# 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.

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# 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.

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# 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.

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# 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.