Study of intrinsic symmetry groups of links, finding counterexamples.
problem Whether every subgroup of the symmetric group is an intrinsic symmetry group of some link.
method Defined intrinsic symmetry groups and provided counterexamples.
result For n > 5, there does not exist an n-component link L for which S(L) is the alternating group.
This paper identifies all topological symmetry groups for Heawood family graphs.
problem Understanding symmetries of spatial graphs in 3D space.
method Analyzing automorphisms of graphs embedded in S3. result All graphs in the Heawood family are intrinsically chiral.
We consider the "intrinsic" symmetry group of a two-component link L, defined to be the image Σ(L) of the natural homomorphism from the standard symmetry group $\MCG(S^3,L)$ to the product $\MCG(S^3) \cross \MCG(L)$. This group, first defined by Whitten in 1969, records directly whether L is isotopic to a link $L…
We present an elementary derivation of the "intrinsic" symmetry groups for knots and links of 8 or fewer crossings. The standard symmetry group for a link is the mapping class group $\MCG(S^3,L)$ or $\Sym(L)$ of the pair (S3,L). Elements in this symmetry group can (and often do) fix the link and act nontrivially onl…
Bispectral OT improves dataset comparison by preserving intrinsic coherence.
problem Ignoring intrinsic coherence in dataset comparisons using pairwise geometric distances.
method Introduces Bispectral Optimal Transport, a symmetry-aware extension of discrete OT.
result Transport plans computed with Bispectral OT achieve greater class preservation accuracy.
Leveraging the intrinsic symmetries in data for clear and efficient analysis is an important theme in signal processing and other data-driven sciences. A basic example of this is the ubiquity of the discrete Fourier transform which arises from translational symmetry (i.e. time-delay/phase-shift). Particularly important…
We develop a novel Gaussian process method for manifold data.
problem Challenges in Gaussian processes on manifold-based predictors, especially in high dimensions.
method Intrinsic approach for constructing Gaussian processes on general manifolds, using the exponential map for heat kernel estimation.
result Remarkable efficiency gains and applicability to high-dimensional manifolds.
New approach to Lagrangian systems using intrinsic geometry.
problem Developing a new framework for Lagrangian systems.
method Direct reformulation of Hamiltonian formalism, introduction of spatial equation and spatial-gauge symmetry.
result Covariant and non-covariant canonical variational principles demonstrated for Maxwell equations.
Study infinitesimal CR symmetries of accidental CR structures.
problem Accidental CR structures and their infinitesimal CR automorphisms.
method Complete explicit lists of infinitesimal CR automorphisms for specific CR models.
result Explicit generators of infinitesimal CR symmetries in extrinsic holomorphic coordinates.
Proves symmetries of extremal horizons in spacetimes.
problem Proving symmetries of extremal horizons in arbitrary dimensions.
method Analyzes Killing vector fields and near-horizon geometry.
result Enhanced isometry groups and shifted Aretakis instability.
We find the first examples of triply periodic minimal surfaces of which the intrinsic symmetries are all of horizontal type.
Structure-preserving GANs learn distributions with group symmetry efficiently.
problem Learning distributions with group symmetry efficiently.
method Developed structure-preserving GANs by reducing the discriminator space and designing structured generators.
result Significantly improved sample fidelity and diversity in small data regimes.
Reduced sample complexity for group-invariant distributions.
problem Improving sample complexity for estimating divergences of group-invariant distributions.
method Quantified reduction in sample complexity for Wasserstein-1 metric and Lipschitz-regularized α-divergences under finite and infinite groups.
result Sample complexity reduction proportional to group size for finite groups, and convergence rate depends on intrinsic dimension for infinite groups.
Study on Lawson surfaces' first Laplace eigenvalue using symmetry and algebraic methods.
problem Yau's conjecture on first eigenvalue of minimal hypersurfaces in the sphere.
method Symmetry-based approach exploiting discrete reflection symmetries and algebraic structure of reflection groups.
result Equality λ1(ξ_{m,k})=2 for Lawson surfaces with m and k even.
Researchers compute differential invariants for Carrollian spacetimes.
problem Understanding the geometry and symmetries of Carrollian spacetimes.
method Derived from the geometry of the screen bundle, computed differential invariants using jet-spaces and Spencer cohomology.
result Specified how to generate the entire algebra of differential invariants for generic Carrollian structures, focusing on dimension 3.
Characterizes density-valued symplectic forms on multisymplectic manifolds.
problem Understanding density-valued symplectic forms on multisymplectic manifolds.
method Intrinsic characterization and Darboux-type theorems.
result Proves Darboux-type theorems for density-valued symplectic forms.
Two new minor minimal intrinsically chiral graphs identified.
problem Identifying intrinsically chiral graphs in molecular structures.
method Analyzing graph symmetry and embedding properties.
result Found two new minor minimal intrinsically chiral graphs Γ7 and Γ8. Theory of symmetric rigidity in hyperbolic geometry.
problem Symmetric rigidity in hyperbolic frameworks.
method Gain graphs and orbit rigidity matrix, matroidal sparsity conditions.
result Characterization of infinitesimal rigidity for Gamma-symmetric frameworks.
Symmetry in inverse problems leads to multiple solutions, but breaking symmetry helps deep learning.
problem Symmetry in physical systems causes multiple solutions in inverse problems, hindering deep learning.
method Careful symmetry breaking on training data helps solve inverse problems and improve deep learning performance.
result Symmetry breaking on training data significantly improves deep learning performance in inverse problems.
Study uniformly differentiable graphs in Carnot groups, proving area formulas.
problem Characterize uniformly differentiable intrinsic graphs in Carnot groups.
method Characterize uniform intrinsic differentiability via Hölder properties of projections of vector fields.
result Explicit area formula for uniformly intrinsically differentiable maps in Carnot groups.
Weyl's tube formula holds for various cross-sections under symmetry conditions.
problem Can the volume of tubes around submanifolds be calculated for non-round cross-sections?
method Investigated the volume of tubes with general cross-sections D under symmetry conditions.
result The volume of tubes around submanifolds can be calculated for general cross-sections under symmetry conditions.
This paper reinterprets Khovanov-Sano symmetries using BV formalism.
problem Understanding symmetries in equivariant Khovanov homology.
method Identifying Shumakovitch operator as a BV Laplacian and proving L∞-algebra structure. result Construction of an intrinsic L∞-algebra on the Khovanov-Sano complex. Almost perimeter-minimizing boundaries in plentiful groups can be approximated by Lipschitz graphs.
problem Regularity of boundaries in plentiful groups.
method Lipschitz approximation of boundaries.
result Boundary of almost minimizers can be approximated by intrinsic Lipschitz graphs.
We use the notion of intrinsic flat distance to address the almost rigidity of the positive mass theorem for asymptotically hyperbolic manifolds. In particular, we prove that a sequence of spherically symmetric asymptotically hyperbolic manifolds satisfying the conditions of the positive mass theorem converges to hyper…
We focus our attention on the notion of intrinsic Lipschitz graphs, inside a special class of metric spaces i.e. the Carnot groups. More precisely, we provide a characterization of locally intrinsic Lipschitz functions in Carnot groups of step 2 in terms of their intrinsic distributional gradients.
A Carnot group G is a connected, simply connected, nilpotent Lie group with stratified Lie algebra. Intrinsic regular surfaces in Carnot groups play the same role as C^1 surfaces in Euclidean spaces. As in Euclidean spaces, intrinsic regular surfaces can be locally defined in different ways: e.g. as non critical level …
The paper develops an asymptotic theory of self-supervised pre-training.
problem Sharpness of current rates in self-supervised pre-training and their accuracy.
method Two-stage M-estimation and tools from Riemannian geometry.
result Characterization of the limiting distribution of the downstream test risk.
Study shows stable graphs in Heisenberg group are essentially planes.
problem Characterizing stable graphs in the Heisenberg group.
method Analyzes Sobolev intrinsic graphs in the Heisenberg group with sub-Riemannian area stability.
result Stable graphs are cosets of two-dimensional subgroups.
Data analysis and data mining are concerned with unsupervised pattern finding and structure determination in data sets. "Structure" can be understood as symmetry and a range of symmetries are expressed by hierarchy. Such symmetries directly point to invariants, that pinpoint intrinsic properties of the data and of the …
Data symmetries in neural networks can generate conserved quantities.
problem Conservation laws in neural networks
method Using tensorizable networks
result Data augmentation can induce conserved quantities
Study maps in semidirect products of groups, proving Lipschitz properties without intrinsic dilations.
problem Proving Lipschitz conditions in semidirect products of groups without intrinsic dilations.
method Using equivalent conditions and properties of projection maps in metric spaces.
result Proves the same Lipschitz results as in Carnot groups, without intrinsic dilations.
Maps in Carnot groups are equivalent to solutions of a PDE system.
problem Understanding maps in Carnot groups of step 2.
method Equivalence between intrinsic Lipschitz maps and solutions to a PDE system.
result Intrinsic Lipschitz maps are equivalent to weak solutions of a PDE system.
The paper studies maps in the Heisenberg group and their images, called Rickman rugs.
problem Understanding maps and their images in the Heisenberg group.
method Analyzes maps f:WoH, where H is the first Heisenberg group and W is a vertical subgroup. result Rickman rugs in the Heisenberg group admit a corona decomposition by intrinsic bilipschitz graphs.
This paper classifies topological symmetry groups for Petersen family graphs.
problem Understanding symmetries of graphs embedded in 3D space.
method Examined all embeddings of Petersen family graphs in S3 and classified their topological symmetry groups. result Identified all possible groups that can be realized as topological symmetry groups for each graph in the Petersen family.
Study of symmetries in 2D Yang-Mills theory, including orbifolds and higher forms.
problem Understanding symmetries and anomalies in 2D Yang-Mills theory.
method Combining continuum methods, topological defects, and higher gauge theory.
result Unified description of higher and lower form gauge fields, identifying spontaneous symmetry breaking.
This paper mainly contributes to a classification of statistical Einstein manifolds, namely statistical manifolds at the same time are Einstein manifolds. A statistical manifold is a Riemannian manifold, each of whose points is a probability distribution. With the Fisher information metric as a Riemannian metric, infor…
New triangulations of quaternionic projective plane found with various symmetry groups.
problem Classifying triangulations of quaternionic projective plane with 15 vertices.
method Constructing and classifying 15-vertex triangulations with various symmetry groups.
result Exactly 75 triangulations of quaternionic projective plane with 15 vertices and symmetry group of order at least 4.
Criterion for surfaces in Heisenberg group to be graphs using flat cones.
problem Characterizing surfaces in Heisenberg group as graphs.
method Using planar cones to define intrinsic rectifiability.
result Criterion for topological surfaces to be intrinsic Lipschitz graphs.
Lie symmetry group method is applied to study the Born-Infeld equation. The symmetry group and its optimal system are given, and group invariant solutions associated to the symmetries are obtained. Finally the structure of the Lie algebra symmetries is determined.
We study the strong maximum principle for horizontal (p-) mean curvature operator and p-(sub)laplacian operator on subriemannian manifolds including, in particular, Heisenberg groups and Heisenberg cylinders. Under a certain Hormander type condition on vector fields, we show the strong maximum principle holds in higher…
Investigates intrinsic Lipschitz sections in nonlinear quotient maps.
problem Analyzing intrinsic Lipschitz sections in non-linear quotient maps.
method Introduced Leibniz formula for intrinsic slope under weaker conditions, used properties of intrinsic dilations in Carnot groups, and provided conditions for sum of sections.
result Found conditions for sum of intrinsically Lipschitz sections in Carnot groups of step 2.
Explains algebraic tools for understanding invariant differential operators in curved geometries.
problem Understanding invariant differential operators in curved geometries.
method Focuses on the direct link between jets of sections and induced modules.
result Extends concepts and procedures to curved cases.
We present the concept of the topological symmetry group as a way to analyze the symmetries of non-rigid molecules. Then we characterize all of the groups which can occur as the topological symmetry group of an embedding of the complete graph K_{4r+3} in S^3.
We characterize all groups which can occur as the topological symmetry group or the orientation preserving topological symmetry group of some embedding of the Petersen graph in S^3.
We study the stability of the Positive Mass Theorem using the Intrinsic Flat Distance. In particular we consider the class of complete asymptotically flat rotationally symmetric Riemannian manifolds with nonnegative scalar curvature and no interior closed minimal surfaces whose boundaries are either outermost minimal h…
A new method for group invariant machine learning using geometric projections.
problem Supervised group invariant and equivariant machine learning.
method Geometric topology approach involving projection of input data into a geometric space parametrizing symmetry group orbits.
result Improvement in accuracy compared to existing methods.
We study continuous groups of generalized Kerr-Schild transformations and the vector fields that generate them in any n-dimensional manifold with a Lorentzian metric. We prove that all these vector fields can be intrinsically characterized and that they constitute a Lie algebra if the null deformation direction is fixe…
Method improves deep learning models for datasets with mixed approximate symmetries.
problem Improving deep learning models for datasets with mixed approximate symmetries.
method Regularizer-based approach to build models for datasets with mixed approximate symmetries.
result Our method achieves better accuracy than prior approaches while discovering the approximate symmetry levels correctly.