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## Department of Mathematics

## Undergraduate Research-Math 444

Name: 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:

A. 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.

B. 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.