New theory connects string theory to swampland distance conjecture.
problem Connecting string theory to swampland distance conjecture.
method Deformations of the heterotic superpotential, treating separately for large fluxes or large distances, integrating out fields to obtain a new field theory.
result New holomorphic theory defined, connects to swampland distance conjecture.
Geometric flows help solve the swampland problem by preserving Einstein equations.
problem Addressing the swampland conjecture in string theory.
method Analyzing scalar and metric bubble solutions under Perelman's flow, deriving geometric flow equations, and introducing an additional energy-momentum tensor term.
result A supplementary energy-momentum tensor term precisely reproduces the infinite tower of states with exponentially dropping masses.
New geometric flow equations describe how space-time dimensions change.
problem Understanding how the number of space-time dimensions affects geometry.
method Developed D-flow equations to model the variation of space-time geometries.
result Solutions for D-flow equations on D-dimensional spheres and Freund-Rubin Compactification.
Study elliptic operators on glued manifolds, reducing to finite-dimensional systems.
problem Mapping properties of elliptic operators in gluing problems.
method Reduction to finite-dimensional linear systems in the limit Tightarrow∞. result Construction of Fredholm inverses with controlled norms.
Classical 10d string backgrounds with a 4d de Sitter space-time, D-brane and orientifold sources, are commonly believed to satisfy the following: 1. There is no classical de Sitter solution with parallel sources. 2. Classical de Sitter solutions with intersecting sources are unstable. 3. Classical de Sitter solutions c…
Compact gravity models yield tiny spin-two field masses.
problem Understanding the mass of spin-two fields in compactified gravity models.
method Relies on Bakry-Émery geometry, Cheeger constant, and synthetic Ricci lower bounds.
result Proves the existence of a spin-two field with very small mass under general assumptions.
Inversive distance circle packing on surfaces was introduced by Bowers-Stephenson as a generalization of Thurston's circle packing and conjectured to be rigid. The infinitesimal and global rigidity of circle packing with nonnegative inversive distance were proved by Guo and Luo respectively. The author proved the globa…
Lorentzian distances to Cauchy surfaces fail to be locally equi-Lipschitz.
problem Lorentzian distances to Cauchy surfaces
method Conjectures based on Cauchy temporal functions
result Lorentz distances to Cauchy surfaces are not locally equi-Lipschitz
Paper proves circle packings converge to Riemann mapping for Jordan domains.
problem Proving discrete conformal maps converge to Riemann mapping.
method Establishing solvability theorem for inversive distance circle packings.
result Bowers-Stephenson's conjecture for Jordan domains is proven.
Inversive distance circle packing metric was introduced by P Bowers and K Stephenson \cite{BS} as a generalization of Thurston's circle packing metric \cite{T1}. They conjectured that the inversive distance circle packings are rigid. For nonnegative inversive distance, Guo \cite{Guo} proved the infinitesimal rigidity a…
This paper investigates several global rigidity issues for polyhedral surfaces including inversive distance circle packings. Inversive distance circle packings are polyhedral surfaces introduced by P. Bowers and K. Stephenson as a generalization of Andreev-Thurston's circle packing. They conjectured that inversive dist…
Study rigidity by logarithmic capacity and related functions.
problem Rigidity phenomena in kernel functions and capacities.
method Exploration of Bergman kernel, logarithmic capacity, Green's function, and Euclidean distance/volume.
result Established rigidity theorems by logarithmic capacity.
Proves conjecture about geodesic foliations in Riemannian planes.
problem Geodesic foliations with bounded distance in non-flat Riemannian planes.
method Analyzes total curvature and visibility properties to prove conjecture.
result Proves conjecture in two specific cases.
Study validates Michor-Mumford conjecture in infinite dimensional Hilbertian H-type groups.
problem Validation of Michor-Mumford conjecture in infinite dimensional Hilbertian H-type groups.
method Introduced infinite dimensional Hilbertian H-type groups with weak, graded, left invariant Riemannian metrics and proved the vanishing of geodesic distance and local unboundedness of sectional curvature.
result Validation of Michor-Mumford conjecture linking geodesic distance vanishing to local unboundedness of sectional curvature.
We give a counterexample of Bowers-Stephenson's conjecture in the spherical case: spherical inversive distance circle packings are not determined by their inversive distances.
Proves rigidity of circle packings in the plane, generalizing previous work.
problem Rigidity of infinite inversive distance circle packings in the plane.
method Maximal principle for generic weighted Delaunay inversive distance circle packings and ring lemma for inversive distance circle packings in hexagonal triangulated plane.
result Proves Bowers-Stephenson's conjecture for inversive distance circle packings.
The paper disproves the properness conjecture for higher-dimensional minimal hypersurfaces.
problem Properness of complete minimal hypersurfaces in higher dimensions.
method Chord-arc estimates and gluing techniques.
result Construction of a complete, improperly embedded minimal hypersurface in Rn+1 for every n≥3. New bounds on cover degrees for Teichmüller distance between hyperbolic surfaces.
problem Finding optimal cover degrees for Teichmüller distance between hyperbolic surfaces.
method Proved the existence of a constant k>0 depending on M and N such that the covers MεoM and NεoN can be chosen to have degrees less than ε−k. result The bound ε−k is optimal for certain arithmetic Riemann surfaces. Physics: Similar long-distance properties can mask vastly different short-distance metrics.
problem Classifying homogeneous metrics on group manifolds by long-distance properties.
method Apply universality concept to geometry, focusing on metrics on Lie groups.
result Many metrics on low-dimensional Lie groups have similar long-distance properties despite differing short-distance properties.
We show that the distance trisector curve is not an algebraic curve, as was conjectured in the founding paper by T. Asano, J. Matousek and T. Tokoyama: "The distance trisector curve", Advances in Math., 212, 338-360 (2007).
The paper establishes Sobolev inequalities between Riemannian metrics and their distance functions.
problem Establishing a theory of Sobolev inequalities for Riemannian metrics and distance functions.
method Analyzing the sub-critical case $p < rac{m}{2}$, proving a Sobolev inequality linking $L^{ rac{p}{2}}$ bounds on metrics to Lq bounds on distance functions. result A Sobolev inequality exists between Riemannian metrics and their distance functions, leading to a convergence theorem.
Study convexity of Mabuchi functional in big cohomology classes.
problem Convexity of Mabuchi functional in big cohomology classes.
method Defined an invariant related to transcendental Fujita approximations and established convexity under vanishing of this invariant.
result Established almost convexity along weak geodesics in big cohomology classes.
Defines a distance function on a manifold using symplectic embeddings and recovers the metric.
problem Recovering a Riemannian metric from symplectic embeddings in cotangent bundles.
method Defines a distance-like function ρW using symplectic embeddings and recovers the metric when W is the unit disc-cotangent bundle. result The distance function ρW recovers the Riemannian metric when W is the unit disc-cotangent bundle. Study knots with genus one, finds Gordian distance and cosmetic crossing constraints.
problem Understanding knots with genus one and their properties.
method Using HOMFLT polynomials to find obstructions for Gordian distance and cosmetic crossings.
result Proves the (generalized) cosmetic crossing conjecture for genus one pretzel knots.
The paper proves inequalities linking Wasserstein distances and eigenfunctions in RCD(K,∞) spaces.
problem Estimating Wasserstein distances and their bounds in RCD(K,∞) spaces.
method Similar techniques used to prove inequalities involving p-Wasserstein distances and Laplace eigenfunctions. result Proves a conjectured lower bound on p-Wasserstein distance between positive and negative parts of Laplace eigenfunctions. The paper proves diameter bounds and finiteness for amply regular graphs.
problem Proving diameter bounds and finiteness for amply regular graphs.
method Improved curvature estimates and new Bakry-Émery curvature estimates.
result There are only finitely many amply regular graphs with specific parameters.
Paper proves rigidity of discrete conformal structures on polyhedral surfaces.
problem Rigidity of discrete conformal structures on polyhedral surfaces.
method Variational principles.
result Proves Glickenstein's conjecture on the rigidity of discrete conformal structures.
The study confirms conjectures about normals to convex polytopes in 3D space.
problem Concurrent normals problem for convex polytopes in 3D.
method Analyzes the PL concurrent normals problem for convex polytopes, proving conjectures for specific cases.
result Polytopes in 3D have points with 10 normals from interior points, confirmed for all tetrahedra and triangular prisms.
Study on geodesic distances on SE(3)/SO(2) in machine learning.
problem Investigating the efficiency of computationally efficient sections in selecting geodesic distances.
method Analyzing geodesic distances on reductive homogeneous spaces, proving the efficiency of minimal distance sections.
result Minimal distance sections are not always geodesic minimizers, but minimal horizontal geodesics are.
We study the geodesic distance induced by right-invariant metrics on the group Diffc(M) of compactly supported diffeomorphisms, for various Sobolev norms Ws,p. Our main result is that the geodesic distance vanishes identically on every connected component whenever s<min{n/p,1}, where …
Example shows learnable distributions not privately learnable.
problem Learnable distributions under non-private conditions not transferable to differential privacy.
method Example of a distribution class learnable up to constant error in total variation distance but not under differential privacy.
result Contradicts conjecture of Ashtiani on learnability under differential privacy.
A slice distance for the class of weak abelian Lp-bundles in 3 dimensions was introduced in a previous article in collaboration with Tristan Rivière, where it was used to prove the closure of such class of bundles for the weak Lp-convergence. We further investigate this distance here, and we prove more properties of it…
Two Anosov metrics with same boundary distance are isometric.
problem Boundary rigidity for surfaces of Anosov type.
method Transfer principle linking marked length spectrum rigidity to marked boundary distance rigidity.
result Two metrics of Anosov type with the same marked boundary distance are isometric.
The Kneser-Poulsen conjecture says that if a finite collection of balls in a Euclidean (spherical or hyperbolic) space is rearranged so that the distance between each pair of centers does not increase, then the volume of the union of these balls does not increase as well. We give new results about central sets of subse…
The Kobayashi pseudometric on a complex manifold is the maximal pseudometric such that any holomorphic map from the Poincaré disk to the manifold is distance-decreasing. Kobayashi has conjectured that this pseudometric vanishes on Calabi-Yau manifolds. Using ergodicity of complex structures, we prove this conjecture fo…
New distances defined between space-times, proving some definite.
problem Defining distances between space-times.
method Introducing causal-null-compactifiable space-times and using cosmological time and null distance.
result Various definite distances defined, proving convergence of space-times.
We prove that a quasiisometric map between rank one symmetric spaces is within bounded distance from a unique harmonic map. In particular, this completes the proof of the Schoen-Li-Wang conjecture.
Given Mφ, a fibered 3-manifold with boundary, we show that the translation distance of the monodromy φ can be bounded above by the complexity of an essential surface with non-zero slope. Furthermore we prove that the minimal complexity of a surface with non-zero slope in Mφn tends to infini…
This article explains how to construct immersed Lagrangian submanifolds in C^2 that are asymptotic at large distance from the origin to a given braid in the 3-sphere. The self-intersections of the Lagrangians are related to the crossings of the braid. These Lagrangians are then used to construct immersed Lagrangians in…
This paper raises an implicit manifold learning perspective in Generative Adversarial Networks (GANs), by studying how the support of the learned distribution, modelled as a submanifold Mθ, perfectly match with Mr, the support of the real data distribution. We show that optimizing Jensen-Sha…
A proof that the separating curve complex of the closed genus two surface has a quasi-distance formula and is delta hyperbolic using tools of Masur and Schleimer. This answers in the affirmative a Conjecture of Schleimer.
Study sequences of static spacetimes using null distance convergence.
problem How to define convergence for sequences of spacetimes.
method Define null distance metric space structure compatible with Lorentzian structure.
result Prove VADB theorem for sequences of static spacetimes with null distance.
This paper introduces a new distance metric for filtered A-infinity categories, focusing on Lagrangian submanifolds.
problem Measuring the distance between filtered A-infinity categories associated with Lagrangian submanifolds.
method Developed a Gromov-Hausdorff distance to measure the difference between these categories.
result Established that the sequence of filtered A-infinity categories forms a Cauchy sequence in Gromov-Hausdorff distance.
We introduce a coarse combinatorial description of the Weil-Petersson distance d_WP(X,Y) between two finite area hyperbolic Riemann surfaces X and Y. The combinatorics reveal a connection between Riemann surfaces and hyperbolic 3-manifolds conjectured by Thurston: the volume of the convex core of the quasi-Fuchsian man…
We study one parameter degenerations of complex projective manifolds by introducing certain type of Hodge metrics coming from the pluricanonical forms. We show that degenerations with at most canonical singularities are all in the finite distance boundary of moduli spaces. We also propose the converse to be true in the…
Neural models price financial options without assuming underlying price forms.
problem Pricing financial options under flexible price processes.
method Apply neural SDEs as universal approximators, use Wasserstein distance for training.
result Error in option prices bounded by Wasserstein distance used for training.
It was conjectured by Escobar [J. Funct. Anal. 165 (1999), 101-116] that for an n-dimensional (n≥3) smooth compact Riemannian manifold with boundary, which has nonnegative Ricci curvature and boundary principal curvatures bounded below by c>0, the first nonzero Steklov eigenvalue is greater than or equal to $…
Study finds almost contact structures in thermal QCD-like theories at intermediate coupling.
problem Understanding (Almost) Contact Structures in thermal QCD-like theories.
method Explicitly obtained (Almost) Contact Structures and SU(3) structures.
result Subspaces of C3S and AC3S are not mutually 'N-path connected' in the Infra-Red.