## Department of Mathematics and Philosophy

### Undergraduate Research - Logic

#### Faculty

##### John Chisholm

**Areas of Interest:** Primary area of expertise: Mathematical Logic. I'm offering to provide guidance to students interested in researching METHODS OF FAIR DIVISION.

**Description:** The goal is to devise procedures to divide a 'cake' (or any kind of 'goods'), with each recipient being able to ensure that they receive a "fair share". It's not enough to try to divide the cake into "equal pieces", since pieces that have equal value in one person's view could easily be unequal to somebody else. With just two people involved, they can use the well-known method of having one cut the cake into two pieces of equal value and letting the other choose which of the two pieces to receive. If we increase the number of people involved, however, a new method must be devised; we can also examine the effects of changing the setting in other ways such as trying to various different standards of "fairness".

##### Rumen Dimitrov

**Areas of Interest:** Mathematical Logic, Recursion Theory, Recursive Theory, Recursive Model Theory, (Geometric) Lattice Theory

**Description:** I would like to encourage the interested students to explore different aspects of recursion theory and its application in vector spaces. Mathematical studies sets in general, while recursion theory studies their information content. The Turing degrees of the sets are measures of their information content. The most "orderly" sets are called recursive and have Turing degree 0. I would like to propose the following topics:

###### Recursion Theory:

- Post Program: Explore the simple, hypersimple, and hyperhypersimple sets.
- Oracle construction of non-R.E. degrees and the forcing method in recursion theory.
- The Finite Injury priority Method. Present the original Friedberg-Muchnik Theorem.
- Major sets.

###### Recursively Enumerable Vector Spaces:

- Present the Fundamental Theorem of Projective Geometry.
- Prove the Existence of Maximal Computably Enumerable Vector Spaces.
- Present the Existence of R.E. spaces with no extendable basis.

##### 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.

##### Jana Marikova

**Areas of Interest:** Mathematical Logic, Model Theory, o-minimality, Real Geometry

**Description:** If you are interested in working on a project in any of the above you are very welcome to talk to me, so that we can find a problem that suits both your interest and your level of expertise.

To give an example, a student interested in real geometry may want to deepen his/her understanding of the notion of a real closed field. There is a number of, at the first glance very different, definitions of this notion, and you could investigate why all of them yield the same class of objects, and the reasons why the field of real numbers stands out among them (and why people who are interested in the reals find other real closed fields useful).

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