# Undergraduate Research Projects

Mathematics research by undergraduates in the United States has gone from rare in the mid-1980s to commonplace now. Undergraduate contributions range from introductory to cutting edge, with occasional publications in some of the highest level professional journals. Now there are several undergraduate journals, Involve, Rose-Hulman, Minnesota, SIAM, and Pi Mu Epsilon among them, containing significant results in all mathematical specialties.

These days undergraduate research experience is critical to being competitive for entrance into top graduate programs. Numerous summer programs are funded by the National Science Foundation and the Mathematics Association of America, among others. VCU also offers summer support through UROP and HSURP, the latter available to Honors students only. These experiences are also highly valuable for employment at the plethora of National Labs and Federal Agencies (including the National Security Agency, the largest employer of mathematicians in the world), as well as the huge array of technical companies in STEM industries like communications, biotech, information, finance, internet, pharmaceuticals, entertainment, and more.

But there are other, more fundamental reasons to engage in mathematical research. Students often gain a greater understanding of how mathematics works in the modern world, a heightened facility with generating and analyzing data in any realm of life, an exuberance for group and solo problem solving that welcomes hard challenges, a stronger ability to communicate technical information, and a deepened connection with fellow students and professors in the project. Not bad.

The selection of projects with the professors listed below offer students opportunities to work closely with Math Department professors, as an independent study or summer activity, potentially receiving funding through one of the sources mentioned above. It can provide an excellent springboard to some of the other opportunities discussed as well. Please read the project descriptions and contact the professor with whom you would like to study.

**Graph Pebbling - Professor Hurlbert**

This project studies a network model for the transportation of consumable resources. As in most models, a supply of the resource (in this case, "pebbles") is placed on various vertices of the graph, and a demand of pebbles is identified at certain other vertices. In this model, the movement of a pebble across an edge requires the loss or payment of a partner pebble; that is, while two pebbles leave a vertex together, only one arrives at the other end of the edge, as the other pebble vanishes. The central question asks whether a given supply can satisfy a fixed demand, but many other significant questions exist. We study this paradigm on an array of special graph classes, seeking to discover when one can be successful in achieving assorted goals. This field is new enough to have numerous interesting problems within reach of talented undergraduate students, and there have been many strong contributions from students over the years. Students with some background in graph theory, discrete math, combinatorics, and/or computer programming are particularly well suited, but the essential concepts can be learned as needed.

**Fluid Dynamics of Nematocysts - Professor Segal**

A nematocyst is a specialized organelle within cells of jellyfish and other Cnidarians that sting. They contain a barbed, venomous thread that accelerates faster than almost anything else in the animal kingdom. This is counter-intuitive because the organisms themselves generally move very slowly. Using computational flow simulations, we want to investigate the nature of the fluid flow associated with variations in the geometry of the organisms and the configuration of the nematocyst(s). There is a large variety of sizes and shapes of nematocysts and this may have important fluid dynamic consequences, particularly important in the ability of the organism to actually capture prey. Students with a background in MATLAB or other coding platforms will be most successful. Some knowledge of odes and/or pdes is also useful but the essential concepts can be learned as needed.

**Arterial Plaque Formation - Professor Segal**

Chronic disease resulting from a diet high in fat and cholesterol affects over 50% of adults. Although the relationship between atherosclerosis and inflammation is not fully understood, it is known that macrophages and foam cells promote inflammation by secreting pro-inflammatory cytokines following accumulation of cholesterol. We are developing a computational model that accounts for the conversion of macrophages to foam cells and the formation of arterial plaques in atherosclerosis. Understanding the progression of size/shape/density and markers of disease progression are all important to examine with regard to the diagnosis and long term prognosis of individuals with atherosclerosis. Students with some background in MATLAB or other coding platforms will be most successful. Some knowledge of odes and/or pdes is also useful but the essential concepts can be learned as needed.

**Biodiversity - Professor Chan**

**Pattern Formations in Agent-Based Aggregation Models - Professor Topaloglu**

Many phenomena in the physical and life sciences share surprisingly similar complex structures. Pattern formations in these complex systems, including swarming of animal populations, motion of human crowds, and molecular distribution, are driven driven by competing between long-range attractive and short-range repulsive forces. Various agent-based models (defined via systems of simple ODEs) can be used to describe these phenomena. Current research in the literature is, particularly, interested in understanding fine properties of these pattern formations as the number of agents becomes large. The goal of this project is to develop an efficient computational code (either using MATLAB or another coding platform) which can handle large number of particles as well as domain and density constraints. Beyond good computational and coding skills some basic knowledge of ODEs and linear algebra is expected.

**Efficient Algorithms for the Quantum Satisfiability Problem - Professor Aldi**

In quantum computers the fundamental bits of information are states of a quantum system known as qubits. In analogy with classical algorithms, quantum algorithms can be thought of as recipes to transform information stored in qubits. Some algorithms are faster than others and classifying problem based on how fast they can be solved by a quantum computer is of great importance for both theoretical and practical applications, including cryptography and on-line security. This project aims at developing efficient algorithms to efficiently solve specific instances of the quantum satisfiability problem (QSAT), which is expected to be in general intractable even on a quantum computer. Basic knowledge of linear algebra is recommended, some experience with coding (in any language) is desirable but not required.

**Generalized Geometry of Lie Algebras - Professor Aldi**

Lie algebras provide an algebraic framework for encoding infinitesimal symmetries. Besides their many applications in geometry and in the theory of differential equations, Lie algebras are of fundamental importance in high energy physics as they are required in order to formulate the Standard Model of particle physics. The goal of this project is to study a class of geometric symmetries of Lie algebras known as generalized CRF structures, which have applications to the mathematical study of string theory. No prior knowledge of physics or Lie algebras is required. Familiarity with linear algebra is recommended.