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.
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).
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 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.
We show that if a cusped hyperbolic manifold is Platonic, i.e., can be decomposed into isometric Platonic solids, it can also be decomposed into geodesic ideal tetrahedra.
We call a 3-manifold Platonic if it can be decomposed into isometric Platonic solids. Generalizing an earlier publication by the author and others where this was done in case of the hyperbolic ideal tetrahedron, we give a census of hyperbolic Platonic manifolds and all of their Platonic tessellations. For the octahedra…
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.
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.
We construct new embedded self-shrinkers of genus 3, 5, 7, 11 and 19 using variational methods. Our self-shrinkers resemble doublings of the Platonic solids and were discovered numerically by D. Chopp in 1994.
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 vertex-transitive map X is a map on a surface on which the automorphism group of X acts transitively on the set of vertices of X. If the face-cycles at all the vertices in a map are of same type then the map is called a semi-equivelar map. Clearly, a vertex-transitive map is semi-equivelar. Converse of this is …
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…
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…
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…
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.
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…
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.
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…
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.
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.
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…
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.
A single qubit may be represented on the Bloch sphere or similarly on the 3-sphere S3. Our goal is to dress this correspondence by converting the language of universal quantum computing (UQC) to that of 3-manifolds. A magic state and the Pauli group acting on it define a model of UQC as a positive operator-value…
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…
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.
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.
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.
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…
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.
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.
Study of cuspidal edges on focal surfaces of regular surfaces.
problem Clarifying the sign of singular curvature at cuspidal edges.
method Investigation using singularities of parallel surfaces.
result Clarification of the sign of singular curvature at cuspidal edges.
New method glues Scherk surfaces into minimal surfaces, limiting possible outcomes.
problem Limiting the outcomes of gluing Scherk surfaces into minimal surfaces.
method Constructing minimal surfaces by stacking and gluing doubly periodic Scherk surfaces.
result Except for special cases, gluing more Scherk surfaces results in known minimal surfaces.
Study on focal surfaces of lightcone framed surfaces in Lorentz-Minkowski 3-space.
problem Investigate differential geometry properties of focal surfaces of lightcone framed surfaces.
method Introduced lightcone frame to define lightcone framed surfaces, then investigated their differential geometry properties.
result Investigated differential geometry properties of focal surfaces of lightcone framed surfaces.
Study on focal surfaces of tubular surfaces in 3D space, focusing on their flatness and asymptotic properties.
problem Characterizing and understanding focal surfaces of tubular surfaces in 3D space.
method Defined tubular surfaces using Frenet and Darboux frames, analyzed their focal surfaces, and derived conditions for flatness.
result No minimal focal surface exists in 3D space for tubular surfaces.
Generalizes ribbonness result for surface-links.
problem Characterizing ribbon surface-links.
method Analyzes handle-irreducible summands and uses equivalences.
result Every stable-ribbon surface-link is a ribbon surface-link.
Introduces hyperbolic generalized framed surfaces and their properties.
problem None explicitly stated; focuses on introducing new geometric objects.
method Generalization of hyperbolic framed surfaces and curves.
result Established conditions for a surface to be a hyperbolic generalized framed base surface and explored their singularities.
The paper studies special surfaces with a new type of support function.
problem Characterizing surfaces with a specific quadratic support function.
method Developed a Weierstrass type representation involving holomorphic functions.
result Classified surfaces of rotation with this new type of support function.
Minimal Legendrian surfaces found in 5D sphere.
problem Characterizing Willmore Legendrian surfaces in S5. method Analyzing properties of Willmore and csL Willmore surfaces.
result Complete Willmore Legendrian surfaces in S5 are minimal. The paper studies knitted surfaces and surface-links, showing their isotopy and closure properties.
problem Understanding the isotopy and closure properties of knitted surfaces and surface-links.
method Analyzing the structure and closure of knitted surfaces and surface-links in R4. result Any surface-link is ambient isotopic to the closure of a 2-dimensional knit.
Classifies surfaces with constant Gaussian curvature in Euclidean 3-space.
problem Classifying surfaces with constant Gaussian curvature in Euclidean 3-space.
method Analyzing surfaces as implicit equations and proving properties based on Gaussian curvature.
result Surfaces with constant Gaussian curvature are either surfaces of revolution, cylindrical surfaces, conical surfaces, or have specific forms.