## Department of Mathematics and Philosophy

### Undergraduate Research - Algebra

#### Faculty

##### Samson Adeleke

**Areas of Interest:** Algebra, Combinatorics, Differential Equations

**Description:** The material could be on any topic beyond the contents of Math 355 (Combinatorics), Math 421 (Abstract Algebra), and Math 333 (Ordinary Differential Equations). In more detail, the Math 355 course introduces applications of counting and graphs to computer science, chemistry, operations research, etc. The Math 421 course covers the basics of abstract algebra of groups, ranges, and fields, an aesthetically satisfying subject while Math 333 on Differential Equations models down-to-earth phenomena across a broad spectrum.

##### Tom Blackford

**Areas of Interest:** Algebraic coding theory, algebra, combinatorics

**Description:** Coding theory involves the transmission of encoded information across a noisy communications channel. It has been used in telegraphs, computers, CD-players and cash registers. It provides mathematical algorithms to correct errors in transmission. Often information is transmitted as blocks of binary digits, and an algebraic structure is put on these blocks. This leads to formation of special types of codes including cyclic, negacyclic, and quasi-cyclic codes. Techniques in linear and abstract algebra are used in decoding and error-correcting.

I am looking for a student who is interested in studying algebraic coding theory and in particular cyclic codes and their generalizations. I will provide any abstract algebra that is needed.

##### Clifton Ealy

**Areas of Interest:** Logic, Algebra, Group Theory, Nonstandard analysis

**Description:** I would invite students to take an undergraduate class they enjoyed and attempt to deepen their understanding of that subject by pursuing a project that picks up where the class leaves off. Such a project could explore connections of that subject to other subjects, or applications of that subject, or just explore the same subject from a different standpoint. For example, a student who had enjoyed Math 421 (Group Theory) could explore connections between group theory and graph theory. Or such a student's project might deal with the applications of group theory to cryptography or error correcting codes. A student who had enjoyed Math 435 (Real Analysis) might study how the subject could be developed using infinitesimals rather than epsilon-delta proofs.

##### Mei Yang

**Areas of Interest:** Algebra and Combinatorics

**Description:** For students who have taken the introductory combinatorics course and learned the basic enumeration techniques and graph theory, there are many directions you may further explore. Here are a few suggestions:

(1) Compare advantages and disadvantages of different techniques by solving problems using more than one method.

(2) Investigate the relationships among different techniques. For instance, how to get generating functions from recurrence relations, or vice versa.

(3) You may further study the generating function method, which nowadays is the main language of enumerative combinatorics. Topics range from compositions of generating functions, generating functions in several variables, and application of generating functions to enumeration of trees, and various kinds of graphs.

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