# Research Topics

## Professor Forgacs' Groups

Students working with Dr. Forgacs will learn about various techniques used in determining the location of zeros of a special class of analytic functions. Some of the theory we will review is over a century old and is now considered classical complex analysis, some of it is quite current. The following are two potential projects for students to work on:

**Creating new multiplier sequences from old ones**. In their 1914 paper, Polya and Schur described a number of intrinsic properties of multiplier sequences, including a few ways of obtaining new multiplier sequences from a given one. Recently, Csordas and Forgacs proposed a way to describe connections between two multiplier sequences via generating relations. Students working on this project would investigate the types of multiplier sequences that can be obtained this way, along with the limitations of such an approach.**Multiplier sequences of the second kind**. Multiplier sequences of the second kind have not been studied much since Polya and Schur's original paper. Although a characterization of these sequences is available in the standard basis, such results are nonexistent for any of the other classical orthogonal bases (such as the Hermite, Legendre or the Laguerre bases). Students working on this project would attempt to understand (and attempt to characterize all) such sequences for the Hermite basis.

In addition to these, groups may identify problems they find more interesting during
the first week or so of introductory lectures.

Students will be encouraged to use Mathematica™ or other similar programs to carry
out calculations, and develop conjectures. An ideal member of this research group
should have completed a proof class and a linear algebra class. Being enrolled or
having completed a real or complex analysis course would be a plus.

## Professor Piotrowski's Groups

Students working with Dr. Piotrowski will learn about various techniques used in determining
whether or not a polynomial has some non-real zeros. The following are two potential
projects for students to work on:

**Complex Zero Decreasing Sequences**. Discovering new examples of complex zero decreasing sequences for any basis is challenging. Progress in this direction was made by Piotrowski and his students in the summer of 2013. Students working on this project would attempt to extend these results and find new classes of complex zero decreasing sequences for various bases. As added incentive, a small monetary award is available for the first person who can determine whether or not the sequence {exp(-k^3)} is a complex zero decreasing sequence for the standard basis (this is an open problem!).**Multiplier sequences for non-orthogonal bases**. Multiplier sequences for orthogonal bases have been studied extensively. There is not much known about multiplier sequences for non-orthogonal bases. Students working on this project would select a non-orthogonal basis, such as the Bessel polynomials, and investigate multiplier sequences for that basis with the aim of classifying all such sequences.

In addition to these, groups may identify problems they find more interesting during
the first week or so of introductory lectures and readings.

Students will be encouraged to use computer algebra systems to carry out calculations,
and develop conjectures. An ideal member of this research group should have completed
a proof class and a linear algebra class. Being enrolled in or having completed a
real or complex analysis course would be a plus.