Professor Steven J Miller: Williams College: Polymath JR REU Page

Miller's Homepage: https://web.williams.edu/Mathematics/sjmiller/public_html/

Polymath Jr REU Homepage: https://geometrynyc.wixsite.com/polymathreu

In Summer 2021 I will lead two Polymath Jr REU projects, and assist in two more. Below are some quick descriptions of my projects. The Q&A video is here: https://youtu.be/l03PAmiZmNw (slides here)

• Benford's Law: Video here: https://youtu.be/vs7nQMAF_e8
  • Benford's law of digit bias states that in many data sets each digit 1 thru 9 is not equally likely to be the leading digit, but rather we observe 1 almost 30% of the time, with the probability falling down to about 4.6% for starting with a 9. In addition to being of theoretical interest, this is also used in detecting various types of fraud. We will explore both aspects. On the fraud side, a lot of our work will be related to new tests to use Benford's law to examine whether or not there has been fraud in election data (I have written a short note with a colleague arguing that the claims circulating on the internet that Benford's law showed there was fraud supporting Biden are wrong; see https://web.williams.edu/Mathematics/sjmiller/public_html/kossovskymiller.pdf ). On the theory side, we will explore many topics, including connections with fractal sets (see for example the video here: https://www.youtube.com/watch?v=TMILk79N_Bs ). This will be joint with Dan Stoll of Michigan.
  • Go to https://web.williams.edu/Mathematics/sjmiller/public_html/benfordresources/ for some resources of mine on Benford's law.

• Diophantine m-tuples and elliptic curves (joint with Professor Seoyoung Kim, who is the lead on this group)
  • A Diophantine m-tuples, first studied by Diophantus of Alexandria, is a set of integers with the property of the product any two distinct elements is one less than a square, such as {1,3,8,120}. We are going to explore Diophantine m-tuples using the theory of elliptic curves and K3 surfaces, and moreover, aim to find new instances of Bias conjecture. In my thesis I found biases in the distribution of coefficients of the L-series in families of elliptic curves. My students, Professor Kim and I have extended this to several other families; the goal is to continue to explore this.
    • Lower-Order Biases in Elliptic Curve Fourier Coefficients in Families (with Blake Mackall, Christina Rapti and Karl Winsor), Frobenius Distributions: Lang-Trotter and Sato-Tate Conjectures (David Kohel and Igor Shparlinski, editors), Contemporary Mathematics 663, AMS, Providence, RI 2016. pdf
    • Lower-Order Biases Second Moments of Dirichlet Coefficients in Families of L-Functions (with Megumi Asada, Ryan Chen, Eva Fourakis, Yujin Kim, Andrew Kwon, Jared Lichtman, Blake Mackall, Eric Winsor, Karl Winsor, Jianing Yang, Kevin Yang; appendices with Roger Weng and Michelle Wu), to appear in Experimental Mathematics,  pdf
    • Rank and Bias in Families of Hyperelliptic Curves via Nagao's Conjecture (with Trajan Hammonds, Seoyoung Kim, Benjamin Logsdon, Alvaro Lozano-Robledo), to appear in the Journal of Number Theory  pdf (arxiv)  or pdf
    • Biases in Moments of the Dirichlet Coefficients in One- and Two-Parameter Families of Elliptic Curves (with Y. Yeng and an appendix by  J. Wu), submitted to The PUMP Journal of Undergraduate Research.  pdf
    • Diophantine triples and K3 surfaces, by Matija Kazalicki, Bartosz Naskręcki. pdf

       

   
• Number Theory: Video here: https://youtu.be/Rn64zHXqDVA (slides here)
  • Number theory is a vast area with a range of problems. We will focus on projects with minimal pre-requisites that can be split into many sub-problems, so the problems will be accessible to hopefully everyone. Possibilities include further work or generalizations of teh following.
    • Zeckendorf games: This is a game I invented using Fibonacci numbers; a former student proved that Player 2 has a winning strategy but it is a non-constructive proof. Last year one of my Polymath Jr groups extended this to several people and related sequences. See the following papers:
      • The Zeckendorf Game (with Paul Baird-Smith, Alyssa Epstein and Kristen Flint), to appear in the Proceedings of CANT 2018.  pdf
      • The Generalized Zeckendorf Game (with Paul Baird-Smith, Alyssa Epstein and Kristen Flint), Fibonacci Quarterly (57 (2019) no. 5, 1-14)  pdf
      • The Fibonacci Quilt Game (with Alexandra Newlon), Fibonacci Quarterly (2 (2020), 157-168) pdf
      • Extending Zeckendorf's Theorem to a Non-constant Recurrence and the Zeckendorf Game on this Non-constant Recurrence Relation (with Ela Boldyriew, Anna Cusenza, Linglong Dai, Pei Ding, Aidan Dunkelberg, John Haviland, Kate Huffman, Dianhui Ke, Daniel Kleber, Jason Kuretski, John Lentfer, Tianhao Luo, Clayton Mizgerd, Vashisth Tiwari, Jingkai Ye, Yunhao Zhang, Xiaoyan Zheng, and Weiduo Zhu), Fibonacci Quarterly. (5 (2020), 55-76)  pdf
      • Deterministic Zeckendorf Games (with Ruoci Li, Xiaonan Li, Clay Mizgerd, Chenyang Sun, Dong Xia, And Zhyi Zhou), Fibonacci Quarterly. (58 (2020), no. 5, 152-160)  pdf
      • Winning Strategy for the Multiplayer and Multialliance Zeckendorf Games (with Anna Cusenza, Aiden Dunkelberg, Kate Huffman, Dianhui Ke, Daniel Kleber, Clayton Mizgerd, Vashisth Tiwari, Jingkai Ye, Xiaoyan Zheng), to appear in the Fibonacci Quarterly. pdf
      • Generalizing Zeckendorf's Theorem to a Non-constant Recurrence (with E. Boldyriew, A. Cusenza, L. Dai, P. Ding, A. Dunkelberg, J. Haviland, K. Huffman, D. Ke, D. Kleber, J. Kuretski, J. Lentfer, T. Luo, C. Mizgerd, V. Tiwari, J. Ye, Y. Zhang, X. Zheng and Weiduo Zhu), to appear in the Fibonacci Quarterly. pdf
    • Non-commutative avoidance: the following preprint is an excellent start, but needs some work to finish and hopefully extend:
      • Avoiding 3-Term Geometric Progressions in Non-Commutative Settings (with Megumi Asada, Eva Fourakis, Eli Goldstein, Sarah Manski, Nathan McNew and Gwyneth Moreland), submitted to the Journal of Integer Sequences. pdf
    • Generalizing the Mordell-Weil group of an elliptic curve to other varieties (this will also involve mentoring some high school students who are working with me). An elliptic curve is of the form y² = x³ + ax + b with a, b integers (and some other conditions to avoid degeneracies). It turns out that one can define a group law on pairs of rationals (x, y) which satisfy this; the goal of this project is to ask related questions in other settings.
    • Math outreach:
      • Math riddle page: I maintain a math riddles page, and could use help in updating it and making it a great resource for teachers and students: http://mathriddles.williams.edu/
      • Math outreach: I have given several lectures to my children, continuing education classes for teachers, ..., and would love to convert these to a book. You can see many of the lectures here: https://web.williams.edu/Mathematics/sjmiller/public_html/math/talks/talks.html (the first few are ones to my children; initially these were way too long, but it would be great to redo with better slides / graphics).
      • Book project: I am working on a book involving applications of L-functions with a colleague, and if people are interested there is a lot of work that can be done there.
    • Other: Depending on the interest of students and the time that I have, I may add some other problems to the list.

• Urns: Exploring taking items out of an urn with different types of replacement (joint with Professor Enrique Treviño, who is the lead on this group)
  • Suppose an urn has R red balls and W white balls for a total of N = R+W balls. Suppose that when you take out a white ball, you keep it with probability w (and return it otherwise), and if you take out a red ball, you keep it with probability r.  Fix a number n smaller than R,W. Take a sample of n balls from the urn one at a time with the replacement rules described above. Let X_{(r,w)} be the number of white balls that are in the sample. When r = 0, w= 0, this is the classical "sampling with replacement" and X_{(0,0)} corresponds to the binomial model while r=w=1 is "sampling without replacement" and X_{(1,1)} corresponds to the hypergeometric model. In their paper "To replace or not to replace" Engbers and Hammett study X_{(0,1)}. In this project we will work on some open questions regarding X_{(0,1)} and will try to find nice expressions for the density, expectation and variance of X_{(r,w)}.