Hochschild cohomology of an associative algebra has a graded ring structure and a compatible graded Lie algebra structure, making it a Gerstenhaber algebra. The structures on Hochschild cohomology reveal information about the algebra and its deformations. One class of algebras with interesting Hochschild cohomology is quantum complete intersections which are a non-commutative analog of truncated polynomial rings. The behavior of Hochschild cohomology of this class of algebras has been shown to vary greatly based on the choice of quantum coefficients, exhibiting behaviors unseen with commutative algebras. In this talk, we explore the behavior of Hochschild cohomology of group extensions of quantum complete intersections.
Hochschild cohomology of group extensions of quantum complete intersections Sponsored by the Meyer Fund
Deciding nonnegativity of real polynomials is a key question in real algebraic geometry with crucial importance in polynomial optimization. Since this problem is NP-hard, one is interested in finding sufficient conditions (certificates) for nonnegativity, which are easier to check. Since the 19th century the standard certificates are sums of squares (SOS); see particularly Hilbert's 17th problem.
In this talk we introduce *polynomials supported on circuits*. For this class, nonnegativity is characterized by an invariant, which can be derived immediately from the initial polynomial. In consequence, we obtain an *entirely new class* of nonnegativity certificates, which are *independent* of SOS certificates.
Our certificates crucially extend geometric programming approaches for the computation of lower bounds in polynomial optimization. Particularly, for polynomials with simplex Newton polytope our approach is significantly faster and often yields better than bounds than semidefinite programming, which is the standard method for polynomial optimization.
These results generalize earlier works by Fidalgo, Ghasemi, Kovacec, Marshall, and Reznick. The talk is based on joint work with Sadik Iliman.
New Certificates for Nonnegativity via Circuit Polynomials and Geometric Programming Sponsored by the Meyer Fund