arXiv research

New inequality for odd-degree flexible curves using surface doubling.

Maps from rational homology solid tori yield rank inequalities in Heegaard Floer homology.

A new simple proof for surface map degree inequality.

Graphs with nonnegative Bakry-Émery curvature have volume doubling and Poincaré inequalities.

We extend the classical Aleksandrov-Fenchel inequality for mixed volumes to functionals arising naturally in hermitian integral geometry. As a consequence, we obtain Brunn-Minkowski and isoperimetric inequalities for hermitian quermassintegrals.

We apply Murasugi-Tristram inequality to real algebraic curves of odd degree on $RP^2$ with a deep nest, i.e. a nest of the depth $k-1$ where $2k+1$ is the degree. For such curves, the ingredients of the Murasugi-Tristram inequality can be computed (or estimated) inductively using the computations for iterated torus li…

Expander graphs have been intensively studied in the last four decades. In recent years a high dimensional theory of expanders has emerged, and several variants have been studied. Among them stand out coboundary expansion and topological expansion. It is known that for every $d$ there are unbounded degree simplicial co…

Unconditional proof of Demailly's transcendental Morse inequality for higher cohomology classes using a general gauge-fixing for the Monge-Ampère-type equation.

We give bounds on the gap functions of the singularities of a cuspidal plane curve of arbitrary genus, generalising recent work of Borodzik and Livingston. We apply these inequalities to unicuspidal curves whose singularity has one Puiseux pair: we prove two identities tying the parameters of the singularity, the genus…

New bounds on HOMFLY polynomial for homogeneous links.

New PL-invariants defined for 4-manifolds with boundary.

Study improves bounds on p-covectors and proves stable systolic inequalities.

Surgery method proves category inequality for specific manifolds.

We establish three rank inequalities for the reduced flavor of Heegaard Floer homology of Seifert fibered integral homology spheres. Combining these inequalities with the known classifications of non-zero degree maps between Seifert fibered spaces, we prove that a map f from Y' to Y between Seifert homology spheres yie…

Study of harmonic functions on infinite penny graphs.

Study harmonic function growth on curved spaces, proving inequalities.

In a recent paper, McMullen showed an inequality between the Thurston norm and the Alexander norm of a 3-manifold. This generalizes the well-known fact that twice the genus of a knot is bounded from below by the degree of the Alexander polynomial. We extend the Bennequin inequality for links to an inequality for all po…

We generalize optimal inequalities of C. Loewner and M. Gromov, by proving lower bounds for the total volume in terms of the homotopy systole and the stable systole. Our main tool is the construction of an area-decreasing map to the Jacobi torus, streamlining and generalizing the construction of the first author in col…

We propose some kinetic models of wealth exchange and investigate their behavior on directed networks though numerical simulations. We observe that network topology and directedness yields a variety of interesting features in these models. The nature of asset distribution in such directed networks show varied results, …

We give examples showing that Kidwell's inequality for the maximal degree of the Brandt-Lickorish-Millett-Ho polynomial is in general not sharp.

In this paper, we prove interior Poincar{é} and Sobolev inequalities in Euclidean spaces and in Heisenberg groups, in the limiting case where the exterior (resp. Rumin) differential of a differential form is measured in L 1 norm. Unlike for L p , p > 1, the estimates are doomed to fail in top degree. The singular integ…

Characterizes diagrams achieving Morton-Franks-Williams inequality for positive knots and links.

Proves Hodge-Riemann relations for mixed valuations and strengthens geometric inequalities.

Formula for harmonic current dimension on foliated surfaces, extending Brunella's inequality.

We prove Cheeger inequalities for p-Laplacians on finite and infinite weighted graphs. Unlike in previous works, we do not impose boundedness of the vertex degree, nor do we restrict ourselves to the normalized Laplacian and, more generally, we do not impose any boundedness assumption on the geometry. This is achieved …

A local cut point is by definition a point that disconnectes its sufficiently small neighborhood. We show that there exists an upper bound for the degree of a local cut point in a metric measure space satisfying the generalized Bishop--Gromov inequality. As a corollary, we obtain an upper bound for the number of ends o…

Researchers prove inequalities for differential forms in Heisenberg groups, extending Euclidean results.

The entropy-degree theorem applies to Alexandrov spaces with curvature constraints.

We describe a procedure for creating infinite families of knots, each having the maximum degree of their HOMFLY polynomial strictly less than twice their canonical genus. These families build upon examples first found by Stoimenow.

New Orlicz Brunn-Minkowski inequalities are established for rigid motion compatible Minkowski valuations of arbitrary degree. These extend classical log-concavity properties of intrinsic volumes and generalize seminal results of Lutwak and others. Two different approaches which refine previously employed techniques are…

This paper proves a Faber-Krahn inequality for trees with given matching number.

It is proved in this note that the analogues of the Bennequin inequality which provide an upper bound for the Bennequin invariant of a Legendrian knot in the standard contact three dimensional space in terms of the lower degree in the framing variable of the HOMFLY and the Kauffman polynomials are not sharp. Furthermor…

Improved Yang-Yau inequality for all orientable surfaces except for specific genera.

Proves a generalized isoperimetric inequality for spheres in dimensions 4 and above.

We use the concept of intrinsic metrics to give a new definition for an isoperimetric constant of a graph. We use this novel isoperimetric constant to prove a Cheeger-type estimate for the bottom of the spectrum which is nontrivial even if the vertex degrees are unbounded.

In a previous article the author extended the Witten deformation to singular spaces with cone-like singularities and to a class of Morse functions called admissible Morse functions. The method applies in particular to complex cones and stratified Morse functions in the sense of the theory developed by Goresky and MacPh…

Defines discrete symmetry of manifolds and proves bounds on its value.

We estimate the linear isoperimetric constants of an n-dimensional ellipse. Using these estimates and a technique of Gromov, we estimate the Hopf and linking invariants of Lipschitz maps from ellipses to round spheres. Using these estimates, we give a lower bound for the k-dilation of degree non-zero maps between ellip…

Using the recent Gauss diagram formulas for Vassiliev invariants of Polyak-Viro-Fiedler and combining these formulas with the Bennequin inequality, we prove several inequalities for positive knots relating their Vassiliev invariants, genus and degrees of the Jones polynomial. As a consequence, we prove that for any of …

This paper presents a dynamic model to study the impact on the economic outcomes in different societies during the Malthusian Era of individualism (time spent working alone) and collectivism (complementary time spent working with others). The model is driven by opposing forces: a greater degree of collectivism provides…

We show that for closed orientable manifolds the $k$-dimensional stable systole admits a metric-independent volume bound if and only if there are cohomology classes of degree $k$ that generate cohomology in top-degree. Moreover, it turns out that in the nonorientable case such a bound does not exist for stable systoles…

We prove that a bounded open set U in Euclidean n-space has k-width less than C(n) Volume(U)^{k/n}. Using this estimate, we give lower bounds for the k-dilation of degree 1 maps between certain domains in Euclidean space. In particular, we estimate the smallest (n-1)-dilation of any degree 1 map between two n-dimension…

We prove a conjecture saying that complex projective space has maximal volume (degree) among all toric Kaehler-Einstein manifolds of dimension n. The proof is inspired by our recent work on sharp Moser-Trudinger and Brezis-Merle type inequalities for the complex Monge-Ampere operator, but is essentially self-contained.

The study proves inequalities for area and boundary length of disks in convex manifolds.

We study a metric version of the simplicial volume on Riemannian manifolds, the Lipschitz simplicial volume, with applications to degree theorems in mind. We establish a proportionality principle and a product inequality from which we derive an extension of Gromov's volume comparison theorem to products of negatively c…

The classical isoperimetric inequality in R^3 states that the surface of smallest area enclosing a given volume is a sphere. We show that the least area surface enclosing two equal volumes is a double bubble, a surface made of two pieces of round spheres separated by a flat disk, meeting along a single circle at an ang…

The paper establishes inequalities for Chern classes and Riemann-Roch type inequalities for projective manifolds.

We prove new local inequality for divisors on surfaces and utilize it to compute $α$-invariants of singular del Pezzo surfaces, which implies that del Pezzo surfaces of degree one whose singular points are of type $\mathbb{A}_{1}$, $\mathbb{A}_{2}$, $\mathbb{A}_{3}$, $\mathbb{A}_{4}$, $\mathbb{A}_{5}$ or $\mathbb{A}_{6…