Project: Arithmetic of elliptic curves
Faculty supervisor: Alina Cojocaru
Graduate mentor: Cara Mullen
Student researchers: Gallen Ballew, James Duncan
Description
What do prime numbers and tori have in common? We will answer this question by exploring primes in the context of elliptic curves from both theoretical and computational perspectives, and by using visualization techniques such as computer graphics and 3d printing. The project will be adjusted according to the mathematical background and interests of the students.
Materials
Project: Statistics of class groups
Faculty supervisor: Nathan Jones
Graduate mentor: Kevin Vissuet
Student researchers: Matthew Fitzpatrick, Ayman Hussein, Shayne Officer
Description
There are number rings whose elements fail to have unique prime factorizations, for instance the ring ℤ[√-5] := { a + b √-5 : a, b ∈ ℤ}, wherein one has 6 = 2·3 = (1+√-5)(1-√-5). In general, there is an abelian group called the class group that governs this issue. We will explore various statistical aspects of class groups associated to number fields.
Materials
Project: Tracking solution paths with PHCpy
Faculty supervisor: Jan Verschelde
Graduate mentor: Nathan Bliss
Student researcher: Konrad Kadzielawa
Description
Many polynomial systems arising in application from science and engineering involve several parameters. Of interest is to know for which values of the parameters do the solutions of the polynomial collide into multiple solutions or degenerate into positive dimensional solution sets. Special values of the parameter where the solutions are singular define the discriminant variety.
The goal of the project is to develop Python scripts to explore the parameter space of systems arising in applications using the sweep homotopies implemented in PHCpy. For the formulation of the polynomial systems we will use the computer algebra systems sympy and Sage.
Materials
Project: Topology of algebraic varieties
Faculty supervisor: Benjamin Antieau
Graduate mentor: Samuel Cole
Student researchers: Mason Boeman, Daniel Etrata
Description
The goal of this project is to use persistent homology to study the topology of random complex algebraic varieties, geometric figures described as the solution sets of system of polynomial equations. Several students will work together to continue to develop a suite of software packages to numerically find solutions of polynomial equations, compute the persistent features of the resulting cloud of points, and visualize the results, using a virtual reality platform such as Oculus Rift.