arXiv research

Proves stability of cone-volume measure with nearly constant density.

The paper confirms a conjecture linking link bipyramid volume and Mahler measure.

Geometrically proves majorizing measure theorem on Hadamard manifolds.

Volume of unit balls defined by quadratic differentials is not proper and has integrable volume.

The paper introduces new volume measures and volume comparison inequalities for Lorentzian spaces.

Integral foliated simplicial volume is a version of simplicial volume combining the rigidity of integral coefficients with the flexibility of measure spaces. In this article, using the language of measure equivalence of groups we prove a proportionality principle for integral foliated simplicial volume for aspherical m…

We study a class of 2-variable polynomials called exact polynomials which contains $A$-polynomials of knot complements. The Mahler measure of these polynomials can be computed in terms of a volume function defined on the vanishing set of the polynomial. We prove that the local extrema of the volume function are on the …

The general volume of a star body, a notion that includes the usual volume, the $q$th dual volumes, and many previous types of dual mixed volumes, is introduced. A corresponding new general dual Orlicz curvature measure is defined that specializes to the $(p,q)$-dual curvature measures introduced recently by Lutwak, Ya…

Measure homology is a variation of singular homology designed by Thurston in his discussion of simplicial volume. Zastrow and Hansen showed independently that singular homology (with real coefficients) and measure homology coincide algebraically on the category of CW-complexes. It is the aim of this paper to prove that…

We consider the relation between simplicial volume and two of its variants: the stable integral simplicial volume and the integral foliated simplicial volume. The definition of the latter depends on a choice of a measure preserving action of the fundamental group on a probability space. We show that integral foliated s…

We compute the Riemannian volume on the moduli space of flat connections on a nonorientable 2-manifold, for a natural class of metrics. We also show that Witten's volume formula for these moduli spaces may be derived using Haar measure, and we give a new proof of Witten's volume formula for the moduli space of flat con…

This article explores some properties of universal covers of compact Kahler manifolds, under the assumption of Caratheodory measure hyperbolicity. In particular, by comparing invariant volume forms, an inequality is established between the volume of canonical bundle of a compact Kahler manifolds and the Caratheodory me…

New weighted surface area measures for convex bodies with applications.

Sharp inequality in spaces with non-negative Ricci curvature.

Study investor attention using search volume data before and after mobile device popularity.

In this note, I will discuss a possible relation between the Mahler measure of the colored Jones polynomial and the volume conjecture. In particular, I will study the colored Jones polynomial of the figure-eight knot on the unit circle. I will also propose a method to prove the volume conjecture for satellites of the f…

Volume is a natural measure of complexity of a Riemannian manifold. In this survey, we discuss the results and conjectures concerning n-dimensional hyperbolic manifolds and orbifolds of small volume.

Sharp isoperimetric inequality on Finsler manifolds with non-negative Ricci curvature.

We study the asymptotic behavior of volume forms on a degenerating family of compact complex manifolds. Under rather general conditions, we prove that the volume forms converge in a natural sense to a Lebesgue-type measure on a certain simplicial complex. In particular, this provides a measure-theoretic version of a co…

The cone-volume measure of a polytope with centroid at the origin is proved to satisfy the subspace concentration condition. As a consequence a conjectured (a dozen years ago) fundamental sharp affine isoperimetric inequality for the U-functional is completely established -- along with its equality conditions.

Study shows volume constraints lead to isoperimetric constant bounds in specific metric spaces.

We study unimodular measures on the space $\mathcal M^d$ of all pointed Riemannian $d$-manifolds. Examples can be constructed from finite volume manifolds, from measured foliations with Riemannian leaves, and from invariant random subgroups of Lie groups. Unimodularity is preserved under weak* limits, and under certain…

We present computational data and heuristic arguments which suggest that given a hyperbolic knot the volume correlates with its determinant, the Mahler measure of its Alexander polynomial and the Mahler measure of the twisted Alexander polynomial corresponding to the discrete and faithful SL(2,C)-representation.

Defines a geometric dimension for Lorentzian spaces, distinguishing spacelike and null subspaces.

Measures neural network decision boundary volume to predict model performance.

We make several improvements to the mean-variance framework for optimal pre-trade algorithmic execution, by working with volume measures and generic price dynamics. Volume measures are the continuum analogies for discrete volume profiles commonly implemented in the execution industry. Execution then becomes an absolute…

Study measures volume of foliations on surfaces, finding integrability range.

We show that the cone-volume measure of a convex body with centroid at the origin satisfies the subspace concentration condition. This implies, among others, a conjectured best possible inequality for the $\mathrm{U}$-functional of a convex body. For both results we provide stronger versions in the sense of stability i…

Study geometric and topological properties of Finsler manifolds with weighted Ricci curvature bounds.

The paper studies curvature measures and volume-preserving flows on convex bodies.

The paper proves a quantitative rigidity result for spaces with specific curvature bounds.

The volume of a credal set correlates with epistemic uncertainty in binary classification but not in multi-class.

Study simplicial volume via foliated simplices and duality.

We consider a connected smooth $n$-dimensional manifold $M$ endowed with a volume form $Ω$, and we show that an open subset $U$ of $R^n$ of Lebesgue measure $\Vol (U)$ embeds into $M$ by a smooth volume preserving embedding whenever the volume condition $\Vol (U) \le \Vol (M,Ω)$ is met.

We study the convergence of volume forms on a degenerating holomorphic family of log-Calabi-Yau varieties to a non-Archimedean measure, extending a result of Boucksom and Jonsson. More precisely, let $(X,B)$ be a holomorphic family of sub log canonical, log-Calabi-Yau complex varieties parameterized by the punctured un…

We extend the theory of Patterson-Sullivan measure to any regular covering of a compact manifold using the Busemann compactification and derive an integral formula for the volume entropy. As applications we prove some rigidity theorems for the volume entropy.

In this report I discuss the relations between systoles and volumes of hyperbolic manifolds and a conjecture of Lehmer about the Mahler measure of non-cyclotomic polynomials.

Geometric approach to majorizing measures for polyhedra and general compact objects.

In this paper we consider non-compact non-flat simply connected harmonic manifolds. In particular, we show that the Martin boundary and Busemann boundary coincide for such manifolds. For any finite volume quotient we show that (up to scaling) there is a unique Patterson-Sullivan measure and this measure coincides with …

New definition of metric current yields Finsler geometry volume densities.

In this paper, we prove that if a metric measure space satisfies the volume doubling condition and the Caffarelli-Kohn-Nirenberg inequality with same exponent n(n>1), then it has exactly n-dimensional volume growth. As application, we obtain geometric and topological properties of Alexandrov space, Riemannian manifold …

Synthetic Ricci flows defined for metric measure spaces.

The paper defines and proves non-triviality of volume and Euler classes in bounded cohomology of transformation groups.

Study shows how to calculate the volume of pseudoeffective line bundles on Kähler manifolds.

We introduce Gaussian-type measures on the manifold of all metrics with a fixed volume form on a compact Riemannian manifold of dimension $\geq 3$. For this random model we compute the characteristic function for the $L^2$ (Ebin) distance to the reference metric. In the Appendix, we study Lipschitz-type distance betwee…

Measure homology was introduced by Thurston in his notes about the geometry and topology of 3-manifolds, where it was exploited in the computation of the simplicial volume of hyperbolic manifolds. Zastrow and Hansen independently proved that there exists a canonical isomorphism between measure homology and singular hom…

Study shows infimum of dual volume equals convex core volume for hyperbolic 3-manifolds.

We provide a proof for an inequality between volume and L2-Betti numbers of aspherical manifolds for which Gromov outlined a strategy based on general ideas of Connes. The implementation of that strategy involves measured equivalence relations, Gaboriau's theory of L2-Betti numbers of R-simplicial complexes, and other …