Polymath-REU 2022: Miller Proposed Projects
Mentor: Steven Miller, Williams College (email: sjm1@williams.edu)
Miller's homepage: https://web.williams.edu/Mathematics/sjmiller/public_html/
Greetings. I am excited to be involved in the inaugural Polymath-REU Program. I currently serve as the Director of the Williams SMALL REU, and have been mentoring students for over two decades. I earned a PhD from Princeton working with Peter Sarnak and Henryk Iwaniec in analytic number theory (specifically, low-lying zeros for families of elliptic curves). Below are some general areas of problems I am proposing for the Polymath-REU project. Different projects will be supervised with different colleagues of mine. Video of problem description: https://youtu.be/yXuKcI0BfbQ (slides here) I will be assisted by several colleagues, including Ajmain Yamin <ayamin@gradcenter.cuny.edu>.
Ramsey Theory: Years ago some of my SMALL REU students and I looked at non-commutative versions of some standard problems in the field, specifically avoiding 3-term geometric progressions. We have working notes here. Most of a paper is done, and in addition to finishing things off there are opportunities to explore related problems.
Elementary Number Theory: Small/large divisors satisfying recurrence relations: These projects are from a former SMALL REU student of mine, Hung Viet Chu at Illinois; see here.
f-palindromes: These projects are from a colleague of mine, Daniel Tsai, at Nagoya University; see here.
Zeckendorf Games: Baird-Smith, Epstein, Flint and Miller devised a game based on the Fibonacci numbers (1, 2, 3, 5, ... and in general Fn+1 = Fn + Fn-1) and one of their interesting properties, the Zeckendorf Decomposition (every integer can be written uniquely as a sum of non-adjacent Fibonacci numbers). It was proved that every game terminates, and a non-constructive proof shows that Player Two always has a winning strategy if the starting value is at least 3). There are still many open questions about this game and its generalizations. See https://web.williams.edu/Mathematics/sjmiller/public_html/math/papers/ZeckGameCANT10.pdf, https://web.williams.edu/Mathematics/sjmiller/public_html/math/papers/ZeckGameGeneral_FibQ10.pdf and https://web.williams.edu/Mathematics/sjmiller/public_html/math/papers/FQgame30.pdf. I have a 40 minute talk on the subject: From Monovariants to Zeckendorf Decompositions and Games, and Random Matrix Theory, Williams College (7/14/21) and Texas Tech (7/29/21). pdf (video: https://youtu.be/Kayru_V75V8)
For more information about the program see https://geometrynyc.wixsite.com/polymathreu, and go to https://www.mathprograms.org/db/programs/969 to apply.