Zeros of Analytic Functions
Students in this group will learn about various techniques used in determining the location of zeros of a special class of analytic functions. Recently there have been two new approaches proposed in this area of research, and both of these approaches have the potential to answer deep questions concerning reality-preserving operators. We will investigate these new approaches as they apply to the so-called Legendre multiplier sequences, and multiplier sequences for arbitrary simple sets of polynomials. Students will be encouraged to use Mathematica© to carry out calculations, and develop conjectures which they will then set out to prove. An ideal member of this research group should have completed a proof class. Being enrolled in or have completed a real or complex analysis course would be a plus.
Algebraic Knot Theory
The mathematical theory of knots studies collection of circles tangled in 3-space. One of its main problems is finding methods to distinguish these structures from each other. Knot theory has provided models and applications to molecular biology, physics and quantum computing, and incorporates several areas of mathematics including topology, geometry, combinatorics and abstract algebra.
Students in this group will explore various invariants for links and their connections to combinatorics and algebraic structures. Hence, those interested in this topic should have had a course on linear algebra, be able to read and write proofs, and hopefully have some familiarity with abstract algebra.