# Steven Leth, Ph.D.

# Professor

**Contact Information**

**Phone:**(970) 351-2457**Email:**steven.leth@unco.edu**Office:**Ross Hall 2240-D

## Education

- Ph.D., Mathematics, University of Colorado (1985)
- M.S., Mathematics, Stanford University (1978)
- B.S., Mathematics and Physics, University of Colorado (1976)

## Current Research and Interests

My research is in the area of nonstandard analysis, which could be described as a method for applying concepts from logic to other branches of mathematics. I have published several papers in which this method is used to obtain results in such areas as combinatorial number theory and geometric topology. I am also interested in methods of teaching mathematics, and have given a number of talks about calculus reform, writing in mathematics classes and ways to enrich the mathematical curriculum at the elementary and middle school levels, and have worked with teachers in the Adams 14 district in a professional development setting. I would be happy to discuss any of these issues, mathematical or educational, with anyone who is interested.

#### Recent research papers:

- Nonstandard
methods and the Erdos-Turan conjecture, in
*The Strength of Nonstandard Analysis*, van den Berg and Neves (eds.), Springer - Near Arithmetic Progressions in Sparse Sets, Proceedings of the American Mathematical Society (134) no. 6, June 2006, pp. 1579-1589
- Regions in the nonstandard plane that contain no standard set, Journal of Logic & Analysis 4:16 (2012) 1–17
- On a sumset conjecture of Erdős (joint work with Di Nasso, Goldbring, Jin, Lupino and Mahlburg), to appear in the Canadian Journal of Mathematics
- High density piecewise syndeticity of sumsets (joint work with Di Nasso, Goldbring, Jin, Lupino and Mahlburg), Advances in Mathematics 278 (2015) 1–33
- A monad measure space for logarithmic density (joint work with Di Nasso, Goldbring, Jin, Lupini and Mahlburg), under review

Slides for recent research talks:

Sumsets contained in sets of positive density, University of Denver sesquicentennial Ramsey Theory conference, May 2014

A Lebesgue Density Theorem for nonstandard cuts and an application to additive number theory, Colloquium at the University of Pisa, March 2014

A nonstandard approach to fixed point problems in the plane, Chico Topology Conference, Chico, CA May 2012

Nonstandard methods in continuum theory, Spring Topology Conference, Tyler, TX March 2011

Some questions and answers about “fixed point traps” in the plane, AMS Sectional meeting on nonstandard analysis, Honolulu, HI, March 2012

An example of the use of nonstandard methods in continuum theory, Chico Topology Conference, Chico, CA, May 2010

An introduction to nonstandard methods in the plane