Related papers: Rotation numbers and rotation classes on one-dimen…
A spectral sequence is defined which converges to the \v{C}ech cohomology of the Euclidean hull of a tiling of the plane with Euclidean finite local complexity. The terms of the second page are determined by the so-called ePE homology and…
Quasiperiodic patterns described by polyhedral "atomic surfaces" and admitting matching rules are considered. It is shown that the cohomology ring of the continuous hull of such patterns is isomorphic to that of the complement of a torus…
In this paper, we compute explicitly both the $K$-theory and integral cohomology rings of the space of commuting elements in $SU(2)$ via the $K$-theory of its desingularization. We also briefly discuss the different behavior of its…
This paper is concerned with an interpretation of f-cohomology, a modification of motivic cohomology of motives over number fields, in terms of motives over number rings. Under standard assumptions on mixed motives over finite fields,…
If E is a non-isotrivial elliptic curve over a global function field F of odd characteristic we show that certain Mordell-Weil groups of E have 1-dimensional eigenspace relative to a fixed complex ring class character provided that the…
We study the rotational structures of aperiodic tilings in Euclidean space of arbitrary dimension using topological methods. Classical topological approaches to the study of aperiodic patterns have largely concentrated just on translational…
For a continuous map on a topological graph containing a loop $S$ it is possible to define the degree (with respect to the loop $S$) and, for a map of degree $1$, rotation numbers. We study the rotation set of these maps and the periods of…
We apply tilting theory over preprojective algebras $Lambda$ to a study of moduli space of $Lambda$-modules. We define the categories of semistable modules and give an equivalence, so-called reflection functors, between them by using…
We develop silting theory of a noetherian algebra $\Lambda$ over a commutative noetherian ring $R$. We study mutation theory of $2$-term silting complexes of $\Lambda$, and as a consequence, we see that mutation exists. As in the case of…
In previous work we introduced the notion of binomial cup-one algebras, which are differential graded algebras endowed with Steenrod $\cup_1$-products and compatible binomial operations. In this paper we show that binomial cup-one algebras…
A basic assumption of tiling theory is that adjacent tiles can meet in only a finite number of ways, up to rigid motions. However, there are many interesting tiling spaces that do not have this property. They have "fault lines", along which…
Various theorems on convergence of general space homeomorphisms are proved and, on this basis, theorems on convergence and compactness for classes of the so-called ring $Q$--homeomorphisms are obtained. In particular, it was established by…
Let $U$ be the quantum group with divided powers in $l-$th root of unity and let $u\subset U$ be the Frobenius kernel. V.Ginzburg and S.Kumar proved that the cohomology algebra of $u$ with trivial coefficients is isomorphic to the functions…
The singular set of a generic map $f: M\to F$ of a manifold $M$ of dimension $m\ge 2$ to an oriented surface $F$ is a closed smooth curve $\Sigma(f)$. We study the parity of the number of components of $\Sigma(f)$. The image $f(\Sigma)$ of…
We study homeomorphisms of tiling spaces with finite local complexity (FLC), of which suspensions of $d$-dimensional subshifts are an example, and orbit equivalence of tiling spaces with (possibly) infinite local complexity (ILC). In the…
In this paper we study the behaviour of modules over finite dimensional algebras whose endomorphism algebra is a division ring. We show that there are finitely many such modules in the module category of an algebra if and only if the length…
A classic result of Brooks, Smith, Stone and Tutte associates to any finite planar network with distinguished source and sink vertices, a tiling of a rectangle by smaller subrectangles whose aspect ratios are given by the conductances of…
In this paper we identify conditions under which the cohomology $H^*(\Omega M\xi;\k)$ for the loop space $\Omega M\xi$ of the Thom space $M\xi$ of a spherical fibration $\xi\downarrow B$ can be a polynomial ring. We use the Eilenberg-Moore…
Poincare's classification of the dynamics of homeomorphisms of the circle is one of the earliest, but still one of the most elegant, classification results in dynamical systems. Here we generalize this to quasiperiodically forced circle…
In this paper we present a generalization of Poincar\'e's Rotation Theory of homeomorphisms of the circle to the case of one-dimensional compact abelian groups which are solenoidal groups, {\it i.e.}, groups which fiber over the circle with…