Clay Mathematics Institute / PROMYS 2016 Research Projects

Mentoring of Research Projects:

The PROMYS/CMI mentors in 2016 were the following research mathematicians: Professor Victor Rotger (Universitat Politècnica de Catalunya, Barcelona), Professor Lionel Levine (Cornell University), Professor Ira Gessel (Brandeis University), Professor Paul Gunnells (University of Massachusetts, Amherst), Professor Laurent Berger (École Normale Supérieure de Lyon), Dr. Ila Varma (Harvard University), Dylan Yott (UC Berkeley), Dr. Jonathan Hanke, Dr. Sandra Rozensztajn (École Normale Supérieure de Lyon), Dr. John Bergdall (Boston University), Dr. Jeremy Booher (Stanford University), and Professor Kevin Buzzard (Imperial College London). Each submitted research problems for advanced students to work on. The materials presented by the mentors included problem statements, hints for getting started and references to the pertinent literature.

Research Projects in the summer of 2016 were:

An Interesting Congruence About the Ramanujan Tau Function 
Proposed by: Professor Kevin Buzzard (Imperial College London)
Students: Matthew Hase-Liu and Sungwon Kim
Counselor: Kevin Lin 

Bouncing Superball
Proposed by: Dr. Jeremy Booher (Stanford University)
Students: Vandana Agarwala, Sam Tang, and Jena Yun
Counselor: Sam Zbarsky

Fibonacci Words
Proposed by: Professor Lionel Levine (Cornell University)
Students: Candace Chiang, Daniel Kleber, and Philip Lamkin
Counselor: Claudia Feng 

Integral Eigenvalues of Schreier Coset Graphs of Weyl Groups
Proposed by: Professor Paul Gunnells (University of Massachusetts, Amherst)
Students: Dylan Pentland and Vijay Srinivasan
Counselor: Samuel Marks

Monogenic Cubic Fields
Proposed by: Dr. Ila Varma (Harvard University), Dylan Yott (UC Berkeley), and Dr. Jonathan Hanke
Students: Mikaeel Yunus, Hari Pingali, and Stephen New
Counselor: Cailan Li

On Wythoff’s (a, a) Variation
Proposed by: Professor Paul Gunnells (University of Massachusetts, Amherst)
Students: Shubham Aggarwal, Jared Geller, Shuvom Sadhuka, and Max Yu
Counselor: Tim Ratigan

PROMYS Polygon
Proposed by: Professor Victor Rotger (Universitat Politècnica de Catalunya, Barcelona)
Students: Isaac Benghiat and James Leng
Counselor: Art Waeterschoot

Ramified Automorphisms of the Field of Laurent Series
Proposed by: Professor Laurent Berger (École Normale Supérieure de Lyon) and Dr. Sandra Rozensztajn (École Normale Supérieure de Lyon)
Students: Kenz Kallal, Hudson Kirkpatrick, and William Zhang
Counselor: Alyosha Latyntsev 

Representation of of Rational Numbers in Different Numerations Systems
Proposed by: Professor Ira Gessel (Brandeis University)
Students: Byron Chin, Ye Dong, Jongyoul Lee, and Daniel Lin
Counselor: Larsen Linov

Slopes of Some Newton Polygons
Proposed by: Dr. John Bergdall (Boston University)
Students: David Amirault, Vanshika Jain, Roshan Padaki, and Sabir Shaik
Counselor: Vineet Gupta