The connection between Hochschild and cyclic cohomologies with generalized De Rham homology and index theories for arbitrary algebras has long been established by the work of Connes, Karoubi, Loday, Feigin, Tsygan, et al. Here we generalize these cohomology theories even further, essentially creating a theory that establishes a step-wise bridge between the two. This theory can then be used to establish similar geometric result for manifolds with boundaries and may have applications in exterior differential systems, as well as extend to higher K-theories.
On Hochschild and cyclic cohomology
May. 01, 2018 4pm (MATH 350)
Algebraic Geometry
Maksym Fedorchuk (Boston College)
X
Title: Invariant-theoretic Mather-Yau theorem, and applications
Abstract: The famous Mather-Yau theorem says that two isolated hypersurface singularities of the same embedding dimension are biholomorphically equivalent if and only if their moduli algebras are isomorphic. Thus determining whether two given moduli algebras are isomorphic becomes an important problem. In the case of quasi-homogeneous hypersurface singularities, Eastwood and Isaev proposed a purely algebraic approach to this problem, rooted in classical invariant theory. For homogeneous singularities, this approach leads to the associated form morphism that assigns to a singularity the Macaulay inverse system of its moduli algebra. The associated form morphism has several marvelous properties, two of which I will discuss in this talk. In joint work with Isaev, we have shown that the associated form morphism preserves GIT polystability (arXiv:1703.00438). This leads to a purely algebraic invariant-theoretic Mather-Yau theorem for homogeneous hypersurface singularities, and to interesting new compactifications of the moduli space of smooth hypersurfaces. The associated form morphism also detects whether a homogeneous polynomial with a non-vanishing discriminant is a direct sum (of Sebastiani-Thom type), and so leads to an algorithm for finding direct sum decompositions over the rationals (arXiv:1705.03452).
Invariant-theoretic Mather-Yau theorem, and applications Sponsored by the Meyer Fund