Title: On instability of F-Yang-Mills connections

Abstract:  The notion of F-Yang-Mills connections is defined on principal fiber bundles over Riemannian manifolds. This notion is a generalization of Yang-Mills connections, p-Yang-Mills connections and exponential Yang-Mills connections on them. Here, F is a strictly increasing C^{2}-function. In this talk, we introduce some results for the instability of F-Yang-Mills connections. We give an extension of Simons Theorem (1977) for the instability of (usual) Yang-Mills connections over the sphere to F-Yang-Mills connections. In fact, we derive a sufficient condition that any non-flat, F-Yang-Mills connection over the sphere is instable. Then the sufficient condition is expressed by an inequality with respect to the dimension of the sphere and a degree of the differential of the function F. Moreover, we discuss a similar result for the instability of $F$-Yang-Mills connections in the case when the base space is a symmetric R-space. This talk is partially based on joint work with Kazuto Shintani (arXiv:2301.0429).

Title: Ultra-hyperbolic operator on pseudo-H type groups

Abstract: pdf-file

Title: The equivalence problem in sub-Riemannian geometry

Abstract:  A central motivation for the introduction of the Levi-Civita connection and its curvature is that the curvature will be a tensor that is invariant under isometries.As such, it can be used as a tool for the equivalence problem: Determining if two Riemannian manifolds are isometric or not. We want to do a similar construction for sub-Riemannian manifolds, where the metric is only defined on a subbundle. We describe a canonical Cartan connection on the sub-Riemannian frame bundle introduced by T. Morimoto,  and give an explicit description of how such Cartan connection correspond to an affine connection and a complement. We will also give concrete formulas of what this connection and complements look like for contact manifolds and (2,3,5)-manifolds.

Title : Totally complex submanifolds and R-spaces

Abstract: We study totally complex submanifolds immersed in a quaternionic projective space and related minimal submanifolds in its quaternionic Kaehler geometry. In this talk our main result is the construction of a (non Levi-Civita) canonical connection on the inverse image of a totally complex submanifold under the Hopf fibration over the quaternionic projective space such that the parallelism of the second fundamental form of the inverse mage with relative to our canonical connection is equivalent to the second fundamental form of a totally complex submanifold.  From this result we obtained as a certain R-space associated with a quaternionic Kaehler symmetric space. This talk is based on joint work with Jong Taek Cho and Yoshihiro Ohnita.

Title : Some remarks on the Dirac operator on symmetric spaces

Abstract:  We develop the spectral analysis for the Dirac operator on symmetric spaces of noncompact type based on harmonic analysis on  Lie groups. In harmonic analysis on Lie groups, the Dirac operator has been deeply studied to characterize discrete series representations of Lie groups.
In this talk, we compute the continuous spectrum of the Dirac operator on irreducible symmetric spaces of noncompact type. We show that the continuous spectrum has a spectral gap if and only if the symmetric space is isomorphic to a coset space of the special pseudo-unitary group of odd matrix size. Furthermore, we give a uniform weighted resolvent estimate for the Dirac operator under a certain assumption on the symmetric space.

Title : Surjectivity of mean value operators on the sphere

Abstract: In this talk, we deal with the mean value operator on the sphere and explain when it is surjective on the space of smooth functions. We also mention a related small denominator problem. This is a joint work with J. Christensen, F. Gonzalez, and J. Wang.

Title : Convergence of Dirichlet forms induced on boundaries of compact domains in open weighted Riemannian manifolds

Abstract:  We consider Markovian closed symmetric forms on the Hilbert space onsisting of functions with finite Dirichlet integrals on an open,  weighted Riemannian manifold. We introduce an ideal boundary of the manifold, called the Kuramochi boundary, and define the traces of Markovian forms on the ideal boundary which act on the traces of functions in the Hilbert space on it. It is shown that the Dirichlet forms induced on the boundaries of relatively compact smooth domains endowed with harmonic measure Mosco converge (in the sense of Kuwae and Shioya) to those on the Kuramochi boundary with harmonic measure whenever the domains increase to exhaust the manifold. The Dirichlet forms which associate the Dirichlet-to-Neumann operators on the boundaires of compact domains, for example, Mosco converge to that on the Kuramochi boundary.

Title: Calabi-Yau structures on the complexifications of rank two symmetric spaces

Abstract: We have recently proved that there exists a G-invariant Calabi-Yau structure on the complexification of any rank two symmetric space G/K of compact type. In this talk, we state the outline of its proof. The key of the proof is to show the global regularity of a W-invariant strictly convex weak solution of a Monge-Ampere type equation on the two-dimensional Euclidean space R^2, where W is the Weyl group associated to G/K.

Title: Cheeger-Müller theorem for singular spaces

Abstract: The Cheeger-Müller theorem is an important theorem in global analysis, stating the equality between the analytic (or Ray-Singer) torsion and the topological torsion of a smooth compact manifold equipped with a unitary flat vector bundle. Using local index techniques and the Witten deformation Bismut and Zhang gave the most general comparison theorem of torsions for a smooth compact manifold. 
The aim of this talk is to present generalisations of the Cheeger-Müller and  the  Bismut-Zhang theorem to singular spaces. In the first part of the talk I will give a gentle introduction to both global analysis on singular spaces as well as to the definition of analytic torsion. 

Title: Nilpotent Lie algebras obtained by quivers

Abstract: Nilpotent Lie groups with left-invariant metrics provide non-trivial examples of Ricci solitons. A quiver is a directed graph where loops and multiple arrows between two vertices are allowed. In this talk, we introduce a new method for obtaining nilpotent Lie algebras from finite quivers without cycles. For all of these Lie algebras, we prove that the corresponding simply connected nilpotent Lie groups admit left-invariant Ricci solitons. This constructs a large family of examples of Ricci soliton nilmanifolds with arbitrarily high nilpotency steps.

Title : Asymptotics of the Porous Medium Equation on Manifolds with Conical Singularities

Abstract: The Porous Medium Equation (PME) is a non-linear variant of the heat equation. The name is derived from the fact that it describes - among other phenomena - the flow of a gas in a porous medium.
In the talk I will explain, how maximal regularity techniques can be used to solve the PME on manifolds with conical singularities, what we can say about the regularity of the solution and how the geometry of the manifold near the conical points is reflected in the structure of the solution.
Based on joint work with N. Roidos (Patras/GR)

Title : Spectral Theory of the Sub-Riemannian Laplacian

Abstract:  Sub-Riemannian (sR)  geometry is the geometry of bracket-generating metric distributions on a manifold. Peculiar phenomena in sR geometry include the exotic Hausdorff dimension describing the growth rate of the volumes of geodesic balls. As well as abnormal geodesics that do not satisfy any variational equation. In this talk I will survey my results which show how both these phenomena are reflected in the spectral theory of the hypoelliptic Laplacian in sR geometry.

Title : Small eigenvalues of the Hodge-Laplacian with sectional curvature bounded below

Abstract:  We consider the positive eigenvalues of the Hodge-Laplacian acting on p-forms on oriented closed Riemannian manifolds.

 For each degree p and each natural number  k≧ 1, we construct a one-parameter family of Riemannian metrics on any oriented closed manifold with volume one and sectional curvature bounded below such that the $k$-th positive eigenvalue of the Hodge-Laplacian acting on p-forms converges to zero.  Furthermore, in the case of the spheres, we can also construct such Riemannian metrics under non-negative sectional curvature.  This talk is based on a joint work with Colette Ann\'e at Nantes Universit\'e in France.