Sven Holtrop (Utrecht):Two Types of Riemannian Groupoids
Abstract: Lie groupoids give a flexible way of modelling symmetries of manifolds and their quotients. Frequently, these symmetries carry geometric structure themselves. For example, there exists notions of symplectic, contact and Riemannian groupoids. In this talk, we will see how Riemannian groupoids have been used and what types of definitions there exist in the literature. I will then introduce transversely multiplicative Riemannian metric and isotropically multiplicative Riemannian metrics. Whereas transversely multiplicative Riemannian metrics tie in nicely with the existing literature on Riemannian groupoids, isotropically Riemannian metrics are a novel notion that generalize bi-invariant metrics on a Lie group.
Kian Shah (RUG):The alternating compositions of weighted differential operators yield the weights' Wronskian with which constant?
Abstract: The alternated composition of N = 2p differential operators w_j(x) d^p/dx^p of strict order p on the line R is again a differential operator of strict order p; its coefficient is the constant const(p), depending only on the parity N, times the Wronskian determinant of the originally taken coefficients w_1, ..., w_N. The case p = 1 of the Lie bracket for two vector fields fixes const(1) = 1, and const(2) = 2 is found easily by hand; const(3) = 90 can still be obtained symbolically. The problem is to determine const(p >= 4). We compute const(p) exactly for all p <= 14 -- a 241-digit integer at p = 14 -- and record the resulting integer sequence as OEIS A392714. We prove that v_p(const(p)) >= p - 1 for every prime p, matching the exact equality observed numerically throughout our range, and conjecture that this equality holds in general. We show that log const(p) grows like alpha*p^2*log(p), with the leading coefficient close to 2, nearly saturating the bound p^2 log p (1 + O(1/log p)) <= log const(p) <= 2 p^2 log p (1 + O(1/log p)) obtained by O. Zaboronski (private communication). A naturally arising reduced constant is found to decay to zero super-exponentially rather than grow, a direct consequence of this near-saturation.
The talk is based on joint work with my supervisor, Arthemy Kiselev; the preprint is available as arXiv:2605.11137.
Robbert Sholtens (RUG):To Frame a Killing - Finding a metric given desired Killing vector fields
Collin Mark Joseph (RU Nijmegen): Factorization of the Dirac Operator for riemannian embeddings
Abstract: We study the relation between the Dirac operators associated to the Euclidean embedding i: ℝᵐ ↪ ℝᵐ⁺ᵏ through the framework of unbounded KK-theory. We construct an explicit unbounded representative of the shriek class associated to the embedding and analyze its Kasparov product with the fundamental class of ℝᵐ⁺ᵏ. By identifying the resulting tensor sum operator with a product involving the Dirac operator on ℝᵐ and a normal-direction operator T, we show that T has index one and represents the identity element in KK(ℂ, ℂ). Consequently, we obtain the factorization
i! ⊗̂ [Dℝᵐ⁺ᵏ] = [Dℝᵐ]
in unbounded KK-theory, thereby recovering the fundamental class of the embedded Euclidean space via the unbounded Kasparov product. We further extend the analysis to embeddings into tubular neighborhoods of the form ℝᵐ × Bᵏε(0), showing that the resulting KK-class is invariant under restriction to ε-balls. The argument combines explicit Clifford-theoretic computations, spectral analysis of Callias-type operators, and homotopy techniques in unbounded KK-theory.