Let G be a finite transitive group on a set , let , and let be the stabilizer of the point in G. In this paper, we are interested in the proportion

that is, the proportion of elements of lying in a suborbit of cardinality at most 2. We show that, if this proportion is greater than , then each element of lies in a suborbit of cardinality at most 2, and hence G is classified by a result of Bergman and Lenstra. We also classify the permutation groups attaining the bound .

We use these results to answer a question concerning the enumeration of Cayley graphs. Given a transitive group G containing a regular subgroup R, we determine an upper bound on the number of Cayley graphs on R containing G in their automorphism groups.

For a given Beurling–Carleson subset E of the unit circle which has positive Lebesgue measure, we give explicit formulas for measurable functions supported on E such that their Cauchy transforms have smooth extensions from to . The existence of such functions has been previously established by Khrushchev in 1978, in non-constructive ways by the use of duality arguments. We construct several families of such smooth Cauchy transforms and apply them in a few related problems in analysis: an irreducibility problem for the shift operator, an inner factor permanence problem. Our development leads to a self-contained duality proof of the density of smooth functions in a very large class of de Branges–Rovnyak spaces. This extends the previously known approximation results.

We investigate quantum lens spaces, , introduced by Brzeziński and Szymański as graph -algebras. We give a new description of as graph -algebras amending an error in the original paper by Brzeziński and Szymański. Furthermore, for , we give a number-theoretic invariant, when all but one weight are coprime to the order of the acting group r. This builds upon the work of Eilers, Restorff, Ruiz, and Sørensen.

Let be a finite Galois-algebra extension of a number field F, with group G. Suppose that is weakly ramified and that the square root of the inverse different is defined. (This latter condition holds if, for example, is odd.) Erez has conjectured that the class of in the locally free class group of is equal to the Cassou–Noguès–Fröhlich root number class associated with . This conjecture has been verified in many cases. We establish a precise formula for in terms of in all cases where is defined and is tame, and are thereby able to deduce that, in general, is not equal to .

An affine variety with an action of a semisimple group G is called “small” if every nontrivial G-orbit in X is isomorphic to the orbit of a highest weight vector. Such a variety X carries a canonical action of the multiplicative group commuting with the G-action. We show that X is determined by the -variety of fixed points under a maximal unipotent subgroup . Moreover, if X is smooth, then X is a G-vector bundle over the algebraic quotient .

If G is of type (), , , , or , we show that all affine G-varieties up to a certain dimension are small. As a consequence, we have the following result. If , every smooth affine -variety of dimension is an -vector bundle over the smooth quotient , with fiber isomorphic to the natural representation or its dual.

We investigate the pluriclosed flow on Oeljeklaus–Toma manifolds. We parameterize left-invariant pluriclosed metrics on Oeljeklaus–Toma manifolds, and we classify the ones which lift to an algebraic soliton of the pluriclosed flow on the universal covering. We further show that the pluriclosed flow starting from a left-invariant pluriclosed metric has a long-time solution which once normalized collapses to a torus in the Gromov–Hausdorff sense. Moreover, the lift of to the universal covering of the manifold converges in the Cheeger–Gromov sense to , where is an algebraic soliton.