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Stephan's comments on undergraduate research

Trends in Undergraduate Research in the Mathematical Sciences conference in Chicago, October 26 - 28, 2012

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Terminology

vector space, subspace, matrix, matrices and linear maps,matrix arithmetic, elementary row operations, (reduced) row echelon form, transpose, symmetric, nonsingular = invertible matrix, similar matrices, span, linear independence, basis, dimension, rank, range, kernel, column space, row space, determinant, properties and elementary methods of computation of determinant, eigenvalue, eigenvector, characteristic polynomial, spectrum, spectral radius, algebraic and geometric multiplicity of an eigenvalue.

Graph (simple, undirected) - note that the definition of a graph depends on the book so the first think you need to do with any book is figure out what is meant by "graph."

Subgraph, induced subgraph, path, cycle, complete graph, connected graph, connected component, graph isomorphism, planar graph, tree, adjacency matrix.

*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.

Terminology

Graph (simple, undirected) - note that the definition of a graph depends on the book so the first think you need to do with any book is figure out what is meant by "graph."

Subgraph, induced subgraph, path, cycle, complete graph, connected graph, connected component, graph isomorphism, planar graph, tree, adjacency matrix.

*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

Terminology

Alternating Group, Symmetric Group, Bipartite Graph, Path Graph, Platonic Solid, Chebyshev Polynomial, Mobius Transformation

*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

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"

Questions or comments to