## Mathematics REU: June 10 – August 2, 2024

We encourage qualified applicants to apply for summer 2024! (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 January 19, 2024, and applications will continue to be accepted until all positions are filled. Please direct all inquiries to Dr. David Milan (dmilan@uttyler.edu).

## Mentors and Subject Areas

Dr. Katie Anders – Graph Splines

Generalized 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 generalized 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.  While the study of graph splines involves algebra and graph theory, their applications include algebraic geometry and topology.

Dr. Maddie Dawsey – Integer Partitions

In number theory, an integer partition is a way of writing an integer as a decreasing sum of positive integers, and the partition function counts the number of partitions of each integer.  For example, one partition of 5 is 3+2, and there are 7 different partitions of 5.  The partition function has lots of interesting size, divisibility, and asymptotic properties and is applicable to many other areas of mathematics, including combinatorics, algebra, and representation theory.  We will explore various properties of partition functions for partitions whose summands are restricted in certain ways.

Past REU projects include: REU 2022.

Dr. David Milan – Inverse Semigroups

We will study various ways to build inverse semigroups having useful algebraic properties. We will be particularly interested in inverse semigroups associated with directed graphs and symbolic dynamical systems, as well as building connections to the theory of C*-algebras.

Past REU projects include: REU 2022, REU 2019, and REU 2018.

## Program Benefits

• 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: