Related papers: Betti numbers of Gaussian fields

We describe and analyze a numerical algorithm for computing the homology (Betti numbers and torsion coefficients) of real projective varieties. Here numerical means that the algorithm is numerically stable (in a sense to be made precise).…

We show that any complex manifold that has a divisor whose normalization has non-zero first Betti number either has a non-trivial holomorphic gerbe which does not trivialize meromorphicly or a meromorphic line bundle not equivalent to any…

This is the first article of a series of two where we study the problem of bounded deviations for homeomorphisms of closed surfaces of genus $\ge 2$. This first part studies bounded deviations with respect to closed geodesics. As a…

The one-skeleton of a G-manifold M is the set of points p in M where $\dim G_p \geq \dim G -1$; and M is a GKM manifold if the dimension of this one-skeleton is 2. Goresky, Kottwitz and MacPherson show that for such a manifold this…

In this paper, we investigate some geometric functionals for band limited Gaussian and isotropic spherical random fields in dimension 2. In particular, we focus on the area of excursion sets, providing its behavior in the high energy limit.…

For line bundles on arithmetic varieties we construct height functions using arithmetic intersection theory. In the case of an arithmetic surface, generically of genus g, for line bundles of degree g equivalence is shown to the height on…

We investigate the evolution of the force network in experimental systems of two-dimensional granular materials under impact. We use the first Betti number, $\beta_1$, and persistence diagrams, as measures of the topological properties of…

There have been several recent articles studying homology of various types of random simplicial complexes. Several theorems have concerned thresholds for vanishing of homology, and in some cases expectations of the Betti numbers. However…

In terms of the excursion set model, we used Monte Carlo methods in order to study the non-Markovian stochastic evolution of the smoothed overdensity $\delta$ at scale $S$. For a Gaussian density field, smoothed by the top-hat filter, in…

We estimate from below the expected Betti numbers of real hypersurfaces taken at random in a smooth real projective n-dimensional manifold. These random hypersurfaces are chosen in the linear system of a large d-th power of a real ample…

We present a rough classification of differential forms on a Riemannian manifold, we consider definitions and properties of conformal Killing forms on a compact Riemannian manifold and define Tachibana numbers as an analog of the well known…

We compute the rational Betti cohomology groups of the coarse moduli spaces of geometrically marked Del Pezzo surfaces of degree three and four as representations of the Weyl groups of the corresponding root systems. The proof uses a blend…

We give a survey of algorithms for computing topological invariants of semi-algebraic sets with special emphasis on the more recent developments in designing algorithms for computing the Betti numbers of semi-algebraic sets. Aside from…

Let X be a complex smooth quasi-projective variety with an epimorphism $\nu \colon \pi_1(X)\twoheadrightarrow \mathbb{Z}^n$. We survey recent developments about the asymptotic behaviour of Betti numbers with any field coefficients and the…

In this paper we discuss two general models of random simplicial complexes which we call the lower and the upper models. We show that these models are dual to each other with respect to combinatorial Alexander duality. The behaviour of the…

There has been considerable recent interest, primarily motivated by problems in applied algebraic topology, in the homology of random simplicial complexes. We consider the scenario in which the vertices of the simplices are the points of a…

The interplay between the quantum state space and a specific set of measurements can be effectively captured by examining the set of jointly attainable expectation values. This set is commonly referred to as the (convex) joint numerical…

In this survey article we describe the geometry of toric hyperk\"ahler varieties, which are hyperk\"ahler quotients of the quaternionic vector spaces by tori. In particular, we discuss the Betti numbers, the cohomology ring, and variation…

For any numerical semigroup $S$, there are infinitely many numerical symmetric semigroups $T$ such that $S=\frac{T}{2}$ is their half. We are studying the Betti numbers of the numerical semigroup ring $K[T]$ when $S$ is a 3-generated…

We prove that the independence complexes of some grids have exponential Betti numbers. This corresponds to the number of ground states in the hard-core model in statistical physics with fermions in the vertices of the grid.