Stable bundles on foliated manifolds
Resumo: It is well-known that the existence of canonical metrics in complex geometry corresponds to a stability condition. I will present a version of this correspondence for transverse holomorphic bundles on compact manifolds with a Hermitian foliation. As it turns out, the natural notion of stability requires that the foliation is taut. I will discuss the relation to higher dimensional instantons on Sasakian manifolds and mention other applications. If time permits, I will also indicate how the result extends to the moduli space of transverse Higgs bundles.
Coisotropic Triples and their reduction: towards $K$-theory
Resumo: In my talk I will explain the notion of a coisotropic triple which governs many geometric situations related to various sorts of reductions like coisotropic reduction in Poisson geometry. It reformulates the reduction in an entirely algebraic way and allows for non-commutative examples from deformation quantization as well. Having such an algebraic formulation based on bicategories, the notion of Morita equivalence is immediate. After characterizing Morita equivalence bimodules explicitly, I explain which options for the notion of projective modules are available and how they can be reduced. The project is based on joint work with Marvin Dippell and Chiara Esposito.
Representing Periodic Tilings of Regular Polygons
Resumo: We present a simple representation for periodic tilings of the plane by regular polygons. Our approach is to represent explicitly a minimal subset of the vertices from which we systematically generate all vertices in the tiling by translations.We then deduce the edges and the faces using the constraint that all edges have the same length. Our representation can be used to synthesize tilings manually and automatically from images.
Representation theory of Vertex algebras
Resumo: During this semester we will have a weekly seminar on representation theory of vertex algebras, as a preparation for the trimester program in 2020. We will cover among others the following articles
• Y. Zhu. Modular invariance of characters of vertex operator algebras. J. Amer. Math. Soc., 9(1):237–302, 1996• I. Frenkel and Y. Zhu. Vertex operator algebras associated to representations of affine and Virasoro algebras. Duke Mathematical Journal, 66(1):123–168, 1992• C. Dong, H. Li, and G. Mason. Vertex operator algebras and associative algebras. Journal of Algebra, 206:67–96, 1998• Y.-Z. Huang and J. Yang. On functors between module categories for associative algebras and for N-graded vertex algebras. Journal of Algebra, 409:344–361, 2014• M. Miyamoto. Modular invariance of vertex operator algebras satisfying c2-cofiniteness condition. Duke Mathematical Journal, 122:51–91, 2004• Y.-Z. Huang and J. Yang. Associative algebras for (logarithmic) twisted modules for a vertex operator algebra. Transactions of the American Mathematical Society, 371:3747–3786, 2019• Y.-Z. Huang. Differential equations, duality and modular invariance. Communications in contemporary mathematics, 7:649–706, 2005 • X. He. Higher level Zhu algebras are subquotients of universal enveloping algebras.
Higher Willmore energies, Q-curvatures, and related global geometry problems.
The Willmore energy and its functional gradient (under variations of embedding) have recently been the subject of recent interest in both geometric analysis and physics, in part because of their link to conformal geometry. Considering a singular Yamabe problem on manifolds with boundary shows that these these surface invariants are the lowest dimensional examples in a family of conformal invariants for hypersurfaces in any dimension. The same construction and variational considerations shows that (on even dimensional hypersurfaces) the higher Willmore energy and its functional gradient are analogues of the integral of the celebrated Q-curvature conformal invariant and its function gradient (now with respect to metric variations) which is known as the Fefferman-Graham obstruction tensor (or the Bach tensor in dimension 4). In fact the link is deeper than this in that the Willmore energy we consider is an integral of an invariant that actually generalises the Branson Q-curvature. This is part of fascinating unifying picture that includes some interesting open problems in global geometry.