ALGORITHMIC COMPLEXITY OF LINEAR NONASSOCIATIVE ALGEBRA

Algorithmic Challenges in Polynomial and Linear Algebra

I will also introduce some very recent results, joint work with M. Shub, that suggest the existence of much faster algorithms, with average running time almost linear in the size of the input. The search of such an algorithm is linked to the study of a geometrical problem: The geodesics in the “condition number metric”. Open questions, both from complexity and geometrical perspectives, will be ...

The Proof Complexity of Linear Algebra

We introduce three formal theories of increasing strength for linear algebra in order to study the complexity of the concepts needed to prove the basic theorems of the subject. We give what is apparently the first feasible proofs of the Cayley-Hamilton theorem and other properties of the determinant, and study the propositional proof complexity of matrix identities such as AB = I → BA = I.

On the Descriptive Complexity of Linear Algebra

Dimension Formulas for the Free Nonassociative Algebra

The free nonassociative algebra has two subspaces which are closed under both the commutator and the associator: the Akivis elements and the primitive elements. Every Akivis element is primitive, but there are primitive elements which are not Akivis. Using a theorem of Shestakov, we give a recursive formula for the dimension of the Akivis elements. Using a theorem of Shestakov and Umirbaev, we ...

Using Automated Theorem Provers in Nonassociative Algebra

We present a case study on how mathematicians use automated theorem provers to solve open problems in (non-associative) algebra.