55 papers
This survey reviews the computation of Weil-Petersson and Masur-Veech volumes for moduli spaces of Riemann surfaces, highlighting their profound impact on combinatorial enumeration, intersection theory, and recursion relations, while drawing parallels between the methods used to calculate these two distinct types of volumes.
This paper establishes sufficient conditions for the existence of longest arcs in left-invariant three-dimensional contact sub-Lorentzian structures on solvable Lie groups and the universal cover of SL(2, R), addressing the nontrivial existence question inherent in such optimal control problems with unbounded control sets.
This paper addresses an Erdős-Herzog-Piranian problem by proving that the maximum product of distances for diameter-2 point sets can be found among convex polygons, providing improved constructions over regular -gons and demonstrating that a general characterization of extremal polygons is impossible for even orders.
This expository article explores the phenomenon where different definitions of fractal dimension yield conflicting results for simple examples and introduces "dimension interpolation" as a framework to unify these isolated numerical answers into a coherent geometric picture by viewing classical dimensions as boundary points of continuous families.
This paper establishes a new lower bound of 11948 for the kissing number in 19 dimensions by combining Cohn and Li's odd-sign construction with an explicit nonlinear binary code of length 19 derived from nested codes within a punctured extended binary Golay code.
This paper investigates the construction of positive definite thresholding functions for correlation matrices that preserve positive semidefiniteness, establishing existence results and bounds while proving that any geometrically unbiased soft-thresholding operator necessarily induces a geometric collapse in the feature space, thereby limiting the recoverable signal.
This paper establishes that for a set of points in the plane with diameter at most 1, the ratio of pairs with distance to pairs with distance is bounded below by , thereby improving upon Steinerberger's previous bound and nearly confirming the conjectured asymptotic behavior.
This paper demonstrates that quantum cellular automata constitute the degree-zero component of a coarse homology theory, thereby providing a formal framework that explains why the space of such automata forms an Omega-spectrum.
This paper establishes the Mosco-convergence of Cheeger energies on Gromov-Hausdorff converging spaces satisfying curvature-dimension conditions by employing a Lagrangian approach that combines Wasserstein geodesic stability with nonsmooth calculus duality, thereby ensuring the continuity of Neumann eigenvalues even in infinite-dimensional settings.
This paper establishes that in doubling metric measure spaces supporting a -Poincaré inequality, domains with uniformly thick boundaries possess a large "visible" portion accessible via John curves, and that the traces of Sobolev functions on these domains belong to the Besov class of the visible boundary.
This paper introduces a canonical -algebra of operators with finite dynamical propagation for non-singular group actions, establishing its isomorphism to the algebraic crossed product under free actions to characterize ergodicity and to describe the Roe algebras of warped spaces and warped cones.
This paper establishes that under upper timelike sectional curvature bounds in Lorentzian length spaces, the space of directions at a point exists as a metric space with curvature bounded above by , while its metric cone forms a Lorentzian length space with non-positive timelike sectional curvature, thereby extending synthetic comparison geometry to causal settings.
This paper solves the Snellius-Pothenot problem by determining the number of points in the plane of a fixed triangle that correspond to any given point on the surface defined by the cosines of the angles subtended by the triangle's sides.
This paper investigates the minimal conditions required to establish Lipschitz and uniform bounds for sequences of bounded rough Riemannian metrics, providing counterexamples to demonstrate the sharpness of these conditions while exploring their underlying geometric intuition.
This paper proves the nonexistence of Heronian triangles with three integer medians by deriving a universal identity for arbitrary triangles and establishing a lemma demonstrating that such triangles, if they existed, would necessarily occur in non-similar pairs.
This paper introduces the -Gromov-Wasserstein distance as a unified framework for comparing -networks (measure spaces with -valued kernels), proving that it forms a metric space with desirable topological properties while providing computable bounds for practical applications.
Using the compression method, this paper generalizes lower bounds for the Erdős unit distance and distinct distance problems to higher dimensions ( for ), providing alternative proofs and establishing new estimates for the number of distinct distances.
This paper establishes that ultralimits of bounded Lipschitz maps extend naturally to Sobolev maps, a result used to prove the stability of Dehn functions under ultraconvergence of pointed length spaces and to provide a simpler proof characterizing spaces of curvature bounded above via isoperimetric inequalities.
This paper introduces the family of -affine dual curvature measures for convex bodies, establishes their connections to classical measures as limiting cases, and investigates the associated Minkowski problems by providing existence and necessary conditions for solutions, particularly in the even and smooth cases.
This paper establishes a dimension-independent lower bound on the cotype constant for normed spaces generated by random polytopes with Gaussian vertices, demonstrating that under specific density conditions on the number of vertices relative to dimension, these spaces satisfy a cotype inequality with high probability.