MATH 465 (Fall 2010)

# MATH 465: Geometric Algebra

Catalogue Description: General linear groups. Projective geometry and projective linear groups. Bilinear forms. Symplectic and orthogonal geometries. Symplectic groups. Orthogonal groups. Hermitian forms and unitary groups.

Prerequisite Courses: Math 262 and Math 367

Textbook: Larry C. Grove, Classical Groups and Geometric Algebra, AMS, 2001.

Other Books: Robert A. Wilson, The Finite Simple Groups, Springer, 2009. (Chapter 3)
Steven Roman, Advanced Linear Algebra, Springer, 2005. (Chapter 11)
Donald E. Taylor, The Geometry of Classical Groups, Heldermann, 1992.
Emil Artin, Geometric Algebra, Intersicence, 1957.

Grading: Two sets of homework (2 x 30 = 60 points), and one final exam (40 points). (The final exam will be open-book.)

Done in the class:
Week 1: (2 hrs) A quick review of basic algebraic structures. General linear group. Affine geometry, affine general linear group.
Week 2: Multiple transitivity. Projective lines. GL(2,F) acts 2-transitively on the projective line over F.
Week 3: (1 hr) PGL(2,F) acts sharply 3-transitively on the projective line over F.
Week 4: Orders of PGL(n), PSL(n) over finite fields. Projective planes. Number of points and lines in a projective plane over a finite field. Fano plane. Introduction to bilinear forms. Radicals.
Week 5: Types of bilinear forms, reflexive bilinear forms are either symmetric or alternate. Structure theorem for alternate forms.
Week 6: Symplectic groups, transvections. Primitve actions.
Week 7: Sp(V) acts primitively on the projective space P(V). Order of Sp(2m,q).
Week 8: Representation of symmetric bilinear forms, special case over various fields, quadratic forms.
Week 9: Hyperbolic planes, orthogonal sums of hyperbolic planes. Orthogonal group, special orthogonal group, Witt's Cancellation Theorem.
Week 10: Witt's Extension Theorem, maximal totally istropic spaces, Witt index, structure theorem for quadratic spaces. Description of O(V), where V is a hyperbolic plane.
Week 11: Z(O(V))={1,-1}. Projective cones. Action of O(V) on its projective cone.
Week 12: (2 hrs) Hermitian forms, unitary spaces.
Week 13: Unitary group, special unitary group, transvections, their action on the projective cone.
Week 14: More on SU(V) and PSU(V). Applications in physics by the guest lecturer Yusuf �peko�lu.

Last updated on 5 January 2011.