiplication like we did vector addition? Or what about determinants -- did you ever want to know how someone came up for the formula for the determinant? Well, now you can find out. Part 1 of "Math on a Desert Isle".
Dec 6, 2019
	This is the entire collection of homeworks I did for MATH 334 (arranged last to first); the first homeworks focused on point-set topology and we slowly moved on to continuity, differentiation, and integration. Overall I'm pretty happy with it (especially the last and first homework sets), but it certainly was a struggle.
	Similarly, this is the entire collection of homeworks I did for MATH 521, my first graduate class (though I audited), on a subject I had no prior experience in (probability/measure theory and statistics). The homeworks were really hard but a lot of fun -- I think I learned a lot of cool and beautiful stuff. I only managed to do about a half of the homework every week, but I'm pretty happy with the stuff I was able to do.
May 25, 2018
	This paper is about very unintuitively diverging series, culminating in the proof that the sum of the reciprocals of the primes is infinite, by using the sum of the reciprocals of the natural numbers and the sum of the reciprocals of the squared numbers

Drafts of Upcoming Papers

Sep 25, 2019
	This is the compilation of all my papers relating to the Gamma Function. I'm including the one on the Reflection Formula, Stirling's Approximation, and Bertrand's Postulate. Of course I will continue to edit it and maybe add one more topic, but here's a sneak peek at two years of work I did in high school + additional insights from college.
	This is what I've been working on over my break and a bit during Autumn quarter -- basically reading through Axler's measure theory textbook and distilling it and adding bits and pieces to help me learn. Maybe solutions to some problems too if you're looking for that.

For older work, visit the Archive