Research Experience for Undergraduates
Mathematics REU: May 23 – July 15, 2022
We encourage qualified applicants to apply for summer 2022! (Click the link to register on the NSF Education and Training website and follow the application instructions from there.) We will begin reviewing applications on February 18, 2022, and applications will continue to be accepted until all positions are filled. Please direct all inquiries to Dr. David Milan (firstname.lastname@example.org).
Mentors and Subject Areas
Dr. Madeline Dawsey – Number Theory
A partition of a nonnegative integer n is a sequence of positive integers in decreasing order which sum to n, and the partition function counts the number of partitions of n. Euler proved the first bijective partition identity: the number of partitions of n into odd parts is equal to the number of partitions of n into distinct parts. There is now a large sub-field of combinatorics dedicated to proving further partition bijections and studying new restricted partition functions. We will investigate “sequentially congruent partitions," or SCPs, which are partitions such that the m-th part is congruent to the (m+1)-th part modulo m for all m, and the smallest part is divisible by the number of parts. Surprisingly, SCPs are in bijection with both unrestricted partitions and partitions whose parts are perfect squares. Are SCPs in bijection with any other restricted partition functions? Do SCPs exhibit any interesting additive, multiplicative, or algebraic properties?
Dr. Christina Graves – Reliability Polynomials
The all-terminal reliability polynomial of a graph is the probability that a graph remains connected if each edge is removed independently with fixed probability 1-p. A graph G is uniformly more reliable than a graph H if its reliability polynomial is greater than the reliability polynomial of H for all values of p. We will consider how various graph operations affect the comparison of reliability polynomials.
Dr. David Milan – Inverse Semigroups
We will study various ways to build inverse semigroups having interesting and useful algebraic properties. Inverse semigroups are a bit like groups in that they are sets with an associative multiplication and an "inverse" operation. Where groups are naturally represented by permutations, inverse semigroups are represented by partial permutations. That is, any inverse semigroup can realized as a set of bijections on subsets of some fixed set X. Inverse semigroup theory has found a number of applications, including the study of local symmetry present in quasi-crystals, and in the theory of partial isometries on vector spaces (both finite and infinite dimensional).
- A stipend of $4,800 for the eight-week program.
- Some travel funding to Tyler and partial funding to attend a national conference.
- Free on-campus housing at the Patriot Village or Eagle's Landing apartments, only a short walk from the mathematics department facilities in Ratliff Building North.
- Work daily with faculty during the research experience.
- Learn how to write and typeset a professional mathematics article using LaTeX.
- Learn how to prepare and deliver an excellent talk on mathematics.
- Participate in fun team-building social activities. Past activities have included: