Related papers: Aspherical neighborhoods on arithmetic surfaces: t…

Assume that R is a local regular ring containing an infinite perfect field, or that R is the local ring of a point on a smooth scheme over an infinite field. Let K be the field of fractions of R and the characteristic of K is not 2. Let X…

In this summary paper, we present the key ideas behind the recent proof of the $K(\pi, 1)$ conjecture for affine Artin groups, which states that complements of locally finite affine hyperplane arrangements with real equations and stable…

We compute the mapping class group orbits in the homotopy set of framings of a compact connected oriented surface with non-empty boundary. In the case $g > 1$ the computation is some modification of Johnson's results and certain arguments…

We prove several positive results regarding representation of homotopy classes of spheres and algebraic groups by regular mappings. Most importantly we show that every mapping from a sphere to an orthogonal or a unitary group is homotopic…

We use the reflection group trick to glue manifolds with corners that are Borel-Serre compactifications of locally symmetric spaces of noncompact type and obtain aspherical manifolds. We call these \emph{piecewise locally symmetric}…

The author proposes a method for investigating actions of finite groups on aspherical spaces. Complete homotopy classification of free actions of finite groups on aspherical spaces is obtained. Also there are some results about non-free…

We explicitly calculate an arithmetic adelic quotient group for a locally free sheaf on an arithmetic surface when the fiber over the infinite point of the base is taken into account. The calculations are presented via a short exact…

We study topological group theoretic properties of algebraic groups over local fields. In particular, we find conditions under which such groups have closed images under arbitrary continuous homomorphisms into arbitrary topological groups.

Anabelian geometry with etale homotopy types generalizes in a natural way classical anabelian geometry with etale fundamental groups. We show that, both in the classical and the generalized sense, any point of a smooth variety over a field…

This paper is the second part of arXiv:0707.1766. We develope harmonic analysis in some categories of filtered abelian groups and vector spaces over the fields R or C. These categories contain as objects local fields and adelic spaces…

Universal algebraic geometry is generalised from solutions of equations in a single algebra to the study of $\varphi$- or $K$-spectra, akin to the prime spectrum of a ring. We explore their basic properties and constructions, give a…

Let K be a ring and let A be a subset of K. We say that a map f:A \to K is arithmetic if it satisfies the following conditions: if 1 \in A then f(1)=1, if a,b \in A and a+b \in A then f(a+b)=f(a)+f(b), if a,b \in A and a \cdot b \in A then…

We describe the algebraic K-theory of the $K(1)$-local sphere and the category of type 2 finite spectra in terms of K-theory of discrete rings and topological cyclic homology. We find an infinite family of 2-torsion classes in the $K_0$ of…

We prove that any geometrically connected curve $X$ over a field $k$ is an algebraic $K(\pi,1)$, as soon as its geometric irreducible components have nonzero genus. This means that the cohomology of any locally constant constructible…

Type A surfaces are the locally homogeneous affine surfaces which can be locally described by constant Christoffel symbols. We address the issue of the geodesic completeness of these surfaces: we show that some models for Type A surfaces…

We study the large-scale geometry of mapping class groups of surfaces of infinite type, using the framework of Rosendal for coarse geometry of non locally compact groups. We give a complete classification of those surfaces whose mapping…

We develop a sequential-topological study of rational points of schemes of finite type over local rings typical in higher dimensional number theory and algebraic geometry. These rings are certain types of multidimensional complete fields…

We show the existence of group-theoretic sections of the "etale-by-geometrically abelian" quotient of the arithmetic fundamental group of hyperbolic curves over $p$-adic local fields relative to a proper and flat model which are…

We examine the local geometry of affine surfaces which are locally symmetric. There are 6 non-isomorphic local geometries. We realize these examples as Type A, Type B, and Type C geometries using a result of Opozda and classify the relevant…

We give a complete classification of Riemannian and Lorentzian surfaces of arbitrary codimension in a pseudo-sphere whose pseudo-spherical Gauss maps are of 1-type or, in particular, harmonic. In some cases a concrete global classification…