Research Experiences for Undergraduate Faculty (REUF4@ICERM)

SAGE

1. Minimum rank, maximum nullity, zero forcing (Matrices and Graphs)

References

Any standard undergraduate linear algebra book should be fine, but it is easier to read matrix oriented ones, such as:
  Howard Anton, Elementary Linear Algebra, John Wiley
  David C. Lay, Linear Algebra and Its Applications, Addison Wesley
  Steven .J. Leon, Linear Algebra with Applications, Prentice Hall.

The basic graph terms can be found in any elementary graph theory book or combinatorics book that has a chapter on graph theory. For example, Richard Brualdi, Introduction to Combinatorics, Prentice Hall. In the 5th edition the relevant subsection is 11.1.   Here is a PDF version of that chapter.

2. Prime and co-prime labeling of graphs (Graph theory)

References

The basic terms can be found in any elementary graph theory book or combinatorics book that has a chapter on graph theory. For example, Richard Brualdi, Introduction to Combinatorics, Prentice Hall. In the 5th edition the relevant subsection is 11.1.   Here is a PDF version of that chapter.

For more detailed information, just look up the terms in the index of any of these books:
  Schaum's Outline of Graph Theory: Including Hundreds of Solved Problems (Paperback) by V. Balakrishnan
  Introductory Graph Theory by Gary Chartrand
  Pearls in Graph Theory: A Comprehensive Introduction by Nora Hartsfield and Gerhard Ringel

3. Dessins d'enfants (Algebra, Graphs, Geometry)

References

This short article gives an overview of Dessins d'Enfants but may be difficult to read carefully before the workshop- it may be fun to skim it:  Notices Amer. Math. Soc. 2003 Zapponi.pdf

If you get hooked by Dessins d'enfants and want to know more, here is background that will be used in the project:  background.pdf

4. Fractionally dense sets (Analysis and Diophantine Equations)

Stephan's math slides

Terminology

This project is designed to be accessible to a variety of students from various backgrounds.  The basic questions can be understood by a solid calculus student, while more advanced questions can be addressed by more sophisticated students and faculty.  For the basics, we require only knowledge of real numbers, rational numbers, and limits. Knowledge of other topics (e.g., prime numbers, Fibonacci numbers, matrices, countable sets, complex numbers, etc.) can be used to develop other problems.

References

The basics of the real number system and limits can be found in many calculus books or introductory analysis texts.  A useful blend of calculus and number theory at the undergraduate level is: Shahriari; "Approximately Calculus"

Some basic familiarity with elementary number theory may be useful, but is not strictly necessary.  Some possible references (in increasing order of sophistication) are:
Pommersheim, Marks & Flapan; "Number Theory: A Lively Introduction with
Proofs, Applications, and Stories"
Burton; "Elementary Number Theory"
Niven, Zuckerman & Montgomery; "An Introduction to the Theory of Numbers"
Hardy & Wright; "An Introduction to the Theory of Numbers"