UT Tyler

## Mathematics REU: May 21 - July 13, 2018

We encourage qualified applicants to apply for summer 2018! We will begin reviewing applications on February 23, 2018, and applications will continue to be accepted until all positions are filled. Please direct all inquiries to Dr. David Milan (dmilan@uttyler.edu).

Students will work on research in groups of three that are led by a faculty advisor. The research groups will be centered around three exciting areas of mathematical research.

Mentors and Subject Areas

• 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 included independently with fixed probability p. This concept can be generalized to the two-terminal reliability polynomial of a graph where we find the probability that two specified vertices remain connected. We will look at all possible graphs on a fixed number of vertices and edges and see if there exists a graph whose two-terminal reliability polynomial is greater than all others.
• Dr. Lindsey-Kay LauderdaleAlgebraic Graph Theory
• The automorphism group of a graph is the set of adjacency preserving permutations of the vertices, and this group gives a description of the graph's symmetries. We will study symmetry breaking in graphs, which arises naturally in data structures for computing permutations, localization of wireless sensor networks, robotic manipulation, switching circuit theory and the enumerating chemical compounds. Our primary interest will be to destroy these symmetries by labeling a subset of the vertex set, resulting in a graph that has a trivial automorphism group.
• 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).

Program Benefits

• A stipend of \$4,500 for the eight-week program.
• 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.
• Full access to The University of Texas at Tyler's facilities, including the Herrington Patriot Center (fitness facilities, available at a discounted rate) and the Muntz Library.
• Work closely with faculty in gaining valuable 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.
• Get information on the process of selecting and applying for graduate programs
• Participate in fun team-building social activities. Past activities have included:
UT Tyler