January 19 (Prof. Enriqueta Carrington):
Introduction to groups:
basic definitions, examples familiar and unfamiliar.
Reading: Chapters 1-3,
and the following seminar notes:
some basic
algebraic structures,
groups and fields,
try these
simple algebra problems.
January 26 (Prof. Carrington):
Cyclic groups (finite and infinite),
the dihedral groups, generators and relations, subgroups.
Reading: Chapters 4,5, and the seminar notes on cyclic groups.
Also read the following seminar note from 2007:
Tim Lou's proof
that the odd dihedral groups can already be generated by two relations.
February 2 (Liz Cox and David Osterman):
Permutations, symmetric and alternating groups.
Reading: Chapter 6, and a short summary
about permutations.
February 9 (Catherine Connolly, Sarah Goodman, Taylor Smith):
The group of rotational symmetries of each Platonic solid.
Reading: Chapters 7,8, and the website Symmetry groups of Platonic solids.
See also the
summary chart used by the speakers in class.
February 16 (Gyu Eun Lee and Asher Wasserman):
Lagrange's theorem for Abelian groups, cosets,
Lagrange's theorem in its full glory.
Reading: Chapters 11,12, and
Prof.
Lyons's 351 class notes.
February 23 (Louis Brown and Matthew Stone):
Conjugacy, normal subgroups of a group, quotient groups.
Reading: Chapters 14,15,16 and
Prof.
Lyons's 351 class notes.
March 1 (Alex Ganescu and Tianyi Liu):
Matrix groups, orthogonal matrices, On and SOn.
Proof that rotations in R3
fixing `the origin' form a group.
Reading: Chapter 9,
Orthogonal matrices.
Topics in Geometry part 2.
March 8 (Prof. Simon Thomas):
Killing the Hydra!
A fascinating story of mythology and set theory.
Reading: The Lernaean_Hydra.
Spring break
March 22 (Gyu Eun Lee):
The Euclidean group, isometries in Rn
and global positioning.
Reading:
Orthogonal
matrices (summary),
Orthogonal matrices
(long version).
Topics in Geometry parts 2 and 4.
Seminar notes on isometries.
March 29 (Kai Loell):
Finite rotation groups
in R2 and R3;
another glorious appearance of the Platonic solids.
Reading: Chapter 19;
Sideboard summary,
Topics in Geometry by John O'Connor,
especially lectures
11 and
10.
Outline of the proof.
April 5 (Lavi Blumberg and Louis Brown): The Banach-Tarski Paradox.
April 12 (Prof. Richard Lyons): Planes and groups.
April 19 (Peter Chi and Richard Wong):
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.
April 26 (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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