This semester the seminar will be dedicated to the study of groups and their relation to other branches of mathematics, in particular to geometry and graph theory.

Our textbook will be Groups and Their Graphs by Israel Grossman and Wilhelm Magnus (The Mathematical Association of America, New Mathematics Library). The book is unfortunately out of print, but some used copies are available on-line, and bound photocopies can be obtained as 640:196 text at the Student Cooperative Store on Douglass College, 57 Lipman Drive, New Brunswick, NJ 08901, 732-932-9017.

A great optional book is M.A. Armstrong's Groups and Symmetries, published by Springer Verlag.

The chapters assigned as reading below are chapters of this book. Don't be dismayed at being assigned two or even three chapters in one week: most chapters in this book are very short, in some cases under three pages.

We will also rely on seminar notes. Some of these are already posted at this site; further notes will be added by professors and students as we go along, so do check this site frequently. [All links on this page are html files or pdf files, the latter ones explicitly indicated as such.]

There are some useful notes posted by students of this seminar. As an inspiring example from an earlier semester, look at the student webpage [a pdf file] about Non-Euclidean Geometries by Aron Samkoff (Fall '03).

For bibliographical data, go to the excellent Scottish website of the University of St Andrews. You may want to start at the history page and browse, or go directly to the index of biographies.

January 22 (Prof. Enriqueta Carrington): Introduction to groups: basic definitions, examples familiar and unfamiliar.
Reading: Chapters 1, 2, and 3; and the following seminar notes: some basic algebraic structures, groups and fields, try these simple algebra problems.

January 29 (Prof. Carrington): Cyclic groups (finite and infinite), the dihedral groups, generators and relations, subgroups.
Reading: Chapters 5, 6, and 7; seminar notes on cyclic groups.

February 5 (Helen Janiszewski and Vicky Yu): Modular arithmetic, equivalence relations.
Reading: Chapters 3 and 8, and the following links.

February 12 (Mark Kim and Aaron Rosenberg): Symmetric and alternating groups.
Reading: Chapters 10, 13; a short summary, and the Appendix.

February 19 (Prof. Simon Thomas): Killing the Hydra!
A fascinating story of mythology and set theory.
Reading: The Lernaean_Hydra.

February 26 (William Hoese): The group of symmetries of each Platonic solid.
Reading: The Appendix, and the website Symmetry groups of Platonic solids.

March 5 and 12 (Itai Feigenbaum and Dakota Killpack): The Euler-Descartes Formula for polyhedra and for plane graphs, and two of its consequences: (1) all plane maps can be colored with at most five colors, (2) there are only five Platonic solids.
Reading: Nineteen proofs; Platonic graphs; Planar graphs and plane graphs; Mathworld on the Four-Color Theorem; coloring maps on other surfaces.
See also the pdf file: Are these proper definitions?

Remark: The standard way to translate questions about polyhedra into questions about planar graphs is by using a bijection called stereographic projection.   See how it was perceived by Peter Paul Rubens in his illustration to the book "Opticorum libri sex philosophis juxta ac mathematicis utiles" by François d'Aiguillon.

Spring break

March 26 (Helen Janiszewski and Margarita Molodan): Subgroups, Lagrange's Theorem for Abelian groups, cosets, Lagrange's Theorem in its full glory; normal subgroups of a group, conjugacy.
Reading: Chapters 8 and 11; Prof. Lyons's 351 class notes.

April 2 (Dustin Richwine and Dan Sroczynski): Some basic tools from linear algebra: orthogonal matrices. Proof that rotations in R³ fixing `the origin' form a group.
Reading: Orthogonal matrices. Topics in Geometry parts 2 and 4.

April 9 (Darlayne Addabbo and Patrick Kramer): Isometries in Rⁿ and global positioning   (this talk can be moved anywhere in the schedule).
Reading: Seminar notes on isometries.

April 16 (John Marcus and Dustin Richwine): Finite groups of symmetry in R² and R³; another glorious appearance of the Platonic solids.
Reading: Topics in Geometry by John O'Connor, especially lectures 11 and 10.
Outline of the proof.   Sideboard summary.

April 23 (Dakota Killpack and Margarita Molodan): Groups and wallpaper designs.
Reading: Chapter 15; The 17 plane symmetry groups.

For artful wallpapers from various cultures, see Wikipedia's wallpaper article, and the Japanese site mathmuse.

April 30 (Prof. Roe Goodman): Alice Through Looking Glass after Looking Glass; the Mathematics of Mirrors.
Reading:  The Mathematics of Mirrors and Kaleidoscopes.

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