NSF Research Experience for Undergraduates

I have had extensive experience with the National Science Foundation's Research Experience for Undergraduates (REU) program, both as an undergraduate participant and as a director of undergraduate research projects. As a student, I participated in the Trinity University (Algebra, 2001), University of Idaho (Graph Theory, 2002), and Pennsylvania State University (Number Theory, 2003) REUs. These rewarding experiences not only introduced me to mathematical research, but they allowed me to see several different models of running an REU (small groups with oversight, independent research, and lecture style) from the student's point of view. I saw the advantages and disadvantages of each model and what kind of students they served best (more background vs. less background, confident vs. needing some reassurance, etc). In the following years, I applied this experiential knowledge to mentoring my own students. I returned to Trinity University in the summers of 2005, 2006 and 2008 to assist Dr. Scott Chapman in the running of his program. Together, we had several successful groups run discover their talents for mathematical research; their subsequent paths of study can be found here.

For publications listed below, an asterisk * indicates an undergraduate author.

Projects Directed

Trinity University REU, Summer 2005

During the summer of 2005, I returned to the Trinity University REU as a full-time graduate mentor. I assisted with the general operations of the program and became great friends with the students. I also collaborated with Dr. Chapman's group (Chris Crutchfield, Grace Kennedy and Matthew Wright) on factorization properties of polynomial and power series rings. Specifically, we were studying the property of full elasticity in numerical semigroup rings, a generalization of the group ring construction, where one uses a numerical semigroup instead of the group (any semigroup is possible, but this is the one we focused on). We also considered the power series analogue of this construction. Chris, Grace and Matthew presented our research at the Young Mathematicians Conference at the Ohio State University in Columbus, Ohio in August 2005. Chris and Grace also presented our research during the undergraduate poster competition of the AMS and MAA Joint Meeting in January 2006 in San Antonio, Texas. I gave a lecture on our work during the special session on Commutative Rings and Monoids at the same conference.

There is one paper already published from this research, and a second paper is foreseen once some additional results are proven.

1. P. Baginski, Scott T. Chapman, Christopher Crutchfield*, K. Grace Kennedy*, and Matthew Wright*. Elastic Properties and Prime Elements Results Math. 49 (2006), 187-200.

Trinity University REU, Summer 2006

I reprised my role as graduate mentor during the summer of 2006. There were 7 students under my and Dr. Chapman's direction, split into three research groups. The topics studied were: Scott Chapman and myself presented this summer's research at University of Graz in Graz, Austria during December 2006.

1. Paul Baginski, Scott T. Chapman, Natalie Hine,* and João Paixão.* On the Asymptotic Behavior of Unions of Sets of Lengths in Atomic Monoids Involve 1 (2008), No. 1, 101-110.

2. Paul Baginski, S. T. Chapman and George J. Schaeffer.* On the Delta Set of a Singular Arithmetical Congruence Monoid J. Théor. Nombres Bordeaux, 20 (2008), 45-59.

3. Paul Baginski and Ross Kravitz.* A New Characterization of Half-Factorial Krull Monoids J. Algebra Appl. 9 (2010), No. 5, 825-837.

Trinity University REU, Summer 2008

In 2008, the Trinity REU was run solely by Scott Chapman and three three graduate mentors, myself, George Schaeffer and Ross Kravitz. As the senior graduate mentor, I oversaw the general workings of all the projects, though I most closely mentored a project by Yiwei She and Ryan Rodriguez. These two undergraduates worked to understand the length sets of Krull monoids whose class group is isomorphic to Z, the integers. They obtained a characterization of when the catenary degree and Delta set are finite, and also worked towards a constructive proof of Kainrath's theorem for this particular group Z. Kainrath's theorem states that in the full block monoid over any infinite abelian group, any length set can be realized.

The group's research resulted in one publication and a second one is in preparation.

1. Paul Baginski, S. T. Chapman, Ryan Rodriguez,* George J. Schaeffer, and Yiwei She.* On the Delta Set and Catenary Degree of Krull Monoids with Infinite Cyclic Divisor Class Group J. Pure Appl. Alg., 214 (2010) 1334-1339.

Projects as an Undergraduate

Trinity University REU, Summer 2001

During the summer of 2001, I spent two months under an NSF Research Experience for Undergraduates (REU) grant working at Trinity University in San Antonio, Texas. My project advisor was Dr. Scott T. Chapman and my research partners were Kathryn McDonald and Lara Pudwell. Our research focused on minimal zero sequences of additive finite Abelian groups. Our results have been published and can be viewed under the first publication listed on my publications page. A more in-depth explanation of our work and related results can be found here. Posters based on this research won prizes at the undergraduate poster competition of the Joint Meetings of the AMS and MAA in January 2003, and at the Meeting of the Minds undergraduate poster competition at Carnegie Mellon University.

University of Idaho REU, Summer 2002

During the summer of 2002, I spent two months at the University of Idaho in Moscow, Idaho under an NSF REU grant. My project advisor was Dr. Arie Bialostocki, who has provided a description of the program and potential areas of research on the following link. My research focused on a combination between zero-sum Ramsey theory and the Ramsey theory of sparse graphs. The final version of the paper may be found under the second link on my publications page. A poster based on this research won first place at the Meeting of the Minds undergraduate poster competition at Carnegie Mellon University.

Pennsylvania State University REU, Summer 2003

During the summer of 2003, I spent two months at Penn State doing research on elliptic curves with a fellow student, Elena Fuchs. Dr. Yuri Zarhin directed us to a problem on calculating the coefficients of the modular equation. As a result of our work, Elena and I obtained an algorithm for calculating the coefficients of the modular equation of degree n for n=pq the product of two distinct primes. This algorithm operates in linear time as a function of the number of coefficients. Our research is the first collection of notes listed on my publications page. Elena and I presented our research at the Undergraduate Mathematics Day at the University of Dayton in Dayton, Ohio in November 2003.

(c) All material accessible through this page is copyright by Paul Baginski and his coauthors. Permission is granted for fair use in personal, noncommercial, and academic projects.