Mike Krebs: Research

 Below is a brief description of my current research; hopefully, it should be comprehensible, even to non-mathematicians. This PowerPoint presentation has a few more details, for a general math audience. For more details, click the image on the right for our book, Expander Families and Cayley Graphs: A Beginner's Guide, from Oxford University Press. The theory of expander graphs is a hot topic in mathematics and computer science, and with good reason: these objects enjoy a wide range of applications. Think of a communications network as a graph, by which we mean a collection of dots (vertices) and line segments (edges) connecting them. Each vertex represents a person (or a telephone, or a computer) in the network; two vertices are connected by an edge if they can communicate directly with one another. When designing a communications network, one naturally wishes that any message originating somewhere within it will propagate as rapidly as possible. There are various measurements which describe how well a graph spreads messages. One such quantity is called the spectral gap of the graph. Roughly speaking, the larger the spectral gap is, the better the graph is as a communications network. If one attempts to create regular graphs with more and more vertices but without increasing the degree, the spectral gap may very well become infinitesimally small as the communications networks become poorer and poorer. A family of expander graphs is an infinite sequence of regular graphs, each with the same degree, such that the spectral gap does not vanish altogether in the limit. Roughly speaking, graphs with the best possible spectral gap are called Ramanujan graphs; families of Ramanujan graphs are the most desirable families of expander graphs.

All files below are in pdf format. Some are preprints, not necesarily the final version.

Toledo invariants of Higgs bundles on elliptic surfaces associated to base orbifolds of Seifert fibered homology 3-spheres. (Appears in Michigan Math. J. Volume 56, Issue 1 (2008), 3-27.)

On the Spectra of Johnson Graphs. Joint work with A. Shaheen. (Appears in Electronic Journal of Linear Algebra, Volume 17, pp. 154-167, March 2008)

Think Around The Box, a crossword puzzle of sorts. The Mathematical Intelligencer, Volume 31, Number 2, 59-60. Here is the solution.

I'm the faculty adviser for Fixed Points of Number Derivatives Modulo n by CSULA student F. Bains. Rose-Hulman Undergraduate Math Journal, Vol. 10 (2009), Issue 1.

How to differentiate an integer modulo n. Joint work with C. Emmons and A. Shaheen. College Mathematics Journal, November 2009.

On Cantor's first uncountability proof, Pick's theorem, and the irrationality of the golden ratio. Joint work with T. Wright. Amer. Math. Monthly 117 (2010), no. 7, 633–637

mini-Sudokus and Groups. Joint work with C. Arcos (CSULA student) and G. Brookfield. Math. Mag. 83 (2010), no. 2, 111–122

Expander Families and Cayley Graphs: A Beginner's Guide, co-authored by A. Shaheen, Oxford University Press, New York, Oct. 2011

K-quasiderivations. (K-quasiderivations are maps that satisfy both the Product Rule and Chain Rule.) Central European Journal of Mathematics, Vol. 10 (2012), no. 2, 824–834

A combinatorial trace method: Counting closed paths to assay graph eigenvalues. Joint work with K. Dsouza (CSULA student). Rocky Mountain Journal of Mathematics, Vol. 43 (2013), no. 2, 469–478

Simple examples of the combinatorial trace method in action. Joint work with N. Martinez (CSULA student). College Mathematics Journal, Vol. 44 (2013), no. 1, 32–36

Expander families, group structure, and semidirect products. Joint work with M. Aivazian (CSULA student). Bulletin of the Irish Mathematical Society, Number 71 (2013), 21–30

Expander graphs. In J. Gross and J. Yellen (Eds.), Handbook of Graph Theory, Second Edition. CRC Press, 2013.

ZERO SUMS ON UNIT SQUARE VERTEX SETS, AND PLANE COLORINGS. Joint work with R. Katz and A. Shaheen. The American Mathematical Monthly. Vol. 121, No. 7 (August–September 2014), pp. 610-618.

Al Fares, A. (CSULA student), E. Golvin (CSULA student), M. Krebs. A class of 2-groups of derived length three. Serdica Mathematical Journal, Volume 41, Numbers 2-3, 2015, pp. 329-332

Kevin Cramer (CSULA student), Mike Krebs, Nicole Shabazi (CSULA student), Anthony Shaheen and Edward Voskanian (CSULA student). The Isoperimetric constant and Kazhdan constant of a Paley graph, to appear in Involve.

The slideshows below need some narration, but hopefully it's possible to get something out of them even without all the accompanying explanations. One of these days, I'll add some audio.

"Using Groups and Graphs to Build a Better Communications Network: A Brief Introduction to Expanders and Ramanujan Graphs":   Slideshow—short version    Slideshow—longer version

"Coloring the Plane with Rainbow Squares":   Slideshow   Goes with the "unit square" paper above.

"Number derivatives: A treasure trove of student research projects":   Slideshow

"The No WAY Moment in Mathematics" (A surprising pattern emerges from a slight twist on Cantor's original uncountability argument.): Download both the slideshow and the accompanying file into the same folder before viewing the slideshow.

"Beaucoup de Sudoku":   Slideshow—short version    Slideshow—longer version

"Toldedo invariants on 2-orbifolds":   Abstract

"Spectra of Johnson Graphs":   Abstract    Slideshow

"Tell Me What You Can About a Group of Order n":   A slideshow describing a project I did in an Abstract Algebra class a few years ago

"Fractals":   A slideshow on fractals for a course I taught a few years ago

I graduated from Johns Hopkins in May 2005. My thesis advisor was Dr. Richard Wentworth.

This website was last updated on Feb. 4, 2011.