Problémy kvaternionické integrální geometrie

• Thesis title in Czech: Problémy kvaternionické integrální geometrie
• Thesis title in English: Problems in quaternionic integral geometry
• Academic year of topic announcement: 2025/2026
• Thesis type: dissertation
• Department: Mathematical Institute of Charles University (32-MUUK)
• Supervisor: Dr. rer. nat. Ing. Jan Kotrbatý

Guidelines

The goal of the project is to study integral-geometric kinematic formulas in quaternionic spaces using the tools of algebraic theory of valuations on convex bodies. The general strategy that proved to be extremely successful in complex [2, 4, 7] and octonionic spaces [3, 9] is to first determine the algebra of valuations invariant under the group of symmetries of the space (in the quaternionic case some version of the symplectic group) and then use the duality between this algebra and the coalgebra given by the corresponding kinematic formulas.

The candidate should acquire the techniques of the aforementioned articles and adapt them to the quaternionic case. This should lead to the explicit determination of the valuation algebras and the kinematic formulas in low dimensions, extending substantially the only currently known case of the quaternionic plane [5, 6]. Based on these results, the candidate might be able to make various conjectures about the general case and possibly try to prove some of them. Moreover, motivated by parallel developments in complex integral geometry, there are other problems closely related to the main line of the project that the candidate may optionally choose to work on, such as to determine the so-called local kinematic formulas, to extend the results to other (curved) quaternionic space forms, or to describe the spaces of symplectic-invariant valuations on convex functions.

References

[1] S. Alesker and J. H. G. Fu, Integral geometry and valuations, Birkhäuser/Springer, Basel, 2014.

[2] A. Bernig, A Hadwiger-type theorem for the special unitary group, Geom. Funct. Anal. 19 (2009), no. 2, 356–372.

[3] A. Bernig, Integral geometry under G2 and Spin(7), Israel J. Math. 184 (2011), 301–316.

[4] A. Bernig and J. H. G. Fu, Hermitian integral geometry, Ann. of Math. (2) 173 (2011), no. 2, 907–945.

[5] A. Bernig and G. Solanes, Classification of invariant valuations on the quaternionic plane, J. Funct. Anal. 267 (2014), no. 8, 2933–2961.

[6] A. Bernig and G. Solanes, Kinematic formulas on the quaternionic plane, Proc. Lond. Math. Soc. (3) 115 (2017), no. 4, 725–762.

[7] J. H. G. Fu, Structure of the unitary valuation algebra, J. Differential Geom. 72 (2006), no. 3, 509–533.

[8] J. Kotrbatý, Octonion-valued forms and the canonical 8-form on Riemannian manifolds with a Spin(9)-structure, J. Geom. Anal. 30 (2020), No. 4, 3616–3640.

[9] J. Kotrbatý and T. Wannerer, Integral geometry on the octonionic plane, Indiana Univ. Math. J. (2024), to appear.

[10] R. Schneider, Convex bodies: the Brunn-Minkowski theory, Cambridge University Press, Cambridge, 2014.