UT Tyler

# Research Experience for Undergraduates

## Mathematics REU: May 27 – July 19, 2019

We encourage qualified applicants to apply for summer 2019! We will begin reviewing applications on February 22, 2019, 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. Katie Anders – Graph Splines
• Graph splines combine algebra and graph theory. Consider a triangle with each edge labeled by an integer. Now try to label the vertices of the triangle with integers in such a way that the difference between the labels of any two vertices is divisible by the integer labeling the edge between them. Such a vertex labeling is a simple example of a graph spline. One can change the triangle to a more complicated graph, and one can choose the labels from a ring other than the integers. We will explore families of edge-labeled graphs and seek to characterize the splines on these graphs. Although the study of graph splines involves algebra and graph theory, their applications include algebraic geometry and topology.
• 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.
• 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:
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