In a 2009 paper, Anderson, Fels and Vassiliou showed that, for a class of Darboux-integrable (DI) hyperbolic systems of PDE, a canonical integrable extension exists and is constructed using the action of a novel invariant called the Vessiot group; moreover, the extension splits as the product of two simpler differential systems both of Lie type. Each solution of the DI system thus arises as a `superposition' of a pair of solutions to the simpler systems.
In this preliminary report on joint work with Mark Fels, we outline a construction of a canonical integrable extension for elliptic DI systems. By contrast, in this case the extension doesn't split but is a holomorphic Lie equation involving the complexified Vessiot group. In several examples the extension is contact-equivalent to a prolongation of the Cauchy-Riemann equations, leading to solution formulas for the PDE in terms of holomorphic data.
Geometry of Elliptic Darboux-Integrable Systems Sponsored by the Meyer Fund
Apr. 11, 2023 12:20pm (MATH …
Kempner
Andrew Linshaw (Denver University)
X
W-algebras are an important class of vertex algebras associated to a Lie (super)algebra g and an even nilpotent element in g. Trialities of W-algebras are isomorphisms between the affine cosets of three different W-algebras. A large family of such isomorphisms were conjectured in physics by Gaiotto and Rapcak in 2017, and were recently proven in my joint work with Thomas Creutzig. This result vastly generalizes both Feigin-Frenkel duality and the coset realization of principal W-algebras of classical types. In this talk, I will give an overview of vertex algebras and discuss the Gaiotto-Rapcak trialities. Time permitting, I will describe some possible generalizations of this result to a larger class of W-algebras.
This is a joint work with Thomas Creutzig (University of Alberta) and Vladimir Kovalchuk (University of Denver).
I’ll survey our current knowledge of the lattice of varieties of Heyting algebras. Towards the end of the talk, I’ll discuss some recent developments, including some open problems that continue to fight back.
In this largely expository talk, we will overview homotopical and geometric motivations for constructing Brown-Gitler spectra, as well as describe a number of different constructions of these spectra due to a range of authors. We will discuss how Brown-Gitler spectra are used in homotopy computations and indicate areas of current research interest. Come to this talk to find out what abelian groups ordinary homology spits out when it eats Brown-Gitler spectra!
Brown-Gitler Spectra: An introduction & current directions
Geometry of Elliptic Darboux-Integrable Systems