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 $16 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. [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 24 (Prof. Enriqueta Carrington): Introduction to groups: basic definitions, examples familiar and unfamiliar.
Reading: Chapters 1, 2, and 3; and the following seminar notes [pdf files]: some basic algebraic structures, groups and fields, try these simple algebra problems

January 31 (Prof. Carrington): Cyclic groups (finite and infinite), the dihedral groups, generators and relations, subgroups.
Reading: Chapters 5, 6, and 7, and the following pdf files: a seminar note on cyclic groups, and Tim Lou's proof that the odd dihedral groups can be generated by two relations.

February 7 (Kristen Lew and Emily Sergel): Modular arithmetic, equivalence relations.
Reading: Chapters 3 and 8, and the following links.

February 14 (Matt Edwards and Ashwin Ramaswamy): Symmetric and alternating groups.
Reading: Chapters 10, 13; a short pdf summary, and the Appendix.

February 21 (Billy Frank and Jonathan Sloane): Isomorphism of groups [a pdf file], the group of symmetries of each Platonic solid.
Reading: The Appendix, and the website Symmetry groups of Platonic solids. See also the summary chart used by the speakers in class [a pdf file].

Remark: Soccer balls are truncated icosahedra.   The truncated icosahedron easily verifies the Euler characteristic: 32 + 60 - 90 = 2 (see a later talk).
The so-called buckyballs are also shaped as truncated icosahedra, are also made of twenty hexagons and twelve pentagons with a carbon atom at the vertices of each polygon and a bond along each polygon edge.

February 28 (Kristen Lew and Emily Sergel): Subgroups, Lagrange's Theorem for Abelian groups, cosets, Lagrange's Theorem in its full glory. Normal subgroups of a group.
Reading: Chapters 8 and 11; Prof. Lyons's 351 class notes [in a pdf file].

March 6 (Scott Schneider, grad student): Killing the Hydra!
Reading: The Lernaean_Hydra.

March 13 (Daniel Leven and Cristina Mamolea): 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 27 (Walter Simons-Rose): Geometric aspects of isometries (a.k.a. congruences) in Euclidean spaces. Global positioning in R² and R³.
Reading: The seminar note on isometries [a pdf file], and   the Wikipedia article on global positioning.

April 3 (Prof. Michael Vogelius):

April 10 (Matt Leone-Zwillinger and David Trethewey): Linear algebraic aspects of isometries in R^d. Some basic tools from linear algebra: orthogonal matrices. Proof that rotations in R³ fixing `the origin' form a group.
Reading: Topics in Geometry parts 4, 2, and our seminar note Orthogonal matrices [a pdf file].

April 17 (Prof. József Beck): The mathematics of ``well-spreadness".

April 24 (Sam Coskey, grad student): The Banach-Tarski Paradox -- an application of the group SO(3) to create madness.

May 1 (Prof. Roe Goodman): Alice Through Looking Glass after Looking Glass; the Mathematics of Mirrors.
Reading:  The Mathematics of Mirrors and Kaleidoscopes [a pdf file].
 

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.

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 website mathmuse.

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