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Academic Research in Mathematics & Computer Science

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As a Loyola student, you have the opportunity to work alongside our talented professors to partner in collaborative research. Learn more about some recent research and projects currently underway.

    Zero Divisor Graphs

    My research involves using graph theory to study rings. A ring is a set within which we can add, subtract, and multiply. Strangely enough, sometimes in a ring the product of two non-zero elements is zero! For example, this can happen when we multiply matrices.

    We can represent this situation using a graph. A graph is just a bunch of point with lines connecting them. Graphs can be used to describe many situations, such as the patterns of city streets, computer networks, and the structure of molecules, among other things.  In my research, we write down all the zero divisors in a ring and connect two of them if their product is zero. We can then study the graph to find patterns which can give information about the ring. Here are two fairly simple graphs.  

    Over the last two years two of my students have received $4,000 grants to study these graphs. The study of these graphs provides a good introduction to undergraduate research.

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    Michael Kelly Published Research

    Dr. Michael Kelly is currently working on research in the area of topology known as Fixed Point Theory. A collaboration with Professor D. Goncalves of the University of Sao Paulo, Brasil has lead to a research article, which will appear in the journal Bulletin of the Mexican Math. Society. This collaboration is ongoing and also involves a research project which is related to the graphic on the left. 

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    Mathematics Investigation

    Second year mathematics major Linda Hexter is currently working under the supervision of Dr. Kelly.  She is working on a project exploring idempotent matrices and implications to a problem in fixed point theory regarding homotopy idempotents.

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    Zero Divisor Graphs

    Zero divisors are objects that arise in one of the most abstract areas of mathematics. Surprisingly, investigators are able to study zero divisors using computational and geometric techniques. One of the geometric techniques involves diagrams called zero divisor graphs. Since 1988 there has been a plethora of articles on this topic. Dr. Thibodeaux and Dr. Tucci have published a joint paper in Communications in Algebra (Volume 42, Issue 9, 2014), “Zero Divisor Graphs of Finite Direct Products of Finite Rings,” which adds the continuing discussion on Zero divisors, and they are completing a second. Both of them are submitting papers which are co-authored with students. 

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    Honors Mathematics student Andrew Brinsko completed his honors thesis under the supervision of Dr. Saxton. The title of the thesis is “Diverse Cases of Singularity Formation in Hyperbolic Partial Differential Equations”. Through exploration of continuous and discontinuous initial conditions for a diverse cases of Hyperbolic Partial Differential Equations, the authors concluded that in some cases, a solution becomes unbounded at some point, while in others the solution becomes a 𝛿-solution.

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    Honors Mathematics student Coleman D. Green completed his honors thesis under the supervision of Dr. Kelly. The title of the thesis is “Fixed Point Linking For Self-Mappings of the Disk”. The author investigated the behavior of the fixed points of homeomorphisms of the disk admitting a periodic orbit, and found that the particular class of maps studied in this thesis fill out the linking difference set very often. Further research will be needed to relate the current results back to periodic orbits in the torus construction.

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