Geodesic tetrahedra found for Platonic cusped manifolds.
problem Finding geodesic decompositions for Platonic cusped manifolds.
method Decomposing Platonic cusped hyperbolic manifolds into geodesic ideal tetrahedra.
result Geodesic triangulations exist for Platonic cusped manifolds.
New mixed-platonic 3-manifolds from different polyhedra types.
problem Finding new hyperbolic knot complements with hidden symmetries.
method Defined mixed-platonic 3-manifolds and studied their properties.
result No mixed-platonic hyperbolic knot complement has hidden symmetries.
Census of hyperbolic Platonic manifolds and their complements.
problem Classifying hyperbolic Platonic manifolds and their complements.
method Generalized earlier work on ideal tetrahedra to octahedra, identifying complements of augmented knotted trivalent graphs.
result Identification of complements of augmented knotted trivalent graphs in hyperbolic Platonic manifolds.
The problem of classifying, upto isometry (or similarity), the orientable spherical, Euclidean and hyperbolic 3-manifolds that arise by identifying the faces of a Platonic solid is formulated in the language of Coxeter groups. In the spherical and hyperbolic cases, this allows us to complete the classification begun by…
We carry out the harmonic analysis on four Platonic spherical three-manifolds with different topologies. Starting out from the homotopies (Everitt 2004), we convert them into deck operations, acting on the simply connected three-sphere as the cover, and obtain the corresponding variety of deck groups. For each topology…
New self-shrinkers of Platonic solids found.
problem Finding new embedded self-shrinkers of specific genus.
method Variational methods, numerical discovery by D. Chopp.
result Constructed self-shrinkers resembling doublings of Platonic solids.
New method studies automorphism groups of Platonic surfaces.
problem Understanding the automorphism groups of Platonic surfaces.
method Unfolding construction and monodromy group analysis.
result Explicit bounds on cyclic quotient groups of Platonic surfaces.
Skeleta of Platonic solids are factored into spheres.
problem Factor Platonic polytope skeletons into canonical spheres.
method Explicit construction and application of Keevash's design theorem.
result Existence and construction of sphere factorizations for Platonic polytope skeletons.
Constant Mean Curvature n-noids with Platonic Symmetries
We interpret Coxeter's truncated braid groups in terms of Platonic solids.
problem Understanding the structure of truncated braid groups.
method Topological interpretation using orbifolds.
result Connection between truncated braid groups and Platonic solids.
Proof confirms perfect representation in deep learning models.
problem Tackles the perfect Platonic Representation Hypothesis in deep learning models.
method Detailed proof using stochastic gradient descent (SGD) and analysis of global minima.
result SGD trains EDLNs to learn the same representation up to rotation, suggesting emergent entropic forces.
The study of Platonic solids' unfoldings leads to high genus Teichmüller curves.
problem Understanding the topology and geometry of Teichmüller curves from Platonic solids.
method Computing Teichmüller curves using lattice surfaces and algorithmic approaches.
result The Teichmüller curve of the unfolded dodecahedron has genus 131 with specific singularities and cusps.
Study of translation covers of platonic solids reveals monodromy group structures.
problem Understanding monodromy groups of translation covers of platonic solids.
method Computed Zariski closures using generators, constraints, and Lyapunov spectrum analysis.
result Zariski closures of monodromy groups are powers of SL(2, R).
A spherical topological manifold of dimension n-1 forms a prototile on its cover, the (n-1)-sphere. The tiling is generated by the fixpoint-free action of the group of deck transformations. By a general theorem, this group is isomorphic to the first homotopy group. Multiplicity and selection rules appear in the form of…
From the homotopy groups of two cubic spherical 3-manifolds we construct the isomorphic groups of deck transformations acting on the 3-sphere. These groups become the cyclic group of order eight and the quaternion group respectively. By reduction of representations from the orthogonal group to the identity representati…
Minimal surfaces span periodic curves in 3D space.
problem Existence of minimal surfaces spanning periodic curves.
method Proof of existence for minimal surfaces using periodic curves in R3. result Existence of noncompact simply connected periodic minimal surfaces.
New theory explains how self-supervised learning converges, advancing AI research.
problem Lack of precise theoretical explanation for self-supervised learning convergence.
method Synthesized Identifiability Theory with empirical evidence to propose Singular Identifiability Theory (SITh).
result SITh provides deeper insights into SSL's implicit data assumptions and advances representation learning.
New semi-equivelar map found on sphere, not vertex-transitive.
problem Characterizing semi-equivelar maps on sphere.
method Combinatorial proofs and vertex-transitivity analysis.
result Exactly one semi-equivelar map on sphere is not vertex-transitive.
New periodic polyhedra found in curved spaces.
problem Existence of periodic polyhedra in curved spaces.
method Using Archimedean solids and transformations, constructing polyhedra with specific properties.
result Existence of compact polyhedral surfaces in spaceforms.
Develops a stochastic approach to financial market delays.
problem Modeling delays in financial markets with multiple assets.
method Introduces a general stochastic framework for information and order execution delays.
result Delayed markets maintain fundamental asset pricing theorems and no asymptotic free lunch condition.
The paper connects quantum computing to 3-manifolds using knots and links.
problem Exploring the relationship between quantum computing and 3-dimensional manifolds.
method Mapping quantum computing operations to geometric structures of knots and links.
result Quantum operations correspond to specific 3-manifold coverings and Dehn fillings.
If all but two vertices of a triangulated sphere have degrees divisible by k, then the exceptional vertices are not adjacent. This theorem is proved for k=2 with the help of the coloring monodromy. For k=3,4,5 colorings by the vertices of platonic solids have to be used. With a coloring monodromy one can asso…
By regular tessellation, we mean any hyperbolic 3-manifold tessellated by ideal Platonic solids such that the symmetry group acts transitively on oriented flags. A regular tessellation has an invariant we call the cusp modulus. For small cusp modulus, we classify all regular tessellations. For large cusp modulus, we pr…
Semi-Equivelar maps are generalizations of Archimedean Solids (as are equivelar maps of the Platonic solids) to the surfaces other than 2−Sphere. We classify some semi equivelar maps on surface of Euler characteristic -1 and show that none of these are vertex transitive. We establish existence of 12-covered triangula…
A classical result of H. S. M. Coxeter asserts that a certain quotient B(m,n) of the braid group B(m) on m strands is finite if and only if (m,n) corresponds to the type of one of the five Platonic solids. If k is a knot or virtual knot, one can study similar quotients G(k,n) for the correspond…
Abstract perspective on quadratic programming for optimal portfolio allocation.
problem Optimal allocation problems in long portfolio theory.
method Using maximum principles and distinguished boundaries in reproducing kernel Hilbert spaces.
result Support of an optimal distribution lies in a variety intersecting a distinguished boundary.
Revisits Jarrow & Turnbull model for credit and liquidity risk.
problem Modeling credit and liquidity risk in financial markets.
method Uses foreign exchange analogy and partially observable exchange rate.
result Derives tractable term structure models and explicit valuation formulae.
We identify 998 closed hyperbolic 3-manifolds whose volumes are rationally related to Dedekind zeta values, with coprime integers a and b giving a/bvol(M)=(−D)3/2/(2π)2n−4(ζK(2))/(2ζ(2)) for a manifold M whose invariant trace field K has a single complex place, discriminant D, degree n, and Dedekin…
In this paper, we will construct an example of a closed Riemann surface X that can be realized as a quotient of a triply periodic polyhedral surface Π⊂R3 where the Weierstrass points of X coincide with the vertices of Π. First we construct Π by attaching Platonic solids in a periodic manner a…
Study flat metrics from right prisms, finding non-lattice surfaces with translation coverings.
problem Analyzing flat metrics from right regular prisms.
method Viewing prisms as n-differentials and analyzing unfoldings, proving translation coverings to hyperelliptic surfaces.
result Non-lattice surfaces admit translation coverings to hyperelliptic surfaces, allowing explicit computation of orbit closures and counting problems.
Parallelizes DEC on curved meshes using group actions.
problem Efficiently solving DEC operators on curved and 3D meshes.
method Universal block-diagonalization framework for d and ⋆ operators, exploiting group actions. result Block-diagonal structure inherited by operators, enabling parallel solvers.
New methods compare Steklov eigenspaces of free boundary minimal surfaces in balls.
problem Comparing Steklov eigenspaces of free boundary minimal surfaces.
method Developed new methods to compare span of coordinate functions with Steklov eigenspace.
result Proved congruence of free boundary minimal annuli in 3D unit ball.
The paper explains emergent phenomena in deep learning using entropic forces.
problem Understanding the cause of emergent phenomena in deep learning and large language models.
method Proposes a rigorous entropic-force theory for neural networks trained with SGD and variants.
result Shows that representation learning is governed by emergent entropic forces that break continuous symmetries and preserve discrete ones.
Develops new shape metrics for high-dimensional objects.
problem Lack of single metrics to describe shape in high dimensions.
method Introduces hyper-Sphericity and hyper-Shape Proportion metrics.
result Discriminates between different shapes in high dimensions.
This work reveals a new scaling law for optimal design of multirotor aerial vehicles.
problem Designing optimal configurations for fully-actuated multirotor aerial vehicles.
method Formulated on the product manifold of Projective Lines \RP^2^N, minimizing a coordinate-invariant Log-Volume isotropy metric.
result The topology of the global optima is governed by the symmetry of the chassis, leading to a N-5 Scaling Law.
3D RadViz improves 3D data visualization of multidimensional datasets.
problem Tackles the challenge of visualizing multidimensional datasets in 3D.
method Develops RadViz3D, a 3D radial visualization tool with uniform anchor points.
result Improves the display of multidimensional datasets, especially for uncorrelated variables.
With the developments of the last decade on complete constant mean curvature 1 (CMC 1) surfaces in the hyperbolic 3-space H3, many examples of such surfaces are now known. However, most of the known examples have regular ends. (An end is irregular, resp. regular, if the hyperbolic Gauss map of the surface has an ess…
Cyclification of orbifolds explained in cohesive higher topos theory.
problem Understanding cyclification of orbifolds in geometric and algebraic contexts.
method Cohesive higher topos theory and transgression of cohomological charges.
result Cyclification of orbifolds is a fundamental base-change construction.
New class of metric f-manifolds introduced.
problem No specific problem stated; focuses on introducing new class.
method Definition and properties of new class of metric f-manifolds.
result Properties and examples of new class of metric f-manifolds.
Characterizes and describes selfsimilar Hessian manifolds with homothetic vector fields.
problem Understanding the structure and properties of selfsimilar Hessian manifolds.
method Characterization and description of selfsimilar manifolds with homothetic vector fields.
result Any selfsimilar Hessian manifold with a potential homothetic vector field is locally isomorphic to a product of radiant Hessian manifolds.
Extends Tian theorem to Vaisman manifolds for approximations.
problem Approximating Vaisman metrics by immersions/embeddings.
method Study Vaisman metrics on compact manifolds.
result Extend Tian's theorem to Vaisman manifolds.
The paper defines trans-para-Sasakian manifolds and explores their geometric properties.
problem Exploring the geometry of trans-para-Sasakian manifolds.
method Definition and study of curvature properties.
result Conditions for η−Einstein and Einstein manifolds. In 1972, K. Kenmotsu studied a class of almost contact Riemannian manifolds. Later, such a manifold was called a Kenmotsu manifold. This paper, we studied Kenmotsu manifolds with (2n+s)-dimensional s−contact metric manifold and this manifold, we have called generalized Kenmotsu manifolds. Necessary and sufficient c…
Study on a new type of manifolds that generalize almost C-manifolds.
problem Understanding weak nearly C-manifolds and their properties.
method Analyzing conditions for local Riemannian product structures and characterizing specific dimensions.
result Conditions for a weak nearly C-manifold to become locally a Riemannian product and characterization of specific dimensions.
New class of complex manifolds defined, properties studied.
problem Understanding properties of complex manifolds.
method Introduced and studied wHHR manifolds, proved metric equivalence.
result Bergman and Kobayashi metrics are biLipschitz equivalent for wHHR Stein manifolds.
Stabilized convex symplectic manifolds are equivalent to flexible Weinstein manifolds.
problem Understanding the equivalence between stabilized convex symplectic manifolds and flexible Weinstein manifolds.
method Analyzing the homotopy type and symplectic properties of the manifolds.
result Stabilized convex symplectic manifolds are symplectomorphic to flexible Weinstein manifolds.
New criteria for Sasakian manifolds and classification of nearly cosymplectic manifolds.
problem Characterizing Sasakian and cosymplectic manifolds.
method Analyzing nearly Sasakian and nearly cosymplectic manifolds of dimensions greater than five.
result Every nearly Sasakian manifold of dimension greater than five is Sasakian.
Smoothly embed 3-manifolds in 5-manifolds, simplifying topological to smooth.
problem Embedding 3-manifolds smoothly in 5-manifolds.
method Homotopy and small homotopy to achieve smooth embeddings.
result Locally flat embeddings are homotopic to smooth ones.