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Events
Analysis Seminar
Zhihao Zhang, University of Waterloo
Spectra of Beurling Algebras of Discrete Abelian Groups
We will discuss a variant of the group algebra, called the Beurling algebra. These algebras differ from their classical counterpart through the addition of a weight function modifying the norm. The Gelfand spectrum of the group algebra of absolutely integrable functions on an abelian group, G, is well known to be the Pontryagin dual of G. In the case of a Beurling algebra, the Gelfand spectrum can be much larger for suitable weights. We will focus on the Beurling algebra of a discrete abelian group, G, and give a description of its Gelfand spectrum in terms of a seminorm constructed from a symmetric weight.
MC 5417 or Join on Zoom
Differential Geometry Working Seminar
Viktor Majewski, University of Waterloo
Dirac Operators on Orbifold Resolutions
In this talk we discuss Dirac operators along degenerating families of Riemannian manifolds that converge, in the Gromov-Hausdorff sense, to a Riemannian orbifold. Such degenerations arise naturally when analysing the boundary of Teichmüller spaces of special Riemannian metrics as well as moduli spaces appearing in gauge theory and calibrated geometry. Here sequences of smooth geometric structures on Riemannian manifolds may converge to an orbifold limit. To understand and control these degenerations, we introduce smooth Gromov-Hausdorff resolutions of orbifolds, that are, smooth families (X_t,g_t), which collapse to the orbifold (X_0,g_0) as t goes to 0.
The central analytic problem addressed in this paper is to understand the behaviour of Dirac operators along such resolutions, in particular in collapsing regimes where classical elliptic estimates fail. We develop a uniform Fredholm theory for the family of Dirac operators on the Gromov-Hausdorff resolution. Using weighted function spaces, adiabatic analysis, and a decomposition of X_t into asymptotically conical fibred (ACF), conically fibred (CF) and conically fibred singular (CFS), we obtain uniform realisations of the model operators and prove a linear gluing exact sequence relating global and local (co)kernels. As a consequence, we construct uniformly bounded right inverses for D_t, and derive an index additivity formula.
The theory developed here provides the analytic foundation for nonlinear gluing problems in gauge theory and special holonomy geometry, including torsion-free G-structures, instantons, and calibrated submanifolds of Riemannian manifolds close to an orbifold limit.
MC 5403