We determine the Hopf Galois structures on a field extension of degree twice an odd prime square and classify corresponding skew left braces. Besides we separable extensions allowing cyclic structure number these structures.

Quantifying dependence between extreme values is a central problem in many theoretical and applied studies. The main distinction is between asymptotically independent and asymptotically dependent extremes, with important theoretical examples of these general limiting classes being the extremal behaviour of a bivariate Normal distribution, for asymptotic independence, and of the bivariate t dist...

This thesis concerns quantum cluster algebras. For skew-symmetric acyclic quantum cluster algebras, we express the quantum F -polynomials and the quantum cluster monomials in terms of Serre polynomials of quiver Grassmannians of rigid modules. Then we introduce a new family of graded quiver varieties together with a new t-deformation, and generalize Nakajima’s t-analogue of q-characters to thes...

The Mackey–Zimmer theorem classifies ergodic group extensions $X$ of a measure-preserving system $Y$ by compact $K$, showing that such are isomorphic to skew-product $X \equiv Y \rtimes _\rho H$ for some closed subgroup $

We study the problem of lifting ergodicity to certain skew-product extensions over shifts of nite type. This can be done by computing the asymptotic range of the deening cocycle. For a class of ergodic shift-invariant measures and a class of functions (deening conservative extensions) we give a characterisation of ergodicity in terms of weights of periodic orbits in the shift space.

We develop the homological theory of KLR algebras of symmetric affine type. For each PBW basis, a family of standard modules is constructed which categorifies the PBW basis.