Abstract:
The book is devoted to study so-called irregular subsets of the Grassmannian manifold $G^{n}_{k}(V)$ (this class of sets was introduced by author). In the previous variant of the book we restrict ourself only to the case when $V$ is an $n$-dimensional vector space under the field $R$. Now we consider irregular subsets of $G^{n}_{k}(V)$, where $V$ is an $n$-dimensional vector space under arbitrary field.

Abstract:
Let $\Omega$ be a non-singular syplectic form on some vector space $V$. Denote by $S^{n}_{k}(\Omega)$ the set of all $k$-dimensional planes $s$ in $V$ such that the restriction of $\Omega$ onto $s$ is singular. For the cases when $k=2,n-2$ a simple geometric characteristic of $S^{n}_{k}(\Omega)$ will be described.

Abstract:
Let $P$ and $P'$ be projective spaces having the same dimension $n$ assumed to be finite. Let $\Delta$ and $\Delta'$ be buildings of type $A_n$. We consider mappings of chambers of $\Delta$ into chambers of $\Delta'$ which send appartments to appartments. It is shown that shuch mappings are induced by strong embeddings of $P$ into $P'$ or into the dual space $P^{'*}$.

Abstract:
Let $V$ and $V'$ be $2n$-dimensional vector spaces over fields $F$ and $F'$. Let also $\Omega: V\times V\to F$ and $\Omega': V'\times V'\to F'$ be non-degenerate symplectic forms. Denote by $\Pi$ and $\Pi'$ the associated $(2n-1)$-dimensional projective spaces. The sets of $k$-dimensional totally isotropic subspaces of $\Pi$ and $\Pi'$ will denoted by ${\mathcal G}_{k}$ and ${\mathcal G}'_{k}$, respectively. Apartments of the associated buildings intersect ${\mathcal G}_{k}$ and ${\mathcal G}'_{k}$ by so-called base subsets. We show that every mapping of ${\mathcal G}_{k}$ to ${\mathcal G}'_{k}$ sending base subsets to base subsets is induced by a symplectic embedding of $\Pi$ to $\Pi'$.

Abstract:
Let $H$ be a separable real Hilbert space. Denote by ${\mathcal G}_{\infty}(H)$ the Grassmannian consisting of closed subspaces with infinite dimension and codimension. This Grassmannian is partially ordered by the inclusion relation. We show that every order preserving transformation of ${\mathcal G}_{\infty}(H)$ can be extended to an automorphism of the lattice of closed subspaces of $H$. It follows from Mackey's result \cite{Mackey} that automorphisms of this lattice are induced by invertible bounded linear operators.

Abstract:
This book is dedicated to Grassmannians associated with buildings of classical types: usual, polar, and half-spin Grassmannians. Grassmannians of vector spaces and Grassmannians consisting of totally isotropic subspaces of non-degenerate alternating, Hermitian, and symmetric forms are special cases of these "building" Grassmannians.

Abstract:
Let $\Pi$ be a polar space of rank $n$ and let ${\mathcal G}_{k}(\Pi)$, $k\in \{0,\dots,n-1\}$ be the polar Grassmannian formed by $k$-dimensional singular subspaces of $\Pi$. The corresponding Grassmann graph will be denoted by $\Gamma_{k}(\Pi)$. We consider the polar Grassmannian ${\mathcal G}_{n-1}(\Pi)$ formed by maximal singular subspaces of $\Pi$ and show that the image of every isometric embedding of the $n$-dimensional hypercube graph $H_{n}$ in $\Gamma_{n-1}(\Pi)$ is an apartment of ${\mathcal G}_{n-1}(\Pi)$. This follows from a more general result (Theorem 2) concerning isometric embeddings of $H_{m}$, $m\le n$ in $\Gamma_{n-1}(\Pi)$. As an application, we classify all isometric embeddings of $\Gamma_{n-1}(\Pi)$ in $\Gamma_{n'-1}(\Pi')$, where $\Pi'$ is a polar space of rank $n'\ge n$ (Theorem 3).

Abstract:
We consider the {\it infinite Johnson graph} $J_{\infty}$ whose vertex set consists of all subsets $X\subset {\mathbb N}$ satisfying $|X|=|{\mathbb N}\setminus X|=\infty$ and whose edges are pairs of such subsets $X,Y$ satisfying $|X\setminus Y|=|Y\setminus X|=1$. An automorphism of $J_{\infty}$ is said to be {\it regular} if it is induced by a permutation on $\mathbb{N}$ or it is the composition of the automorphism induced by a permutation on $\mathbb{N}$ and the automorphism $X\to {\mathbb N}\setminus X$. The graph $J_{\infty}$ admits non-regular automorphisms. Our first result states that the restriction of every automorphism of $J_{\infty}$ to any connected component ($J_{\infty}$ is not connected) coincides with the restriction of a regular automorphism. The second result is a characterization of regular automorphisms of $J_{\infty}$ as order preserving and order reversing bijective transformations of the vertex set of $J_{\infty}$ (the vertex set is partially ordered by the inclusion relation). As an application, we describe automorphisms of the associated {\it infinite Kneser graph}.

Abstract:
Let $V$ and $V'$ be vector spaces over division rings. Suppose $\dim V$ is finite and not less than 3. Consider a mapping $l:V\to V$ with the following property: for every $u\in {\rm GL}(V)$ there is $u'\in {\rm GL}(V')$ such that $lu=u'l$. Our first result states that $l$ is a strong semilinear embedding if $l|_{V\setminus{0}}$ is non-constant and the dimension of the subspace of $V'$ spanned by $l(V)$ is not greater than $n$. We present examples showing that these conditions can not be omitted. In some special cases, this statement can be obtained from Dicks and Hartley (1991) and Zha (1996). Denote by ${\mathcal P}(V)$ the projective space associated with $V$ and consider the mapping $f:{\mathcal P}(V)\to {\mathcal P}(V')$ with the following property: for every $h\in {\rm PGL}(V)$ there is $h'\in {\rm PGL}(V')$ such that $fh=h'f$. By the second result, $f$ is induced by a strong semilinear embedding of $V$ in $V'$ if $f$ is non-constant and its image is contained in a subspace of $V'$ whose dimension is not greater than $n$, we also require that $R'$ is a field.