We define and study multiplicative connections in the tangent bundle of a Lie groupoid. Multiplicative are linear satisfying an appropriate compatibility with groupoid structure. Our definition is natural sense that connection on if only its torsion tensor Bursztyn–Drummond [Lie theory tensors, Mat. Ann. 375 (2019) 1489–1554, arXiv:1705.08579] geodesic spray vector field. identify obstruction t...

An alternative version of the theory of multiplicative unitaries is presented. Instead of the original regularity condition of Baaj and Skandalis we formulate another condition selecting manageable multiplicative unitaries. The manageability is the property of multiplicative unitaries coming from the quantum group theory. For manageable multiplicative unitaries we reproduce all the essential re...

In this paper we settle a special case of the Grassmann convexity conjecture formulated earlier by B.and M.Shapiro. We present conjectural formula for maximal total number real zeros consecutive Wronskians an arbitrary fundamental solution to disconjugate linear ordinary differential equation with time. show that gives lower bound required equations order and, using our results on convexity, pr...