Sciences: Topology
Department: Mathematics
Supervising Faculty Member: Julie Bergner
Specialization: Topology
Research Focus: The project I would suggest for a student is concerned with developing number sequences from a pattern of directed graphs. In a previous project with students, we showed that a certain family of directed graphs produces a new way to obtain the Catalan numbers, a wellknown sequence that appears in many different ways throughout mathematics. A student could either investigate this family and its relation to the Catalan numbers more deeply, or consider the number sequences which arise from different families of directed graphs.
Position Description: A student would meet with me, probably for about an hour each week, but most of the time would be spent working independently (or possibly with another research student). That time would likely be spent reading background material, exploring patterns, and trying to develop proofs which makes those patterns explicit, and then eventually writing up the results. Our meetings would be a time to share what you've learned, where you are stuck, and what questions you have. I can give you feedback and answer questions which hopefully will help you in your progress in the following week.
Required skills: Courses beyond calculus (in particular, some exposure to proofbased mathematics) are helpful but not required.
Computer software: none required
Training/certification: no
What you will learn:
 Explore patterns and make conjectures
 Develop good proofbased strategies for verifying or disproving those conjectures
 Write about mathematics in a way a fellow student could understand
Web site link to research: http://people.virginia.edu/~jeb2md/catalan.pdf
