The paper calculates delta invariants for specific geometric structures.
problem Computing delta invariants for projective bundles and cones of Fano type.
method Provides a precise formula for delta invariants.
result A formula to compute delta invariants for projective bundles and cones of Fano type.
The paper examines conditions for stochastic invariance of cones in SPDEs with jumps.
problem Stochastic invariance of cones in SPDEs with jumps.
method Sufficient conditions for stochastic invariance of closed convex cones in abstract L2-spaces. result Conditions for stochastic invariance of cones are provided and analyzed.
We construct $\sorth{p} \times \sorth{q}$-invariant special Lagrangian (SL) cones in $\C^{p+q}$. These SL cones are natural higher-dimensional analogues of the $\sorth{2}$-invariant SL cones constructed previously by MH and used in our gluing constructions of higher genus SL cones in $\C^{3}$. We study in detail the ge…
Researchers found a family of Sp(2)-invariant solitons for Laplacian flow.
problem Analyzing Sp(2)-invariant solitons for Laplacian flow on the 4-sphere.
method Used mathematical analysis and asymptotic cone determination.
result Identified a 1-parameter family of Sp(2)-invariant expanding solitons with specific asymptotic behavior.
Study proves Volume Conjecture for Reshetikhin-Turaev invariants.
problem Volume Conjecture for Reshetikhin-Turaev invariants.
method Hyperbolic cone metrics and discrete Fourier transforms.
result Proves Volume Conjecture for most figure-8 knot configurations.
Holomorphic tensors on algebraic cones are invariant under certain group actions.
problem Holomorphic tensors on products of algebraic cones
method Using algebraic structures and embeddings
result Holomorphic tensors are invariant under group actions
CoNES optimizes blackbox functions using convex optimization and information geometry.
problem Optimizing high-dimensional blackbox functions efficiently.
method Formulated as a convex program that adapts evolutionary strategies gradient estimates.
result Vastly outperforms conventional blackbox optimization methods on benchmarks and MuJoCo tasks.
The Lee-Gauduchon cone is a convex cone of cohomology classes for complex manifolds.
problem Understanding the Lee-Gauduchon cone for complex manifolds.
method Analyzing the Lee-Gauduchon cone as a convex cone of cohomology classes.
result The Lee-Gauduchon cone is a bimeromorphic invariant.
Study left-invariant pseudo-Riemannian metrics on Lie groups using moving bracket approach.
problem Characterize left-invariant pseudo-Riemannian metrics on Lie groups with vanishing scalar curvature invariants.
method Using the moving bracket approach, analyze Lie algebras of dimensions ≤ 6 and semi-simple Lie algebras.
result All Lie algebras of dimension ≤ 6 except 3-dimensional solvable Lie algebra are in the null cone, leading to VSI metrics.
Study left-invariant pseudo-Riemannian metrics on Lie groups focusing on null cone Lie algebras.
problem Characterize left-invariant pseudo-Riemannian metrics on Lie groups in the null cone.
method Use bracket flow on Lie algebra to study metrics on Lie groups.
result Classify all cases of null cone Lie algebras in signatures (1,q) and (2,q).
The paper proves the existence and uniqueness of Calabi-Yau metrics on affine spherical varieties.
problem Existence and uniqueness of Calabi-Yau metrics on affine spherical varieties.
method Explicit K-stability condition, degeneration, and asymptotic cone analysis.
result Uniqueness of K-invariant Calabi-Yau metrics on affine spherical manifolds. The study describes special real manifolds and invariant admissible cubics in Vinberg cones.
problem Understanding special real manifolds and invariant admissible cubics in Vinberg cones.
method Simplified Vinberg theory using Nil-algebras to describe invariant functions and polynomials.
result Examples of continuous families of non-homogeneous special real manifolds.
In this paper we prove that the space of differential invariants for curves with arc-length parameter in the light cone of Lorentzian R4, invariants under the centro-affine action of the Lorentzian group, is Poisson equivalent to the space of conformal differential invariants for curves in the Möbius sphere…
The paper connects 3D manifold invariants to hyperbolic cone metrics and discrete Fourier transforms.
problem Volume conjecture for Reshetikhin-Turaev invariants of 3-manifolds with links.
method Volume conjecture, hyperbolic cone metrics, discrete Fourier transforms, change-of-pair operations.
result Volume conjecture proven for specific cases, provides approach to solving Volume Conjecture for hyperbolic 3-manifolds.
New Ricci flow solutions found with rotational symmetry and cone-like singularities.
problem Finding Ricci flow solutions with specific symmetry and singularity properties.
method Rotationally symmetric Ricci flow with scaling-invariant curvature bounds, using approximation method.
result Complete Ricci flow solution with cone-like singularity at the origin.
Study verifies asymptotic expansion for Reshetikhin-Turaev invariants of fundamental shadow link pairs.
problem Verifying the asymptotic expansion conjecture for Reshetikhin-Turaev invariants of fundamental shadow link pairs.
method Using logarithmic holonomies of meridians and hyperbolic cone structures, the study verifies the conjecture for pairs where M∖L is homeomorphic to a fundamental shadow link complement. result The asymptotic expansion conjecture is true for pairs (M,L) with sufficiently small cone angles and M∖L homeomorphic to a fundamental shadow link complement. Study convex hulls of orbits for compact groups, defining new invariants related to polynomial degrees.
problem Understanding properties of convex hulls of coadjoint orbits of compact groups.
method Introduce partial convex hulls and use them to define numerical invariants.
result Orbits with new invariants form rational convex polyhedral cones related to Littlewood-Richardson cones.
In this paper we compute the discrete fundamental groups of warped cones. As an immediate consequence, this allows us to show that there exist coarsely simply-connected expanders and superexpanders. This also provides a strong coarse invariant of warped cones and implies that many warped cones cannot be coarsely equiva…
Constructs moduli spaces for Calabi-Yau cones and Sasaki-Einstein manifolds.
problem Proper moduli spaces for K-polystable Q-Fano cones and their links.
method Algebraic construction using local normalized volume and higher Θ-stable reduction.
result Alternative algebraic proof of proper moduli spaces for Q-Fano varieties.
The volume conjecture is extended for surface diffeomorphisms with quantum invariants.
problem Extending the volume conjecture for quantum invariants of surface diffeomorphisms.
method Relating asymptotics of quantum invariants to hyperbolic cone structures on mapping tori.
result The conjecture is proven for a specific case of the once-punctured torus bundle.
Minimal hypersurfaces with cylindrical tangent cones constructed and analyzed.
problem Constructing minimal hypersurfaces with specific geometric properties.
method Constructing minimal hypersurfaces with cylindrical tangent cones and proving unique continuation results.
result Existence and properties of minimal hypersurfaces with cylindrical tangent cones.
Continuity of delta invariant leads to uniform Kähler-Einstein metrics.
problem Continuity of delta invariant in Kähler and twisted Kähler-Einstein metrics.
method Analytic delta invariant and uniform Yau-Tian-Donaldson theorem.
result Uniform Yau-Tian-Donaldson theorem for twisted Kähler-Einstein metrics.
Study shows Futaki invariant vanishes on most Fano threefolds.
problem Analyzing the Futaki invariant on K-polystable Fano threefolds.
method Examining the Futaki invariant as a map from the Kähler cone to the dual of the Lie algebra of the reduced automorphism group.
result Futaki invariant vanishes identically on most Fano threefolds.
Cone structures in quantum field theory linked to information geometry.
problem Understanding geometric structures in quantum field theory.
method Analyzing invariant cones under modular automorphism groups and their relation to Wishart laws.
result Explicit connection between CAH cones and Wishart laws.
We prove a formula for the intersection R-torsion of a finite cone and use it to introduce a family of spectral invariants which is closely related to Cheeger's half torsion.
We construct new families of two-ended O(m)×O(n)-invariant solutions to the Allen- Cahn equation Δu+u-u3=0 in RN+1, with N≥7, whose zero level sets diverge logarithmically from the Lawson cone at infinity. The construction is based on a careful study of the Jacobi-Toda system on a given $O(m)…
Study of line bundles on spherical varieties leads to Calabi-Yau metrics.
problem Understanding linearized line bundles on spherical varieties.
method Formulas for valuative invariants and application to Fano spherical varieties.
result Calabi-Yau metrics on spherical varieties' cone.
The paper studies horofunction compactifications of symmetric cones under Finsler distances.
problem Understanding horofunction compactifications of symmetric cones under Finsler distances.
method Establishing a correspondence between horofunction compactifications of symmetric cones and normed spaces, using Thompson and Hilbert distances.
result Explicit extensions of the exponential map and characterizations of horofunctions for Thompson and Hilbert distances.
We describe the positive cone and the pseudo-effective cone of a non-Kählerian surface. We use these results for two types of applications: - Describe the set σ(X) of possible total Ricci scalars associated with Gauduchon metrics of fixed volume 1 on a fixed non-Kähhlerian surface, and decide whether the assignment $…
New approach linking CR Yamabe invariant to Sasaki structures.
problem Existence of constant transversal scalar curvature Sasaki structures.
method Drawing on CR Yamabe problem ideas, establishing link between invariant, Sasaki structures, and K-stability.
result CR Yamabe invariant value determines K-semistability of Sasaki manifolds.
We find a class of minimal hypersurfaces H(k) as the zero level set of Pfaffians, resp. determinants of real 2k+2 dimensional antisymmetric matrices. While H(1) and H(2) are congruent to a 6-dimensional quadratic cone resp. Hsiang's cubic su(4) invariant in R15, H(k>2) (special harmonic so(2k+2)-invariant cones of degr…
We study the scalar curvature of Kähler metrics that have cone singularities along a divisor, with a particular focus on certain specific classes of such metrics that enjoy some curvature estimates. Our main result is that, on the projective completion of a pluricanonical bundle over a product of Kähler--Einstein Fano …
Study of mean curvature flows with conical singularities using mathematical techniques.
problem Understanding the dynamics of mean curvature flows near conical singularities.
method Feynman-Kac formula and invariant cone method for noncompact settings.
result Generic initial perturbations avoid conical singularities in mean curvature flows.
A polynomial invariant for veering triangulations helps in understanding 3-manifold fibers.
problem Understanding the fibers of 3-manifolds using veering triangulations.
method Introducing a polynomial invariant Vτ associated to veering triangulations and using flow graphs. result The invariant Vτ recovers the Teichmüller polynomial for fibered faces and determines cones in homology. The paper explores cone structures and their connections to parabolic geometries in complex manifolds.
problem Understanding cone structures and their properties in complex manifolds.
method Analyzes cone structures induced by parabolic geometries and VMRT structures, focusing on local invariants.
result Establishes a local differential-geometric version of a global algebraic-geometric recognition theorem.
Novel threefold partitioning of L2-spinor space on cone links.
problem Understanding equivariant Riemann-Roch defects in complex spaces with conic singularities.
method Investigates supertraces over local cohomology groups and spectral asymmetry.
result Novel complex equivariant ξT and ηT invariants defined. Gerbes encode spectral gaps in topological insulators.
problem Capturing spectral gaps in topological insulators.
method Geometric encoding through 'Real' gerbes.
result Gerbes precisely capture spectral gaps.
Due to a result by Gallot a Riemannian cone over a complete Riemannian manifold is either flat or has an irreducible holonomy representation. This is false in general for indefinite cones but the structures induced on the cone by holonomy invariant subspaces can be used to study the geometry on the base of the cone. Th…
We prove some epsilon regularity results for n-dimensional minimal two-valued Lipschitz graphs. The main theorems imply uniqueness of tangent cones and regularity of the singular set in a neighbourhood of any point at which at least one tangent cone is equal to a pair of transversely intersecting multiplicity one n-dim…
We introduce a concept of tree-graded metric space and we use it to show quasi-isometry invariance of certain classes of relatively hyperbolic groups, to obtain a characterization of relatively hyperbolic groups in terms of their asymptotic cones, to find geometric properties of Cayley graphs of relatively hyperbolic g…
Torsion invariants for manifolds which are not simply connected were introduced by K. Reidemeister and generalized to higher dimensions by W. Franz. The Reidemeister torsion, was the first invariant of manifolds which was not a homotopy invariant. The analytic counterpart of the combinatorial Reidemeister torsion was i…
We show that the tangent cone at the identity is not a complete quasiconformal invariant for sub-Riemannian nilpotent groups. Namely, we show that there exists a nilpotent Lie group equipped with left invariant sub-Riemannian metric that is not locally quasiconformally equivalent to its tangent cone at the identity. In…
Defines a new version of Turaev-Viro invariants for 3-manifolds with boundaries.
problem Computing the volume of hyperbolic polyhedral 3-manifolds.
method Introduces a relative version of Turaev-Viro invariants for ideally triangulated compact 3-manifolds with boundaries and a coloring on edges.
result Proves the Volume Conjecture for these invariants, suggesting a method to solve the conjecture for hyperbolic 3-manifolds with totally geodesic boundary.
B Wilking has recently shown that one can associate a Ricci flow invariant cone of curvature operators C(S), which are nonnegative in a suitable sense, to every $Ad_{SO(n,\C)}$ invariant subset $S \subset {\bf so}(n,\C)$. For curvature operators of a Kähler manifold of complex dimension n, one considers $Ad_{GL(n,\…
Coarse homotopy theory connects Euclidean cones to shape theory of compact spaces.
problem Establishing connections between coarse homotopy theory and shape theory.
method Using pointed shape invariants and inverse mapping telescopes.
result Proving two compact spaces are strong shape equivalent if their Euclidean cones are coarsely homotopy equivalent.
We study the analytic torsion of the cone over an orientable odd dimensional compact connected Riemannian manifold W. We prove that the logarithm of the analytic torsion of the cone decomposes as the sum of the logarithm of the root of the analytic torsion of the boundary of the cone, plus a topological term, plus a fu…
A regularization procedure developed in [1] for the integral curvature invariants on manifolds with conical singularities is generalized to the case of squashed cones. In general, the squashed conical singularities do not have rotational O(2) symmetry in a subspace orthogonal to a singular surface Σ so that the surfa…
This paper is devoted to the systematic investigation of the cone construction for Riemannian G manifolds M, endowed with an invariant metric connection with skew torsion ∇c, a `characteristic connection'. We show how to define a Gˉ structure on the cone $\bar M=M\x \R^+$ with a cone metric, and we prov…