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Register a new user$mathrm{Spin}(9)$-invariant valuations on the octonionic plane
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The dimensions of the spaces of $k$-homogeneous $mathrm{Spin}(9)$-invariant valuations on the octonionic plane are computed using results from the theory of differential forms on contact manifolds as well as octonionic geometry and representation theory. Moreover, a valuation on Riemannian manifolds of particular interest is constructed which yields, as a special case, an element of $mathrm{Val}_2^{mathrm{Spin}(9)}$.
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A new class of plurisubharmonic functions on the octonionic plane O^2= R^{16} is introduced. An octonionic version of theorems of A.D. Aleksandrov and Chern- Levine-Nirenberg, and Blocki are proved. These results are used to construct new examples of continuous translation invariant valuations on convex subsets of O^2=R^{16}. In particular a new example of Spin(9)-invariant valuation on R^{16} is given.
We give an explicit classification of translation-invariant, Lorentz-invariant continuous valuations on convex sets. We also classify the Lorentz-invariant even generalized valuations.
We study the space of generalized translation invariant valuations on a finite-dimensional vector space and construct a partial convolution which extends the convolution of smooth translation invariant valuations. Our main theorem is that McMullens polytope algebra is a subalgebra of the (partial) convolution algebra of generalized translation invariant valuations. More precisely, we show that the polytope algebra embeds injectively into the space of generalized translation invariant valuations and that for polytopes in general position, the convolution is defined and corresponds to the product in the polytope algebra.
We introduce the new notion of convolution of a (smooth or generalized) valuation on a group $G$ and a valuation on a manifold $M$ acted upon by the group. In the case of a transitive group action, we prove that the spaces of smooth and generalized valuations on $M$ are modules over the algebra of compactly supported generalized valuations on $G$ satisfying some technical condition of tameness. The case of a vector space acting on itself is studied in detail. We prove explicit formulas in this case and show that the new convolution is an extension of the convolution on smooth translation invariant valuations introduced by J.~Fu and the second named author.
We survey recent results in hermitian integral geometry, i.e. integral geometry on complex vector spaces and complex space forms. We study valuations and curvature measures on complex space forms and describe how the global and local kinematic formulas on such spaces were recently obtained. While the local and global kinematic formulas in the Euclidean case are formally identical, the local formulas in the hermitian case contain strictly more information than the global ones. Even if one is only interested in the flat hermitian case, i.e. $mathbb C^n$, it is necessary to study the family of all complex space forms, indexed by the holomorphic curvature $4lambda$, and the dependence of the formulas on the parameter $lambda$. We will also describe Wannerers recent proof of local additive kinematic formulas for unitarily invariant area measures.
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