We describe the Stein handlebody diagrams of Milnor fibers of Brieskorn singularities $x^p + y^q + z^r = 0$. We also study the natural symplectic operation by exchanging two Stein fillings of the canonical contact structure on the links in the case $p = q = r$, where one of the fillings comes from the minimal resolution and the other is the Milnor fiber. We give two different interpretations of this operation, one as a symplectic sum and the other as a monodromy substitution in a Lefschetz fibration.

We use the language and tools available in model theory to redefine and clarify the rather involved notion of a {\em special subvariety} known from the theory of Shimura varieties (mixed and pure).

In this note we prove two main results. 1. In a rigid braided finite tensor category over C (not necessarily semisimple), some power of the Casimir element and some even power of the braiding is unipotent. 2. In a (semisimple) modular category, the twists are roots of unity dividing the algebraic integer D^{5/2}, where D is the global dimension of the category (the sum of squares of dimensions of simple objects). Both results generalize Vafa's theorem, saying that in a modular category twists are roots of unity, and square of the braiding has finite order. We also discuss the notion of the quasi-exponent of a finite rigid tensor category, which is motivated by results 1 and 2 and the paper math/0109196 of this http URL and the author.

In this paper, we develop a new method to classify abelian automorphism groups of hypersurfaces. We use this method to classify (Theorem 4.2) abelian groups that admit a liftable action on a smooth cubic fourfold. A parallel result (Theorem 5.1) is obtained for quartic surfaces.

We study linearizability properties of finite subgroups of the Cremona group ${\mathrm{Cr}}_n(k)$ in the case where $k$ is a global field, with the focus on the local-global principle. For every global field $k$ of characteristic different from 2 and every $n \ge 3$ we give an example of a birational involution of $\mathbb P^n_k$ (=an element $g$ of order $2$ in ${\mathrm{Cr}}_n(k)$) such that $g$ is not $k$-linearizable but $g$ is $k_v$-linearizable in ${\mathrm{Cr}}_n(k_v)$ for all places $v$ of $k$. The main tool is a new birational invariant generalizing those introduced by Manin and Voskresenski\uı in the arithmetic case and by Bogomolov--Prokhorov in the geometric case. We also apply it to the study of birational involutions in real plane.

We express the rational homotopy type of the mapping spaces $\mathrm{Map}^h(\mathsf D_m,\mathsf D_n^{\mathbb Q})$ of the little discs operads in terms of graph complexes. Using known facts about the graph homology this allows us to compute the rational homotopy groups in low degrees, and construct infinite series of non-trivial homotopy classes in higher degrees. Furthermore we show that for $n-m>2$, the spaces $\mathrm{Map}^h(\mathsf D_m,\mathsf D_n^{\mathbb Q})$ and $\mathrm{Map}^h(\mathsf D_m,\mathsf D_n)$ are simply connected and rationally equivalent. As application we determine the rational homotopy type of the deloopings of spaces of long embeddings. Some of the results hold also for mapping spaces $\mathrm{Map}_{\leq k}^h(\mathsf D_m,\mathsf D_n^{\mathbb Q})$, $\mathrm{Map}_{\leq k}^h(\mathsf D_m,\mathsf D_n)$, $n-m\geq 2$, of the truncated little discs operads, which allows one to determine rationally the delooping of the Goodwillie-Weiss tower for the spaces of long embeddings.