arXiv research

Study positive characteristic Fano 4-folds with nef tangent bundles.

Proves positive Euler characteristic for certain manifolds with positive second intermediate Ricci curvature.

The paper proves simplicial volume positivity for certain nonpositively curved 4-manifolds with nonzero Euler characteristic.

We classify those manifolds of positive euler characteristic on which a lie group G acts with cohomogeneity one, where G is classical simple

We fully classify all Lagrangian submanifolds of a complex Grassmannian which are an orbit of a compact group of isometries and have positive Euler characteristic.

Constructs 4-manifolds with positive Euler characteristic proving a conjecture.

This paper formalizes manifolds in positive characteristic varieties.

The paper develops a theory of Ehresmann structures in positive characteristic.

New proof shows 4-manifolds can't support complex structures.

For $Σ$ an orientable surface of finite topological type having genus at least 3 (possibly closed or possibly with any number of punctures or boundary components), we show that the mapping class group $Mod(Σ)$ has no faithful linear representation in any dimension over any field of positive characteristic.

The paper proves positivity of characteristic forms for certain vector bundles.

Study finds symplectic fillings' properties for specific contact covers.

New methods classify convex lattice polygons for affine dimers.

We show that a compact manifold admitting a Killing foliation with positive transverse curvature fibers over finite quotients of spheres or weighted complex projective spaces, provided that the singular foliation defined by the closures of the leaves has maximal dimension. This result is obtained by deforming the folia…

We prove the existence of a geometric characteristic submanifold for non-positively curved manifolds of any dimension greater than or equal to three. In dimension three, our result is a geometric version of the topological characteristic submanifold theorem due to Jaco, Shalen and Johannson.

We consider linear groups which do not contain unipotent elements of infinite order, which includes all linear groups in positive characteristic, and show that this class of groups has good properties which resemble those held by groups of non positive curvature and which do not hold for arbitrary characteristic zero l…

We prove that the Euler characteristic of an even-dimensional compact manifold with positive (nonnegative) sectional curvature is positive (nonnegative) provided that the manifold admits an isometric action of a compact Lie group $G$ with principal isotropy group $H$ and cohomogeneity $k$ such that $k - (\rank G - \ran…

The Grove-Searle theorem on 2d manifolds with 8 or less symmetry groups has positive Euler characteristic.

New kernels defined for various spaces, including measures.

Improved theorem on curvature and manifold symmetry.

Study L^2-Betti numbers in prime characteristic for a conjecture about 2-complex towers.

We compute the characteristic varieties and the Alexander polynomial of a finitely generated nilpotent group. We show that the first characteristic variety may be used to detect nilpotence. We use the Alexander polynomial to deduce that the only torsion-free, finitely generated nilpotent groups with positive deficiency…

The paper extends a theorem to number fields without infinite places.

We give lower bounds, in terms of the Euler characteristic, for the $L^2$-norm of the Weyl curvature of closed Riemannian 4-manifolds. The same bounds were obtained by Gursky, in the case of positive scalar curvature metrics.

Foams have Lie algebra symmetries that simplify web state spaces.

New invariants detect exotic smooth structures in 4-manifolds.

In this paper we show that a simply connected 8-dimensional manifold M of positive sectional curvature and symmetry rank $\geq 2$ resembles a rank one symmetric space in several ways. For example, the Euler characteristic of M is equal to the Euler characteristic of S^8, H P^2 or C P^4. And if M is rationally elliptic …

The paper finds a geometric lower bound for the first positive eigenvalue of the rough Laplacian on 1-forms.

FI-modules were introduced by the first three authors in [CEF] to encode sequences of representations of symmetric groups. Over a field of characteristic 0, finite generation of an FI-module implies representation stability for the corresponding sequence of S_n-representations. In this paper we prove the Noetherian pro…

The Weyl functional is analyzed on 4-manifolds with positive scalar curvature.

Manifolds admitting positive sectional curvature are conjectured to have rigid homotopical structure and, in particular, comparatively small Euler charateristics. In this article, we obtain upper bounds for the Euler characteristic of a positively curved Riemannian manifold that admits a large isometric torus action. W…

We extend the deep and important results of Lichnerowicz, Connes, and Gromov-Lawson which relate geometry and characteristic numbers to the existence and non-existence of metrics of positive scalar curvature (PSC). In particular, we show: that a spin foliation with Hausdorff homotopy groupoid of an enlargeable manifold…

This is the first part of a series of papers where we compute Euler characteristics, signatures, elliptic genera, and a number of other invariants of smooth manifolds that admit Riemannian metrics with positive sectional curvature and large torus symmetry. In the first part, the focus is on even-dimensional manifolds i…

We classify simply connected, closed cohomogeneity one manifolds with singly generated or 4-periodic rational cohomology and positive Euler characteristic.

Study on metrics on specific nilmanifolds, finding new examples and properties.

A subset of a group is characteristic if it is invariant under every automorphism of the group. We study word length in fundamental groups of closed hyperbolic surfaces with respect to characteristic generating sets consisting of a finite union of orbits of the automorphism group, and show that the translation length o…

Any closed orientable and smooth non-positively curved manifold M is known to admit a geometric characteristic splitting, analogous to the JSJ decomposition in three dimensions. We show that when this splitting consists of pieces which are Seifert fibered or pieces each of whose fundamental group has non-trivial centre…

Investment strategies involving cryptocurrencies and VIX INDEX show positive impact in market performance.

The paper studies spectral properties of Jacobi operator for surfaces with nonpositive Euler characteristic.

Positive curvature manifolds from smaller ones using division algebras and geodesic flow.

The Hopf conjecture states that an even-dimensional, positively curved Riemannian manifold has positive Euler characteristic. We prove this conjecture under the additional assumption that a torus acts by isometries and has dimension bounded from below by a logarithmic function of the manifold dimension. The main new to…

Proves a formula for push-forward of polynomial Chern forms in universal vector bundles.

Study improves HOMFLY polynomial coefficients for positive braid links.

Proves a 1930s Hopf conjecture about positive curvature manifolds.

Study shows how 3+1D cosmologies can evolve to de Sitter space under certain conditions.

The authors give a short survey of previous results on $δ$-homogeneous Riemannian manifolds, forming a new proper subclass of geodesic orbit spaces with non-negative sectional curvature, which properly includes the class of all normal homogeneous Riemannian manifolds. As a continuation and an application of these resul…

Parreau compactified the Hitchin component of a closed surface $S$ of negative Euler characteristic in such a way that a boundary point corresponds to the projectivized length spectrum of an action of $π_1(S)$ on an $\mathbb R$-Euclidean building. In this paper, we use the positivity properties of Hitchin representatio…

By adopting Multifractal detrended fluctuation (MF-DFA) analysis methods, the multifractal nature is revealed in the high-frequency data of two typical indexes, the Shanghai Stock Exchange Composite 180 Index (SH180) and the Shenzhen Stock Exchange Composite Index (SZCI). The characteristics of the corresponding multif…