We classify hyperbolic monopoles with continuous symmetries and construct new examples.
problem Classifying and constructing hyperbolic monopoles with continuous symmetries.
method Developed a Structure Theorem and used representation theory to simplify the problem.
result Found constraints on structure groups and constructed novel spherically symmetric Sp(n) hyperbolic monopoles. Classifies flat projective structures with specific symmetries.
problem Classifying local projective structures with non-trivial Lie symmetries.
method Analyzes flat projective structures with positive-dimensional Lie algebra of projective vector fields.
result Obtained a classification of flat projective structures.
Classifies maximal symmetry models of CR dimension 1.
problem Classifying maximal symmetry models of CR manifolds.
method Classification based on Bloom-Graham and Tanaka types, coordinate realization, and extension principle.
result Classification and coordinate realization of maximal symmetry models.
The paper classifies and proves properties of symmetry breaking operators for specific groups.
problem Classifying and understanding symmetry breaking operators for de Sitter and Lorentz groups.
method Constructing and classifying differential symmetry breaking operators, proving localness, and showing sporadic nature.
result All symmetry breaking operators are differential and sporadic, not obtainable by residue formulas.
Paper classifies symmetries of cross caps using invariants.
problem Classifying symmetries of cross caps.
method Used Bruce-West's normal form and associated functions to create invariants.
result Classified possible symmetries on cross caps.
New symmetry dimensions for higher order ODEs are identified.
problem Determining the maximal and submaximal symmetry dimensions for higher order ODEs.
method Cartan-geometric approach to classify symmetry dimensions.
result Next largest realizable symmetry dimensions for scalar ODEs of order ≥ 4 and vector ODEs of order ≥ 3 are determined.
The paper classifies symmetries and determines exterior types of knotted handlebodies.
problem Classifying symmetries and determining exterior types of knotted handlebodies.
method Analysis of hyperbolic knots with non-integral toroidal Dehn surgeries.
result Determines symmetries and exterior types of knotted genus two handlebodies.
Study of continuous symmetries in Nahm data and BPS monopoles.
problem Solutions to Nahm's equations with continuous symmetries.
method Classification of Ansätze and construction of Nahm data.
result Construction of new BPS monopoles with spherical symmetry.
We classify all groups which can occur as the topological symmetry group of some embedding of the Heawood graph in S3.
We classify closed, simply-connected, non-negatively curved 6-manifolds of almost maximal symmetry rank up to equivariant diffeomorphism.
We classify all groups which can occur as the orientation preserving topological symmetry group of some embedding of a Möbius ladder graph in S3.
Classifies Lie algebras and related spacetimes for a specific type of symmetry.
problem Classifying Lie algebras and related spacetimes for a specific type of symmetry.
method Classification of Lie algebras, homogeneous spacetimes, and coadjoint orbits.
result Classification of three types of Lifshitz spacetimes.
Classifies charge-3 monopoles with symmetry, identifying new spectral curves.
problem Classifying charge-3 monopoles with symmetry.
method Constructing Nahm data and identifying spectral curves with elliptic quotients.
result New monopole spectral curves with D6 and V4 symmetry identified. Classifies Lie symmetry algebras for 2D quasilinear equations, linking symmetry to linearity.
problem Classifying Lie symmetry algebras for 2D quasilinear equations.
method Classification based on abelian Lie symmetry algebras of dimension and rank.
result Equations with specific symmetry algebras are linearizable.
This paper classifies symmetries of biharmonic heat equations on surfaces of revolution.
problem Investigating symmetries of biharmonic heat equations on surfaces of revolution.
method Lie symmetry analysis to classify symmetries and derive invariant solutions.
result The biharmonic heat equation on a surface of revolution has the same Lie symmetries as the harmonic heat equation.
Classifies symmetries of knots using group actions and orthogonal representation theory.
problem Classifying symmetries of knots in 3-sphere.
method Using geometrization and orthogonal representation theory, constructing examples, and distinguishing symmetries.
result Cyclic and dihedral families of symmetries of prime knots and composite knots.
Neural networks exhibit simplex symmetry in their final and penultimate layers.
problem Understanding the symmetry in neural network layers.
method Analytical and numerical studies of toy models and deep neural networks.
result Neural networks map data points from the same class to a single point in a high-dimensional space, forming a simplex.
Since the foundational work of Chenciner and Montgomery in 2000 there has been a great deal of interest in choreographic solutions of the n-body problem: periodic motions where the n bodies all follow one another at regular intervals along a closed path. The principal approach combines variational methods with symmetry…
The paper classifies expanding gradient Yamabe solitons based on scalar curvature.
problem Classifying expanding gradient Yamabe solitons based on scalar curvature.
method Rigorous analysis of scalar curvature in both cases: greater than and less than the soliton constant.
result Complete classification of nontrivial complete expanding gradient Yamabe solitons.
Classifies Einstein metrics on 4-manifolds with specific symmetry groups.
problem Classifying Einstein metrics on 4-manifolds with certain symmetry properties.
method Analyzes cohomogeneity-one Einstein metrics and uses symmetry properties.
result Locally symmetric or homothetic to the Page metric on CP2♯CP2. The paper classifies and constructs differential symmetry breaking operators from a line bundle to a vector bundle over real projective spaces.
problem Classifying and constructing differential symmetry breaking operators.
method Utilizing factorization identities and branching laws of generalized Verma modules.
result Differential symmetry breaking operators from a line bundle to a vector bundle over real projective spaces are classified and constructed.
We classify nonnegatively curved simply connected 4-manifolds with circle symmetry up to equivariant diffeomorphisms. The main problem is rule out knotted curves in the singular set of the orbit space. As an extension of this work we classify all knots in S^3 which can be realized as an extremal set with respect to an …
In this paper the generic bifurcations of the Minkowski symmetry set for 1-parameter families of plane curves are classified and the necessary and sufficient geometric criteria for each type are given. The Minkowski symmetry set is an analogue of the standard Euclidean symmetry set, and is defined to be the locus of ce…
Classifies Toda-type tt*-structures and their fixed points.
problem Classifying Toda-type tt*-structures and their fixed points.
method Fixed point description and reduction of anti-symmetry conditions.
result Reduces possibilities of anti-symmetry condition to two cases.
Classifies non-integrable distributions with simple infinite-dimensional Lie superalgebras of symmetries.
problem Classifying non-integrable distributions with specific Lie superalgebras.
method Classification based on locality assumptions and W-grading.
result 15 series and 7 exceptional Lie superalgebras identified over C, and analogs over K of characteristic p>0. Study 2D viscoelastic equations using Lie group theory.
problem Symmetry classification and reduction of 2D viscoelastic equations.
method Investigation through Lie group theory, including algebra of symmetries and optimal subalgebras.
result Classification of reductions of similarities related to Lie subalgebras.
Classifies Einstein metrics on R^4 with Heisenberg symmetry, finding incomplete Ricci-flat metrics and two complete negative-curvature examples.
problem Classifying Einstein metrics on R4 with Heisenberg symmetry. method Invariant under a four-dimensional group of isometries including the Heisenberg group, analyzing Ricci-flat and negative-curvature metrics.
result Found two complete negative-curvature examples: complex hyperbolic metric and one-loop deformed universal hypermultiplet.
Classifies symmetries of non-flat 3-webs around a point.
problem Understanding symmetries of non-flat 3-webs.
method Classification and construction methods for symmetries.
result Classification of symmetries for non-flat 3-webs.
We find flat band Hamiltonians and Ginsparg-Wilson relations for symmetry classes.
problem Finding flat band Hamiltonians and Ginsparg-Wilson relations for symmetry classes.
method Integrating out the additional bulk direction to obtain effective Dirac operators and then deriving flat and overlap Dirac operators.
result Established Ginsparg-Wilson relations and mod-two index theorems for each symmetry class.
Classifies knots that bound equivariant surfaces with free symmetries.
problem Classifying knots that bound equivariant surfaces with free symmetries.
method Homology cobordism classification of lens spaces using d-invariants.
result Numerical condition determining free periods for torus knots.
We develop a general structure theory for compact homogeneous Riemannian manifolds in relation to the co-index of symmetry. We will then use these results to classify irreducible, simply connected, compact homogeneous Riemannian manifolds whose co-index of symmetry is less or equal than three. We will also construct ma…
We classify smooth del Pezzo surfaces whose alpha-invariant of Tian is bigger than one.
A BV algebra is a formal framework within which the BV quantization algorithm is implemented. In addition to the gauge symmetry, encoded in the BV master equation, the master action often exhibits further global symmetries, which may be in turn gauged. We show how to carry this out in a BV algebraic set up. Depending o…
The paper studies cylindrical handlebody-knots of genus two with unique unknotting annuli and finds trivial symmetry groups.
problem Understanding the topology and symmetry of cylindrical handlebody-knots of genus two.
method Analysis of Thurston's hyperbolization theorem and investigation of unknotting annuli.
result The symmetry group is trivial if the unknotting annulus is unique and of type 2. We study integrable non-degenerate Monge-Ampere equations of Hirota type in 4D and demonstrate that their symmetry algebras have a distinguished graded structure, uniquely determining the equations. This is used to deform these heavenly type equations into new integrable PDE of the second order with large symmetry pseu…
Symmetry, a central concept in understanding the laws of nature, has been used for centuries in physics, mathematics, and chemistry, to help make mathematical models tractable. Yet, despite its power, symmetry has not been used extensively in machine learning, until rather recently. In this article we show a general wa…
The symmetries of complex molecular structures can be modeled by the {\em topological symmetry group} of the underlying embedded graph. It is therefore important to understand which topological symmetry groups can be realized by particular abstract graphs. This question has been answered for complete graphs; it is natu…
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.
Classifies scalar second-order PDEs with low-dimensional symmetry groups.
problem Classifying differential equations with specific symmetry groups.
method Algebraic technique based on covariant form for constructing equations.
result Complete classification of quasi-linear scalar second-order PDEs with free symmetry groups of dimension ≤3.
Summarizes quantum field theories with discrete symmetry, classifying representations and anomalies.
problem Classifying representations and anomalies in quantum field theories with discrete symmetry.
method Classification of representations and anomalies using the ring of profinite integers.
result Rich and complex classification of representations and anomalies.
Study of deformations of Virasoro symmetries using variational bihamiltonian cohomology.
problem Deformations of Virasoro symmetries of principal hierarchies.
method Variational bihamiltonian cohomology.
result Classification of conformal bihamiltonian structures.
Generically an almost complex structure has no symmetries at all, but there exist symmetric structures. In this paper we describe how to guarantee that the pseudogroup of local symmetries is small (finite-dimensional). It will be indicated that a large symmetry pseudogroup (infinite-dimensional) is a signature of some …
Study classifies Kähler-Einstein metrics with rotational symmetries.
problem Classify Kähler-Einstein metrics with rotational symmetries.
method Focus on integrable structures to classify metrics.
result Classify Kähler-Einstein metrics with rotational symmetries.
Classifies patterns of symmetry breaking and vacuum degeneracy in scalar and gauge fields.
problem Understanding patterns of symmetry breaking and vacuum degeneracy in complex field systems.
method Uses mathematical classification of singular foliations to encode and classify patterns of spontaneous symmetry breaking and vacuum degeneracy.
result Mathematical classification provides a qualitative understanding of possible patterns of vacuum degeneracy.
We classify closed, simply connected n-manifolds of non-negative sectional curvature admitting an isometric torus action of maximal symmetry rank in dimensions 2≤n≤6. In dimensions 3k, k=1,2 there is only one such manifold and it is diffeomorphic to the product of k copies of the 3-sphere.
The paper classifies quantizable functions and explores symmetry in quantization methods.
problem Classifying quantizable functions and understanding symmetry in quantization methods.
method Deformation quantization and geometric quantization methods are compared and classified.
result Formal quantizable functions are of a specific form and relate to Hamiltonian Killing vector fields.
Classifies curvature homogeneous hypersurfaces in S^4 and H^4.
problem Classifying curvature homogeneous hypersurfaces in different space forms.
method Analyzing hypersurfaces in S^4 and H^4, identifying FKM examples and Tsukada's isolate example.
result Identifies an isolated hypersurface with a circle of symmetries and a one-parameter family with no continuous symmetries.
We classify all (locally) homogeneous Levi non-degenerate real hypersurfaces in C3 with symmetry algebra of dimension ≥6.