Steven Leth

Contact Information

Education

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

Professional/Academic Experience

• 1996-present, UNC, Professor of Mathematics
• 1991-1996, UNC, Assoc. Professor of Mathematics
• 1988-1991, UNC, Assistant Professor of Mathematics
• 1985-1988, University of Wisconsin, Van Vleck Visiting Assistant Professor of Mathematics

Research/Areas of Interest

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 worked on applying these methods to obtain results in such areas as combinatorial number theory and continuum theory.

I am also interested in the teaching and learning of mathematics, and have given some talks about ways to enrich the mathematical curriculum at the elementary, middle school, high school and college level.  I have worked with high school teachers in the Adams 14 district and with elementary and middle school teachers in Eagle County and other districts, in a professional development setting. 

I would be happy to discuss any of these issues, mathematical or educational, with anyone who is interested.

Publications/Creative Works

• A monad measure space for logarithmic density (with Mauro Di Nasso, Isaac Goldbring, Renling Jin, Martino Lupini and Karl Mahlburg), accepted for publication in Monatshefte für Mathematic, arXiv:1503.03810

• Approximate polynomial structure in additively large sets (with Mauro Di Nasso, Isaac Goldbring, Renling Jin, Martino Lupini and Karl Mahlburg), accepted for publication inIntegers , arXiv:1508.02350

• High density piecewise syndeticity of product sets in amenable groups (with Mauro Di Nasso, Isaac Goldbring, Renling Jin, Martino Lupini and Karl Mahlburg),  accepted for publication in the Journal of Symbolic Logic, November, 2015, arXiv:1505.04701

• High density piecewise syndeticity of sumsets (joint work with Di Nasso, Goldbring, Jin, Lupino and Mahlburg), Advances in Mathematics 278 (2015) 1–33

• On a sumset conjecture of Erdős (joint work with Di Nasso, Goldbring, Jin, Lupino and Mahlburg), to appear in the Canadian Journal of Mathematics

• Regions in the nonstandard plane that contain no standard set, Journal of Logic & Analysis 4:16 (2012) 1–17

• 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

• An asymptotic formula for powers of binomial coefficients (with Jeff Farmer), The Mathematical Gazette, v. 89 no. 516, 2005, pp. 385-391

• The Use of Writing in mathematics classes: the new imperative The Project Calc Newsletter, December 1994

• Meager Sets on the hyperfinite time line, (with H. J. Keisler), Journal of Symbolic Logic,56 Number 1 (March 1991) pp. 71-102.

• Descriptive set theory over hyperfinite sets, (with H.J. Keisler, K. Kunen and A. Miller), Journal of Symbolic Logic. 54, Number 4 (Dec. 1989) pp. 1167-1180.
• Some nonstandard methods in combinatorial number theory, Studia Logica XLVII 3 (Sept. 1988) pp. 85-98.

• Sequences in countable nonstandard models of the natural numbers, Studia Logica XLVII 3 (Sept. 1988) pp. 63-83.

• Applications of nonstandard models and Lebesgue measure to sequences of natural numbers, Trans. Am. Math. Soc. 307, No. 2 (June 1988) pp. 457-468.

•  Some nonstandard methods in geometric topology, in  Developments in Nonstandard Mathematics, Pitman Research Notes in Mathematics No.336, Cutland, Neves, Oliveira and Pinto (eds.), Longman; Wiley and Sons, 1995, pp. 50-60

• A uniqueness condition for sequences, Proc. Am. Math. Soc. 93 (1985) pp. 287-290.

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

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