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 for $18 from Pequod (119 Somerset Street, New Brunswick, 732-214-8787).

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. The names of the student speakers, more lecture titles, and other details will be added to the list of talks as they become known.

There are some useful notes posted by students of this seminar. As an inspiring example from an earlier semester, look at the student webpage 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 19 and 26 (Profs): Introduction to groups: basic definitions, examples familiar and unfamiliar, generators of groups.
Reading: Chapters 1, 2, 3, and 7; seminar notes on basic algebraic structures and exercises.

February 2 (Jordan Ledvina and Scott Robinson): Subgroups, cyclic groups (finite and infinite).
Reading: Chapter 5 and 6; seminar notes on cyclic groups.

February 9 (Milton Bose and Noam Brown): Modular arithmetic and equivalence relations.
Reading: Chapters 3, 5, and 6.

February 16 (Vincent Lam and Sofia Machado): Symmetric and alternating groups.
Reading: Chapters 10, 13, a short summary, and the Appendix.

February 23 (Mark Labrador and Adam Pantel): Isometries in Rⁿ.
Reading: Seminar notes on isometries.

March 2 (Pawel Grzech and Brian Turner): Isomorphism of groups, the group of symmetries of each Platonic solid.
Reading: The Appendix, and the website Symmetry groups of Platonic solids.

March 9 (Jeremy Engel and Jack Hanson): The Euler-Descartes Formula for polyhedra and for plane graphs and two of its consequences: (1) there are only five Platonic solids, (2) all plane maps can be colored with at most five colors.
Reading: Seventeen proofs; Platonic graphs; Planar graphs and plane graphs; Mathworld on the Four-Color Theorem; coloring maps on other surfaces.

March 23 (Chris Skalit): Subgroups, cosets, Lagrange's Theorem, conjugacy.
Reading: Prof. Lyons 351 class notes.

March 30 (Yifan Lin and Sofia Machado): 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.

April 6 (Yifan Lin and Sofia Machado): Some basic tools from linear algebra: orthogonal matrices. Proof that rotations in R³ fixing `the origin' form a group.
Reading: Topics in Geometry parts 2 and 4.

April 6 (last 20 minutes): Adam Pantel and Avi Rashin present their own proof for coloring the dual of Eulerian graphs.

April 13 (Prof. Michael Beals): When Singularities Collide: Some Wave Interactions.

April 20 (Sam Coskey, grad student): The Banach-Tarski Paradox -- an application of the group SO(3) to create madness.

April 27 (Prof. Roe Goodman):  Alice Through Looking Glass after Looking Glass; the Mathematics of Mirrors.
Reading:  The Mathematics of Mirrors and Kaleidoscopes.
          Alice's Adventures in Wonderland and Through the Looking Glass.