Quadratic Forms, Math 8430, Spring 2011
Daniel Krashen

• Lectures : MWF 12:20-1:10 in Boyd 326
• Office Hours: TBD

Video Lectures (need iTunes)

Lecture Notes (scanned PDF):

• Quadratic and Bilinear forms on a vector space (Lecture 1, January 14, 2011)
• Radicals, complements, and representability (Lecture 4, January 24, 2011)
• Diagonalization, determinants and decomposition (Lecture 5, January 26, 2011)
• Hyperbolic spaces (Lecture 6, January 28, 2011)
• Representation, Witt Decomposition (Lecture 7, January 31, 2011)
• Witt Decomposition (continued), chain equivalence (Lecture 8, February 2, 2011)
• Chain equivalence (Lecture 9, February 4, 2011)
• Witt group, virtual vector spaces (Lecture 10, February 7, 2011)
• Vector space constructions, the Witt ring (Lecture 11, February 9, 2011)
• The fundamental ideal and its square, discriminants (Lecture 12, February 11, 2011)
• Q(F), quadratically closed and real closed fields (Lecture 13, February 14, 2011)
• The Witt group of a finite field of odd order (Lecture 14, February 16, 2011)
• Billinear forms in arbitrary characteristic, for example, 2 (Lecture 15, February 18, 2011)
• Quaternion algebras, basic structure (Lecture 16, February 21, 2011)
• Quaternion algebras and their norms (Lecture 17, February 23, 2011)
• Quaternion algebras and their norms, continued (Lecture 18, February 25, 2011)
• Quaternion algebras and their norms, continued again (Lecture 19, February 28, 2011)
• Quaternion algebras and central simple algebras (Lecture 20, March 2, 2011)
• Quaternion algebras and central simple algebras (Lecture 21, March 4, 2011)
• The Brauer group (Lecture 22, March 7, 2011)
• Clifford algebras and super algebras (Lecture 23, March 9, 2011)
• Super simple and super central algebras (Lecture 23, March 11, 2011)
• The Brauer Wall group (Lecture 24, March 23, 2011)
• Structure of super cenral simple algebras part 1 (Lecture 25, March 28, 2011)
• Structure of super cenral simple algebras part 2 -- ideals (Lecture 26, March 30, 2011)
• Structure of super cenral simple algebras part 3 -- even and odd algebras (Lecture 27, April 1, 2011)
• Structure of super cenral simple algebras part 4 -- tensor vs. super tensor, simple modules (Lecture 28, April 4, 2011)
• Structure of super cenral simple algebras part 5 -- Wedderburn-Artin/Jacobson Density and double centralizers (Lecture 29, April 6, 2011)
• Double Centralizer Theorem (Lecture 30, April 8, 2011)
• BW(F) --> Q(F) (Lecture 31, April 18, 2011)
• The map to the Brauer-Wall group (Lecture 32, April 20, 2011)
• The Milnor Conjecture (Lecture 33, April 22, 2011)
• The Pfister philosophy. Rigidity. Multiplicative forms (Lecture 34, April 25, 2011)
• Anisotropic multiplicative forms are Pfister forms (Lecture 35, April 27, 2011)
• Multiplicative forms and torsion (Lecture 36, April 29, 2011)
• Level, pythagoras number, u-invariant, and the Hodge conjecture (Lecture 37, May 1, 2011)