This semester the seminar will be dedicated to the study of groups and their relation to other branches of mathematics, in particular to 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 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 heavily 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 inspiring examples from the Fall '03 semester, look in particular at Aron Samkoff's Non-Euclidean Geometry and Vidya Venkateswaran's Fields and Constructible Numbers.
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 and 29
(Profs):
Introduction to groups: basic definitions, examples familiar and unfamiliar,
subgroups, the importance of groups in math and science.
Reading:
Chapters 1, 2, and 3;
seminar notes on basic algebraic structures,
exercises.
February 5
(Dennis Faynberg and Charles Siegel):
More group examples: modular arithmetic, generators of a group, cyclic groups.
Reading:
Chapters 4 and 5;
seminar notes on cyclic groups.
February 12
(Yuliya Karuchek and Aron Samkoff):
The graph of a group, more on generators, cyclic groups and
the dihedral groups (finite and infinite), product of cycles.
Reading:
Chapters 5 and 6.
February 19
(Pablo Mosteiro and Ilija Zeljkovic):
The symmetric and alternating groups;
the group of symmetries of each Platonic solid.
Reading:
Chapter 13 and the Appendix, and the website
Symmetry groups of Platonic solids.
February 26
(Lars Barquist and Srihitha Yerabaka):
The Euler-Descartes Formula for polyhedra
and for plane graphs, and how it implies
that there are only five Platonic solids.
Reading:
Seventeen proofs;
Platonic graphs.
March 4
(Justin Palumbo and Lars Barquist):
Coloring plane maps with five colors
- another consequence of Euler's Formula.
Reading:
Planar graphs and plane graphs;
Mathworld on the
Four-Color Theorem;
coloring maps on
other surfaces.
March 11
(Matt Meola and Michael Solovyov):
Subgroups, conjugacy, cosets, Lagrange's Theorem.
Reading:
Chapter 8;
seminar notes.
March 25
(Profs):
Basics of linear algebra.
Reading:
seminar notes on vector spaces
and bases.
April 1
(Profs):
Isometries in Rn.
Reading:
seminar notes on isometries.
April 8
(Sonali Phatak and Anne Seery):
Finite groups of symmetry in R2
and R3.
Reading:
Topics in Geometry
by John O'Connor, especially lectures
11,
10,
2, and
4.
(Notice the St Andrews website again!)
April 15
(William Starnick and Jeffrey Ulrich):
Groups and wallpaper designs.
Reading:
Chapter 15;
The 17 plane symmetry groups.
April 22
(Prof. Michael O'Nan):
The game of Nim.
Reading:
April 29
(Prof. Roe Goodman):
Alice Through Looking Glass after Looking Glass;
the Mathematics of Mirrors.
List of participants in this semester
Reading:
The Mathematics of Mirrors and Kaleidoscopes.