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For all $k \ge 2$, we show that there exists a group $G$ and a non-free stably free $\mathbb{Z} G$-module of rank $k$. We use this to show that, for all $k \ge 2$, there exist homotopically distinct finite $2$-complexes with fundamental…

Let $k$ be a field of characteristic $p>0$, and let $W$ be a complete discrete valuation ring of characteristic $0$ that has $k$ as its residue field. Suppose $G$ is a finite group and $G^{\mathrm{ab},p}$ is its maximal abelian $p$-quotient…

Given a reductive group $G$ and a parabolic subgroup $P\subset G$, with maximaltorus $T$, we consider (following Dabrowski's work) the closure $X$ of a generic $T$-orbit in $G/P$, and determine in combinatorial termswhen the toric variety…

We classify the simple quantum group modules with finite dimensional weight spaces when the quantum parameter $q$ is transcendental and the Lie algebra is not of type $G_2$. This is part $2$ of the story. The first part being Irreducible…

Given a finite group $G$ and two unitary $G$-representations $V$ and $W$, possible restrictions on Brouwer degrees of equivariant maps between representation spheres $S(V)$ and $S(W)$ are usually expressed in a form of congruences modulo…

For a positive integer $k$, a group $G$ is said to be totally $k$-closed if for each set $\Omega$ upon which $G$ acts faithfully, $G$ is the largest subgroup of $\mathrm{Sym}(\Omega)$ that leaves invariant each of the $G$-orbits in the…

Let (G) be a connected compact non-abelian Lie-group and (T) a maximal torus of (G). A torus manifold with (G)-action is defined to be a smooth connected closed oriented manifold of dimension (2\dim T) with an almost effective action of (G)…

An orbit polytope is the convex hull of an orbit under a finite group $G \leq \operatorname{GL}(d,\mathbb{R})$. We develop a general theory of possible affine symmetry groups of orbit polytopes. For every group, we define an open and dense…

This thesis is devoted to the study of geometric properties of affine algebraic varieties endowed with an action of an algebraic torus. It comes from three preprints which correspond to the indicated points (1), (2), (3). Let $X$ be an…

We show that when G is a finite group which contains an elementary Abelian subgroup of order p^2 and k is an algebraically closed field of characteristic p, then the study of simple endotrivial kG-modules which are not monomial may be…

A vertex algebra with an action of a group $G$ comes with a notion of $g$-twisted modules, forming a $G$-crossed braided tensor category. For a Lie group $G$, one might instead wish for a notion of $(\mathrm{d}+A)$-twisted modules for any…

We develop the theory of rational ideals for arbitrary associative algebras R without assuming the standard finiteness conditions, noetherianness or the Goldie property. The Amitsur-Martindale ring of quotients replaces the classical ring…

We prove a flat torus theorem for quadric complexes. In particular, we show that if a non-cyclic free abelian group $G$ acts metrically properly on a quadric complex $X$, then $G \cong \mathbb{Z}^2$ and $X$ contains a $G$-invariant…

Trinomial hypersurfaces form a natural class of affine algebraic varieties closely connected with varieties admitting a torus action of complexity one. We investigate orbits of the automorphism group on these hypersurfaces. We prove that…

We classify holomorphic as well as algebraic torus equivariant principal $G$-bundles over a nonsingular toric variety $X$, where $G$ is a complex linear algebraic group. It is shown that any such bundle over an affine, nonsingular toric…

We establish some interactions between uniformly recurrent subgroups (URSs) of a group $G$ and cosets topologies $\tau_\mathcal{N}$ on $G$ associated to a family $\mathcal{N}$ of normal subgroups of $G$. We show that when $\mathcal{N}$…

We define a notion of complexity for modules over infinite groups. We show that if $M$ is a module over the group ring $kG$, and $M$ has complexity $\leq f$ (where $f$ is some complexity function) over some set of finite index subgroups of…

An affine varieties with an action of a semisimple group $G$ is called "small" if every non-trivial $G$-orbit in $X$ is isomorphic to the orbit of a highest weight vector. Such a variety $X$ carries a canonical action of the multiplicative…

Let $G/H$ be a homogeneous variety, and let $X$ be a $G$-equivariant embedding of $G/H$such that the number of $G$-orbits in $X$ is finite. We show that the equivariant Borel-Moore homology of $X$ has a filtration with associated graded…

Let $X$ be a smooth algebraic variety endowed with an action of a finite group $G$ such that there exists the geometric quotient $\pi_X:X\to X/G$. We characterize rational tensor fields $\tau$ on $X/G$ such that the {\it pull back} of $\tau…