Related papers: Cohomology fractals
In this paper we use fractal geometry to investigate boundary aspects of the first homology group for finite coverings of the modular surface. We obtain a complete description of algebraically invisible parts of this homology group. More…
We propose a method for computing the cohomology ring of three--dimensional (3D) digital binary-valued pictures. We obtain the cohomology ring of a 3D digital binary--valued picture $I$, via a simplicial complex K(I)topologically…
A likelihood function on a smooth very affine variety gives rise to a twisted de Rham complex. We show how its top cohomology vector space degenerates to the coordinate ring of the critical points defined by the likelihood equations. We…
The quantum cohomology algebra of a projective manifold X is the cohomology H(X,Q) endowed with a different algebra structure, which takes into account the geometry of rational curves in X. We show that this algebra takes a remarkably…
Using a homological invariant together with an obstruction class in a certain Ext^2-group, we may classify objects in triangulated categories that have projective resolutions of length two. This invariant gives strong classification results…
We show that the first twisted cohomology group associated to closed 1-forms on compact manifolds is related to certain 2-dimensional representations of the fundamental group. In particular, we construct examples of nowhere-vanishing…
Motivated by physics, we propose two conjectures regarding the cohomology ring of the crepant resolutions of orbifolds and cohomological invariants of K-equivalent manifolds.
Bounded-cohomological dimension of groups is a relative of classical cohomological dimension, defined in terms of bounded cohomology with trivial coefficients instead of ordinary group cohomology. We will discuss constructions that lead to…
In this survey, I suggest to approach the problem of functorial properties of quantum cohomology by drawing lessons from several versions of Mirror duality involving deformation spaces.
We investigate the relationship between the symmetric, exterior and classical cohomologies of groups. The first two theories were introduced respectively by Staic and Zarelua. We show in particular, that there is a map from exterior…
In this note, we survey our recent work concerning cohomologies of harmonic bundles on quasi-compact Kaehler manifolds.
We use cross ratios to describe second real continuous bounded cohomology for locally compact topological groups. We also derive a rigidity result for cocycles with values in the isometry group of a proper hyperbolic geodesic metric space.
The notion of a Frobenius manifold appears in relation to various topics in algebraic and analytic geometry, such and quantum cohomology, deformation of meromorphic connections, unfolding of singularities and others. In the local setting…
A family of fractal arrangements of circles is introduced for each imaginary quadratic field $K$. Collectively, these arrangements contain (up to an affine transformation) every set of circles in the extended complex plane with integral…
Let $\mathcal{O}_K$ be a mixed characteristic complete DVR with perfect residue field $k$ and fraction field $K$. It is a celebrated result of Berthelot and Ogus that for a smooth proper formal scheme $X/\mathcal{O}_K$ there exists a…
Aperiodic point sets (or tilings) which can be obtained by the method of cut and projection from higher dimensional periodic sets play an important role for the description of quasicrystals. Their topological invariants can be computed…
Perfect fractals are mathematical objects that, because they are generated by recursive processes, have self-similarity and infinite complexity. In particular, they also have a fractional dimension. Although several proposals for the study…
We exhibit a family of complex manifolds, which has a member at each odd complex dimension and which has the same cohomology groups as the complex projective space at that dimension, but not homotopy equivalent to it. We also analyze the…
We compute the dimensions of the Hochschild cohomology of symmetric groups over prime fields in low degrees. This involves us in studying some partition identities and generating functions of the dimensions in any fixed degree of the…
For image segmentation, the current standard is to perform pixel-level optimization and inference in Euclidean output embedding spaces through linear hyperplanes. In this work, we show that hyperbolic manifolds provide a valuable…