## Department of Mathematics and Philosophy

### Lawrence WelchDepartment of Mathematics and Philosophy

#### Professor

Education:

Ph.D., University of Illinois (Logic, Carl Jockusch)

M.S., University of Illinois (Logic)

B.S., Bucknell University (Music)

Contact Information:

(309) 298-2314
468 Morgan Hall
l-welch@wiu.edu

Spring 2018 Office Hours:

Monday, Tuesday, Wednesday & Thursday: 11:00-12:00
Or by appointment

Courses Taught:

Math 137: Applied Calculus I
Math 435: Introduction to Real Variables I

Personal Web Page:

(This personal web page contains important information about the classes I teach.)

Research Interests:

Research: My principal area of research is computability, with emphasis on computable analysis and computable topology. In particular, I am interested in computable partial functions whose domains contain all computable points, and the shapes that such domains can take.

Research with undergraduate students: There are many advanced functions, like the Jacobi elliptic functions or the sine integral, whose basic properties are within the abilities of an advanced undergraduate student to investigate. I would help a student to grasp the properties of one or two such functions, to understand their uses, and to develop some skill in applying them to mathematical and scientific problems. I would also encourage him or her to investigate similar functions that may arise in some problems but that may not be analyzed in the literature, with an eye to seeing if they are derivable from those under study. In the case of a really ambitious student, I might suggest that one of these other functions become the main topic of research.

This research would lead to the production of a paper detailing what the student has learned. Besides the necessary rigorous mathematical analysis, it would entail the use of a computer algebra system to investigate the graphical properties of the functions, find particular values of them, and so on.

Publications:

• Welch, L., & Kalantari, I. (2013). When series of computable functions with varying domains are computable. Mathematical Logic Quarterly 59, no. 6, 471–493.
• Welch, L., & Kalantari, I. (2008). On degree-preserving homeomorphisms between trees in computable topology. Archive for Mathematical Logic 46, 679–693.
• Welch, L., & Kalantari, I. (2008). On Turing degrees of points in computable topology. Mathematical Logic Quarterly 54, no. 5, 470–482.
• Welch, L., & Kalantari, I. (2007). Turing Degrees & Topology. Computation and Logic in the Real World: Third Conference on Computability in Europe, CiE 2007, Siena, Italy, June 18-23, 2007, Local Proceedings, no. 6, 210-218.
• Welch, L., & Kalantari, I. (2006). Specker's theorem, cluster points, and computable quantum functions. Logic in Tehran: Proceedings of the Workshop and Conference on Logic, Algebra and Arithmetic, held October 18-22, 2003 in the series Lecture Notes in Logic 26, 134–159.
• Welch, L., & Kalantari, I. (2004). A blend of methods of recursion theory and topology: a Π  0 1 tree of shadow points. Archive for Mathematical Logic 43, no. 8, 991–1008.
• Welch, L., &  Kalantari, I. (2004). Density and Baire category in recursive topology. Mathematical Logic Quarterly 50, no. 4-5, 381–391.
• Welch, L. &  Kalantari, I. (2003). A blend of methods of recursion theory and topology. Annals of Pure and Applied Logic 124, no. 1-3, 141–178.
• Welch, L., &  Kalantari, I. (2002). Recursive quantum functions, avoidable points, & shadow points in recursive analysis. Electronic Notes in Theoretical Computer Science 66, no. 1.
• Welch, L., &   Kalantari, I. (1999). Recursive and nonextendible functions over the reals filter foundation for recursive analysis. II. Annals of Pure and Applied Logic 98, no. 1-3, 87–110.
• Welch, L., & Kalantari, I. (1998). Point-free topological spaces, functions and recursive points filter foundation for recursive analysis. I. Annals of Pure and Applied Logic 93, no. 1-3, 125–151.
• Welch, L.V., &   Adeleke, S.A. (1997).  On universal words for automorphism groups of linear orders. Algebra. Ordered Algebraic Structures: Proceedings of the Curacao Conference, Sponsored by the Caribbean Mathematics Foundation, June 26-30, 1995, 315–332.
• Welch, L.V., Downey, R.G., & Remmel, J. B. (1987). Degrees of splittings and bases of recursively enumerable subspaces. Transactions of the American Mathematical Society 302, no. 2, 683-714.
• Welch, L.V., & Downey, R.G. (1986). Splitting properties of r. e. sets and degrees. The Journal of Symbolic Logic 51, no. 1, 88-109.
• Welch, L.V.  (1984). A hierarchy of families of recursively enumerable degrees. The Journal of Symbolic Logic 49, no. 4, 1160-1170.