Let $\Pi$ be an open, relatively compact period annulus of real analytic vector field $X_0$ on an analytic surface. We prove that the maximal number of limit cycles which bifurcate from $\Pi$ under a given multiparameter analytic deformation $X_{\lambda }$ of $X_0$ is finite provided that $X_0$ is either a Hamiltonian or generic Darbouxian vector field

The free abelian group $R(Q)$ on the set of indecomposable representations of a quiver $Q$, over a field $K$, has a ring structure where the multiplication is given by the tensor product. We show that if $Q$ is a rooted tree (an oriented tree with a unique sink), then the ring $R(Q{)}_{\mathrm{red}}$ is a finitely generated $\mathbb{Z}$-module (here $R(Q{)}_{\mathrm{red}}$ is the ring $R(Q)$ modulo the ideal of all nilpotent elements). We describe the ring $R(Q{)}_{\mathrm{red}}$ explicitly by studying functors from the category $\mathrm{rep}(Q)$ of representations of $Q$ over $K$ to the category of finite-dimensional $K$-vector spaces

Shokurov conjectured that the set of all log canonical thresholds on varieties of bounded dimension satisfies the ascending chain condition. In this article we prove that the conjecture holds for log canonical thresholds on smooth varieties and, more generally, on locally complete intersection varieties and on varieties with quotient singularities

We derive two multivariate generating functions for three-dimensional (3D) Young diagrams (also called plane partitions). The variables correspond to a coloring of the boxes according to a finite Abelian subgroup $G$ of $\mathrm{SO}(3)$. These generating functions turn out to be orbifold Donaldson-Thomas partition functions for the orbifold $[{\mathbb{C}}^3/G]$. We need only the vertex operator methods of Okounkov, Reshetikhin, and Vafa for the easy case $G={\mathbb{Z}}_n$; to handle the considerably more difficult case $G={\mathbb{Z}}_2\times {\mathbb{Z}}_2$, we also use a refinement of the author's recent $q$-enumeration of pyramid partitions.

In the appendix, we relate the diagram generating functions to the Donaldson-Thomas partition functions of the orbifold $[{\mathbb{C}}^3/G]$. We find a relationship between the Donaldson-Thomas partition functions of the orbifold and its $G$-Hilbert scheme resolution. We formulate a crepant resolution conjecture for the Donaldson-Thomas theory of local orbifolds satisfying the hard Lefschetz condition

Let $G$ be an exceptional simple algebraic group over a field $k$, and let $X$ be a projective $G$-homogeneous variety such that $G$ splits over $k(X)$. We classify such varieties $X$. This classification allows us to relate the Rost invariant of groups of type $E_7$ and their isotropy and to give a two-line proof of the triviality of the kernel of the Rost invariant for such groups. Apart from this, it plays a crucial role in the solution of a problem posed by Serre for groups of type $E_8$