Prof. Max Alekseyev
Aaron Shanil
Smallest tuples of consecutive Harshad numbers
A Harshad number is a positive integer that is divisible by the sum of its digits. Cooper and Kennedy (Fibonacci Quart. 21, 1993) proved that at most 20 consecutive integers can all be Harshad numbers. The smallest n-tuples of consecutive Harshad numbers are known for n up to 14 as well as for n = 16 and n = 17 (see https://oeis.org/A060159). This project is aimed at closing the gap by finding the smallest n-tuples of consecutive Harshad numbers for the remaining values n = 15, 18, 19, and 20, employing GW HPC cluster facility.
Elementary number theory, computer programming
Prof. Lien-Yung (Nyima) Kao
Yi Nie and Chengyi Yang
Owen Cain, Rohan Singh, and Jiexi Xie
Fractal Geometry and Dynamical Systems
The aim of this project is to use 3D printing to visualize examples in fractal geometry, such as the Sierpinski Triangle, Sierpinski Pyramid, Menger Carpet, Menger Sponge, and Koch Snowflake. Through these examples, students will gain insight into the theory of fractal dimensions and explore its connections with dynamical systems.
Calculus I, II, and III
Faculty Mentor
Prof. Joel Lewis
Sophie Rubenfeld
Chespeak Dowdy, Robin Liam, and Suzane Rai
Inversion hyperplane arrangements in low-rank reflection groups
For purposes of this project, a reflection group is a finite group of symmetries of real Euclidean space generated by the reflections it contains. Elements of reflection groups have a generating set of "simple reflections", and with respect to this set they have a length function (how many simple reflections are needed to generate the particular element). The "inversions" of an element are the reflections that multiply with it to give a shorter (in length) product; the associated arrangement of reflecting hyperplanes is called the "inversion arrangement". These are interesting geometric arrangements, which play a role in various combinatorial and reflection group-theoretic questions. The main goal of the project would be to create (in some way) a complete set of inversion arrangements in the rank-2 and rank-3 real reflection groups. (Rank 2 is the dihedral groups, so we are just talking about lines in the plane; rank 3 is where the real interest lies, and we’re talking mostly about the symmetry groups of the Platonic solids in that case.)
Linear algebra
Prof. Xiaofeng Ren
Mathew Kukla
Oliver Kemper
Confinement property in inhibitory physical and biological systems
When a physical or biological system is studied in a bounded region, the geometry of the region restricts the movement of species and acts as a confinement force. However when the system is posed in an unbounded region, say the entire Euclidean space, one needs a long range attractive interactive mechanism to keep the species close to each other. We propose several interaction kernels with the long range attractive property in this project and study them and resulting patterns by analytical and numerical methods.
Calculus I, II, and III, Linear algebra, Matlab or Python programming language.