Tag - Non-associative rings

Ivan Shestakov: Coordination Theorems for certain non-associative algebras

16th April, 2020

Coordinatization Theorems are very useful for classification problems. The classical Wedderburn Coordinatization Theorem claims that if a unital associative algebra A contains a matrix subalgebra Mn(F) with the same unit then A=Mn(B) for a certain subalgebra B. The Jacobson Coordinatization Theorems in the structure theories of alternative and Jordan algebras state similar results for octonions and Albert algebras. Various coordinatization theorems were proved for noncommutative Jordan algebras, for commutative power associative algebras, for alternative and Jordan superalgebras, etc. In our talk, we consider three coordinatization theorems:

1) for 2x2 matrices in the class of alternative algebras (Jacobson's problem),

2) for Jordan algebra of symmetric 2x2 matrices in the class of Jordan algebras,

3) for octonions in the class of right alternative algebras.

Andrei Okounkov: Weyl groups and their generalizations in enumerative geometry

15th March, 2016

These lectures will be about enumerative K-theory of curves (and more general 1-dimensional sheaves) in algebraic threefolds. In the first lecture, we will set up the enumerative problem and survey what we know and what we conjecture about it. In particular, we will meet the fundamental building blocks of the theory: threefolds fibered in ADE surfaces. In the second lecture, we will learn what geometric representation theory says about these building blocks, and, in particular, meet the present day incarnation of the Weyl group, which is really a fundamental groupoid of a certain periodic hyperplane arrangement, associated to a certain geometrically defined infinite-dimensional Lie algebra. This Weyl group completely determines the curve counts, and so seems like a very fitting topic for Hermann Weyl lectures. In the third lecture, I plan to introduce some of the geometric ideas that go into the actual technical construction of the theory.