The theme of this lecture centers around two famous books, one by Hermann Weyl on “The Classical Groups: Their Invariants and Representations” (1939), another the six-volume work by I.M. Gelfand et al. on “Generalized Functions” (1960’s). I will discuss some broad ideas related to the theme of these two books, as well as some recent development in classical groups and their smooth representations.
The compactification of the moduli space of super Riemann surface has been studied by Deligne (1987) and more recently by Donagi and Witten (2015). We aim to extend those results by constructing a moduli space of stable maps.
Let M be a super Riemann surface with holomorphic distribution D and N a symplectic manifolds with compatible almost complex structure J. We call a map Φ: M-> N a super J-holomorphic curve if its differential maps the almost complex structure on D to J. Such a super J-holomorphic curve is a critical point for the superconformal action and satisfies a super differential equation of first order. Using component fields of this super differential equation and a transversality argument we construct the moduli space of super J-holomorphic curves as a smooth subsupermanifold of the space of maps M->N. The reduced space of the moduli space coincides with the moduli space of J-holomorphic curves from the reduction of M to N.
Gaussian heat kernel bounds play a fundamental role in geometric analysis. We present recent results on explicit Gaussian upper bounds for non-compact manifolds depending on locally uniform Ricci curvature integral assumptions. Furthermore, we discuss generalizations of integral curvature bounds in terms of the so-called Kato class. If time allows, topological applications of those heat kernel upper bounds will be given.
We start giving an overview of Bakry-Emery and Ollivier curvature and corresponding gradient estimates which require global curvature bounds. We then introduce the perpetual cutoff method to give gradient estimates requiring the curvature bound not everywhere. As an application, we give a sharp distance bound to the set where no curvature bound is assumed.
We study the asymptotic behavior of solutions to subcritical $\sigma_{k}$ Yamabe equation in a punctured ball. We prove that an admissible solution to this equation with a non-removable isolated singular point is asymptotic to a radial solution. Then we are able to obtain higher order expansion of solutions using analysis of the linearized operators. These results generalize earlier pioneering work of Caffarelli, Gidas and Spruck.
On a class of closed oriented Riemannian manifolds with lower bounds on Ricci curvature and injectivity radius we study the parameter given by a characteristic number over the volume of the manifold. The domain of this parameter is endowed with the Benjamini-Schramm (BS) topology. The BS-topology is a weak notion of convergence, that originated from graph theory, where it is insensively studied, but also has applications in various other areas. Using an integral characterization (actually, a generalization of Gauss-Bonnet formula) one can show that the parameter explained is continuous and has a continuous extension to the completion of its domain in the BS topology. From the known fact that this completion is compact one can derive that the parameter is testable in constant time.
Damek-Ricci spaces are some one-dimensional extensions of Heisenberg groups. We prove that there does not exist non-geodesic biharmonic curves in a four-dimensional Damek-Ricci space although there exist such curves in three-dimensional Heisenberg groups.
In this talk, we give a new estimate for the eigenvalues of the Dirac operator on a compact spin manifold in terms of an appropriate endomorphism $E^\psi$ of the tangent bundle associated with an eigenspinor $\psi$. We then show that, for isometric immersions and Riemannian flows (local Riemannian submersions), the limiting case could be achieved. In this case the tensor $E^\psi$ is identified with the second fundamental form of the immersion while it is identified with the O'Neill tensor of the flow.
