Average zeros of Finsler functions equals mixed symplectic volume of ellipsoids.
problem Counting isolated common zeros of Finsler functions.
method Construction of ring of normal densities and Crofton formula generalization.
result Average number of zeros equals mixed symplectic volume of Finsler ellipsoids.
The study connects the average number of solutions to mixed volumes of convex bodies.
problem Finding a relationship between the average number of solutions to systems of equations and mixed volumes of convex bodies.
method Developed Banach metrics in vector spaces, constructed Banach convex bodies in the cotangent bundle of X, and calculated the average number of solutions as the mixed symplectic volume of these bodies. result The average number of solutions is equal to the mixed symplectic volume of Banach convex bodies.
We construct an oriented cobordism between moduli spaces of flat connections on the three holed sphere and disjoint unions of toric varieties, together with a closed two-form which restricts to the symplectic forms on the ends. As applications, we obtain formulas for mixed Pontrjagin numbers and Witten's formulas for s…
Study of mixed equation combining gauge theory and symplectic geometry.
problem Regularity and compactness of solutions to the mixed equation.
method Combining Uhlenbeck and Gormov compactness theorems.
result Moduli spaces of solutions to the mixed equation satisfy compactness properties.
Smooth symplectic manifolds can be approximated by PL symplectic manifolds.
problem Understanding the relationship between smooth and piecewise linear symplectic structures.
method Defining PL symplectic manifolds and proving approximations.
result Smooth symplectic manifolds can be C0-approximated by PL symplectic manifolds. Dual pairs constructed for volume preserving diffeomorphisms using symplectic geometry.
problem Understanding the group of volume preserving diffeomorphisms through symplectic geometry.
method Using cotangent bundles of spaces of smooth embeddings, symplectic reduction, and nonlinear Grassmannians of augmented submanifolds.
result Descriptions of coadjoint orbits of the group of volume preserving diffeomorphisms in terms of submanifolds of augmented spaces.
The paper studies a flow of hypersurfaces preserving mixed volumes and finds convergence to a sphere.
problem Evolution of hypersurfaces under mixed volume preserving flow.
method A flow defined by powers of homogeneous curvature functions of degree one.
result If initial hypersurface satisfies a pinching condition, there exists a unique, smooth solution converging to a round sphere.
Teichmüller space realized as symplectic quotient.
problem Realizing Teichmüller space as a symplectic quotient.
method Infinite-dimensional symplectic manifold, volume-preserving diffeomorphisms, momentum map.
result Teichmüller space and moduli space realized as symplectic orbit reduced spaces.
The paper introduces a volume invariant for Hermitian-symplectic metrics and proves its critical points are Kähler.
problem Investigating volume invariants for Hermitian-symplectic metrics.
method Introducing a functional acting on metrics in Aeppli cohomology classes and proving critical points are Kähler.
result The volume invariant generalises the volume of a Kähler class and vanishing is a necessary condition for the existence of a Kähler metric.
This study calculates the average number of common zeros of holomorphic functions on complex manifolds.
problem Calculating the average number of common zeros of holomorphic functions.
method Defined a Hermitian mixed volume for a mix of non-negative Hermitian forms and proved the average number of common zeros equals this mixed volume.
result The average number of common zeros of holomorphic functions equals the mixed volume of the manifold.
New proofs of geometric inequalities using Bochner formulas.
problem Geometric inequalities and mixed volumes in convex geometry.
method Reduction to Bochner formulas via spectral theorem.
result New, simpler proofs of Alexandrov-Fenchel and Alexandrov's inequalities.
The paper derives formulas for symplectic volume forms on surface representation varieties.
problem Calculating symplectic volume forms on surface representation varieties.
method Multiplicative gluing formulas and Heusener-Porti results.
result Symplectic volume form on Σg,0 is a product of forms on Σ2,1 and Σ2,2. Solves a long-standing convex geometry problem about mixed volumes.
problem Characterizing the support of mixed area measures.
method Geometric approach to convex bodies in R^n and R^3.
result Resolved one direction of Schneider's conjecture for arbitrary convex bodies.
Holomorphic volume form on circle representations generalizes Witten's formula for surfaces with boundary.
problem Generalizing Witten's formula to surfaces with boundary.
method Introducing an holomorphic volume form on the space of representations of the circle.
result Holomorphic volume form appears as a peripheral term in the generalized Witten's formula.
Study geodesic equation on mixed-volume forms on balanced manifolds, proving existence of solutions.
problem Existence of solutions to the Calabi-Yau equation for balanced metrics.
method Introduced a L2 metric space of mixed-volume forms and derived a geodesic equation. result Existence of solutions to the Calabi-Yau equation on all balanced manifolds.
Uniform volume estimate for Kähler metrics in big cohomology classes.
problem Estimating volume for singular Kähler metrics in big cohomology classes.
method Generalized mixed energy estimate for functions in complex Sobolev space to big cohomology classes.
result Uniform non-collapsing volume estimate for local Kähler metrics.
Consider a d×d matrix M whose rows are independent centered non-degenerate Gaussian vectors ξ1,...,ξd with covariance matrices Σ1,...,Σd. Denote by Ei the location-dispersion ellipsoid of ξi:Ei=x∈Rd:x⊤Σi−1x⩽1. We sh…
Proves regularity of geodesic equation on Hermitian manifolds.
problem Regularity of geodesic equation in mixed volume forms space.
method Ellipticity conditions, uniform Laplacian estimates, explicit subsolutions.
result Existence of unique C1,1 solution to Donaldson equation. This article studies the abelian analytic torsion on a closed, oriented, Sasakian three-manifold and identifies this quantity as a specific multiple of the natural unit symplectic volume form on the moduli space of flat abelian connections. This identification computes the analytic torsion explicitly in terms of Seifer…
I construct the real counterparts (which I call Borel-Bott classes) of the R/Z classes constructed in "Characteristic classes in symplectic topology", to appear, in the cohomology of volume-preserving and symplectomorhisms of a compact (symplectic) manifold.I show that, for the symplectic action of the mapping class gr…
Proves stable degeneration preserves symplectic forms and confirms Kaledin's conjecture.
problem Symplectic singularities and their degenerations.
method Combining volume minimization, deformation theory, and rigidity results.
result Kaledin's conjecture confirmed for symplectic singularities.
This paper develops basic setting for the dual Orlicz-Brunn-Minkowski theory for star bodies. An Orlicz φ-radial addition of two or more star bodies is proposed and related dual Orlicz-Brunn-Minkowski inequality is established. Based on a linear Orlicz φ-radial addition of two star bodies, we derive a f…
Inequalities found in contact and symplectic geometry.
problem Finding inequalities in contact and symplectic geometry.
method Proving inequalities for Zoll contact and odd-symplectic forms.
result Proves a local systolic-diastolic inequality for Zoll contact and odd-symplectic forms.
Develops a diagrammatic method for symplectic filling classifications.
problem Classifying exact/weak symplectic fillings of 3D contact manifolds.
method Symplectic JSJ decomposition applied to contact surgery diagrams.
result Recover symplectic fillings for certain lens spaces and torus bundles, and classify fillings for a large class of plumbed 3-manifolds.
New weighted surface area measures for convex bodies with applications.
problem Generalizing surface area measures to weighted Borel measures.
method Formulating and analyzing weighted surface area measures, proving integral formula and Bézout-type inequality.
result New integral formula for mixed measure of three bodies, generalizing Bézout-type inequality.
Researchers show symplectic packing by ellipsoids is unobstructed for various manifolds.
problem Symplectic packing constraints for different manifolds.
method Kahler resolution and cohomology classes.
result Symplectic packings by ellipsoids are unobstructed for specified manifolds.
The paper finds lower bounds for volumes of complex geometric structures.
problem Estimating the volume of complex geometric structures.
method Reduction to a counting problem in the unit tangent bundle, solved using exponential multiple mixing for the geodesic flow.
result First known lower bound for the volume of these manifolds in terms of curve length.
Solves Christoffel-Minkowski problem for axially symmetric bodies.
problem Necessary and sufficient conditions for mixed area measures of axially symmetric convex bodies.
method Introduced a new method to transform mixed area measures and mixed volumes of axially symmetric bodies, refining Firey's classification and improving estimates.
result Complete solution to the mixed Christoffel-Minkowski problem for axially symmetric bodies without regularity assumptions.
Researchers solved Minkowski's quadratic inequality extremals.
problem Characterizing the extremals of Minkowski's quadratic inequality.
method Representation of mixed volumes as Dirichlet forms associated to degenerate elliptic operators, with a quantitative rigidity property.
result Completely settled the extremals of Minkowski's quadratic inequality.
In this paper, we study the symplectic volume of the moduli space of polygons by using Witten's formula. We propose to use this volume as a measure for the flexibility of a polygon with fixed side-lengths. The main result of our is that among all the Spherical and Euclidean polygons with fixed perimeter the regular one…
Constructs scalar-flat Kähler metrics on toric symplectic manifolds.
problem Creating scalar-flat Kähler metrics on toric symplectic manifolds.
method Explicit construction and alternative construction with conical singularity.
result Explicit construction of scalar-flat Kähler metrics on toric symplectic manifolds.
Proves Hodge-Riemann relations for mixed valuations and strengthens geometric inequalities.
problem Geometric inequalities and mixed Hodge-Riemann relations for translation-invariant valuations.
method Proves mixed Hodge-Riemann relations for various convex bodies and their mixed volumes.
result Strengthened geometric inequalities for lower dimensional convex bodies.
We consider curvature flows in hyperbolic space with a monotone, symmetric, homogeneous of degree 1 curvature function F. Furthermore we assume F to be either concave and inverse concave or convex. For compact initial hypersurfaces, which are strictly convex by horospheres, we show the long time existence of mixed volu…
Some methods based on simple regularizing geometric element transformations have heuristically been shown to give runtime efficient and quality effective smoothing algorithms for meshes. We describe the mathematical framework and a systematic approach to global optimization-based versions of such methods for mixed volu…
Survey on invariant quasimorphisms and their relation to stable commutator length.
problem Understanding the relationship between invariant quasimorphisms and stable commutator length.
method Review of existing methods and examples in invariant quasimorphisms and their relation to stable commutator length.
result The existence of non-extendable invariant quasimorphisms is closely related to the behavior of stable mixed commutator length.
This paper pursues the study of the Calabi-Yau equation on certain symplectic non-Kaehler 4-manifolds, building on a key example of Tosatti-Weinkove in which more general theory had proved less effective. Symplectic 4-manifolds admitting a 2-torus fibration over a 2-torus base are modelled on one of three solvable Lie …
Study compatible and associated metrics for contact-symplectic structures, showing geodesic integral curves and minimal leaf properties.
problem Characteristics foliations of metric contact-symplectic structures.
method Analysis of compatible and associated metrics, study of geodesic integral curves, and minimal leaf properties.
result Integral curves of the Reeb vector field are geodesics for any compatible metric, and associated metrics share a common volume element.
Let M be a closed symplectic manifold of volume V. We say that M admits an unobstructed symplectic packing by balls if any collection of symplectic balls (of possibly different radii) of total volume less than V admits a symplectic embedding to M. In 1994 McDuff and Polterovich proved that symplectic packings of Kahler…
Let (M,ω) be a symplectic manifold admitting a metaplectic structure (a symplectic analogue of the Riemannian spin structure) and a torsion-free symplectic connection ∇. Symplectic Killing spinor fields for this structure are sections of the symplectic spinor bundle satisfying a certain first order partial dif…
New Lp-Steiner quermassintegrals defined from Steiner formula.
problem Defining new Lp-Steiner quermassintegrals. method Analogy to classical Steiner formula, investigating properties in convex bodies.
result Rotation and reflection invariant valuations in convex bodies.
The paper connects Chern-Simons invariants to mixed Tate motives in hyperbolic 3-manifolds.
problem Understanding the relationship between Chern-Simons invariants and mixed Tate motives in hyperbolic 3-manifolds.
method Constructing a mixed Tate motive over the invariant trace field whose image equals the Chern-Simons invariant and complex volume.
result The mixed Hodge realization of the motive is a quotient of the path torsor of the augmented character variety.
We consider odd Laplace operators arising in odd symplectic geometry. Approach based on semidensities (densities of weight 1/2) is developed. The role of semidensities in the Batalin--Vilkovisky formalism is explained. In particular, we study the relations between semidensities on an odd symplectic supermanifold and di…
We discuss Witten's formulas for the symplectic volumes of moduli spaces of flat connections on 2-manifolds from the viewpoint of Hamiltonian cobordism as introduced by Ginzburg-Guillemin-Karshon.
The goal of this article is to establish estimates involving the Yamabe minimal volume, mixed minimal volume and some topological invariants on compact 4-manifolds. In addition, we provide topological sphere theorems for compact submanifolds of spheres and Euclidean spaces, provided that the full norm of the second fun…
New general volume concept solves Minkowski problem for star bodies.
problem Minkowski problem for star bodies
method General volume concept, new curvature measure, variational formulas, Minkowski-type inequality
result Solution to the Minkowski problem for the new general dual Orlicz curvature measure
Smoothness of Hamiltonian stationary submanifolds in symplectic manifolds proven.
problem Smoothness of Hamiltonian stationary Lagrangian submanifolds in symplectic manifolds.
method Developed a regularity theory for fourth order nonlinear elliptic equations with two distributional derivatives.
result Any C1-regular Hamiltonian stationary Lagrangian submanifold in a symplectic manifold is smooth. We consider the volumes of classical supermanifolds such as the supersphere, complex projective superspace, and Stiefel and Grassmann supermanifolds, with respect to the natural metrics or symplectic structures. We show that the formulas for the volumes, upon certain universal normalization, can be obtained by an analy…
We study the rigidity and flexibility of symplectic embeddings of simple shapes. It is first proved that under the condition rn2≤2r12 the symplectic ellipsoid E(r1,...,rn) with radii r1≤...≤rn does not embed in a ball of radius strictly smaller than rn. We then use symplectic folding to …