We focus on fast-slow systems involving both fractional Brownian motion (fBm) and standard Brownian motion (Bm). The integral with respect to Bm is the standard Itô integral, and the integral with respect to fBm is a generalized Riemann–Stieltjes integral by means of fractional calculus. We establish an averaging principle in which the fast-varying diffusion process of the fast-slow systems acts as a “noise” to be averaged out in the limit. We show that the slow process has a limit in the mean square sense, which is characterized by the solution of stochastic differential equations driven by fBm whose coefficients are averaged with respect to the stationary measure of the fast-varying diffusion. An implication is that one can ignore the complex original systems and concentrate on the averaged systems instead. This averaging principle paves the way for reduction of computational complexity.

In this paper, we derive from deep results due to Clozel and Ullmo a sharp density result of Noether–Lefschetz loci inside the moduli space of marked (polarized) irreducible holomorphic symplectic (IHS) varieties. In particular, we obtain the density of Hilbert schemes of points on projective $K3$ surfaces and of projective generalized Kummer varieties in their moduli spaces. We present applications to the existence of rational curves on projective deformations of such varieties, to the study of the Mori cone of curves and of the associated extremal birational contractions, and a refinement of Hassett’s result on cubic fourfolds whose Fano variety of lines is isomorphic to a Hilbert scheme of two points on a $K3$ surface. We also discuss Voisin’s conjecture on the existence of coisotropic subvarieties on IHS varieties and relate it to a stronger statement on Noether–Lefschetz loci in their moduli spaces.

In this paper, we discuss the asymptotic behavior of the capacity of the range of symmetric random walks on finitely generated groups. We show the corresponding strong law of large numbers and central limit theorem.

We consider an axiomatic treatment of support theory for abelian categories. Many supports of Rosenberg and the atom support of Kanda are examples of support in the sense described here. We introduce the notion of a locally projective object, and we use it to give a generalization of the triangulated category of singularities for schemes. A result of Chen about the triangulated category of singularities is shown to hold in this generality.

Let $\mathcal{C}$ be a locally bounded $\mathbb{k}$-category, where $\mathbb{k}$ is a field. We prove that $\mathcal{C}$ is pure-semisimple, that is, every object of $\mathrm{Mod}\text{-}\mathcal{C}$ is pure-projective if and only if every family of morphisms between indecomposable finitely generated $\mathcal{C}$-modules is Noetherian. Our formalism establishes the pure-semisimplicity of Galois coverings, that is, if $\mathcal{C}$ is a G-category with a free G-action on $\mathrm{ind}\text{-}\mathcal{C}$, then $\mathcal{C}$ is pure-semisimple if and only if $\mathcal{C}∕G$ is so.

In our recent work, homotopy types of nilpotent orbits are explicitly described in the case of real simple classical Lie algebras for which any maximal compact subgroup in the associated adjoint group is not semisimple. Here we extend the above description of homotopy types of nilpotent orbits to the remaining cases of real simple classical Lie algebras for which any maximal compact subgroup in the associated adjoint group is semisimple.

Over complex numbers, the Fourier–Mukai (FM) partners of abelian varieties are well understood. A celebrated result is Orlov’s derived Torelli theorem. In this note, we study the FM partners of abelian varieties in positive characteristic. We notice that in odd characteristic, two abelian varieties of odd dimension are derived equivalent if their associated Kummer stacks are derived equivalent, which is Krug and Sosna’s result over complex numbers. For abelian surfaces in odd characteristic, we show that two abelian surfaces are derived equivalent if and only if their associated Kummer surfaces are isomorphic. This extends the result of Hosono, Lian, Oguiso, and Yau to odd characteristic fields, solving a classical problem originally from Shioda. Furthermore, we establish the derived Torelli theorem for supersingular abelian varieties and apply it to characterize the quasiliftable birational models of supersingular generalized Kummer varieties.