New proof of Khovanov-Rozansky homology base point independence in finite characteristic.
problem Base point independence of Khovanov-Rozansky homology in characteristic p. method New proof using infinitesimal symmetries of gl(p)-homology. result Proof of base point independence in characteristic p. The paper connects link symmetries to finite subgroups of O(3).
problem Understanding symmetries of flat fully augmented links.
method Developed a dictionary linking graph automorphisms to link symmetries, constructing infinite link classes.
result Symmetry groups of b-prime flat fully augmented links match finite subgroups of O(3).
Analyzes symmetries in neural networks to predict learning dynamics.
problem Understanding the dynamics of neural network parameters during training.
method Unified theoretical framework based on symmetries and conservation laws.
result Symmetries impose geometric constraints on gradients and Hessians, leading to conservation laws.
Researchers create BPS monopoles with any desired symmetry breaking.
problem Creating monopoles with specific symmetry breaking patterns.
method Using a new class of Nahm data to construct finite energy BPS monopoles.
result Arbitrary symmetry breaking monopoles can be constructed.
Method implements symmetries in TQFT for finite groups.
problem Implementing symmetries in TQFT for finite groups.
method Cohomology classes and bordisms with a (d−q−2)-skeleton. result Generalizes existing methods to arbitrary dimensions and q.
Study entropy bounds and finiteness for symmetric self-shrinkers.
problem Entropy and finiteness of symmetric self-shrinkers.
method Comparison geometry, entropy bounds, compactness theorem.
result Only finitely many symmetric self-shrinkers with extra symmetry.
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.
Generalizes pseudo-product structures with abnormal extremals.
problem Finiteness of symmetry algebras for non-degenerate pseudo-product structures.
method Modified universal prolongation of graded nilpotent Lie algebras and generalized finiteness criterion.
result Distributions with singularly transitive properties have finite-dimensional symmetries.
Maximizes symmetry of surfaces embedded in 3D space.
problem Finding surfaces with maximum symmetry in 3D space.
method Analyzing finite group actions on surfaces with given genus.
result Identifies surfaces with maximum symmetry and provides embeddings.
Study finite group actions on aspherical manifolds, proving rigidity and symmetry bounds.
problem Understanding actions of finite groups on aspherical manifolds.
method Analyzing the outer automorphism group and homeomorphism group of the fundamental group.
result Proves the homeomorphism group is Jordan and bounds the discrete degree of symmetry.
New finite symmetries found in 4-manifolds.
problem Understanding symmetries in 4-dimensional spaces.
method Analyzing connected sums and finite group actions.
result Found infinitely many non-equivalent smooth structures and actions.
This paper shows how symmetry in special links affects a specific polynomial.
problem Understanding the symmetry in extended strongly periodic links.
method Proving the relationship between the symmetry of these links and the HOMFLYPT polynomial.
result The first coefficients of the HOMFLYPT polynomial reflect the symmetry of extended strongly periodic links.
Develops tests for conditional symmetry under group actions.
problem Testing conditional symmetry in distributions under group actions.
method Nonparametric randomization tests with kernel methods and asymptotic consistency.
result Tests achieve finite-sample Type I error control and power.
New recursive relations for symmetries of special 2-flags discovered.
problem Local finite classification and continuous numerical modulus of special 2-flags.
method Recursive relations derived for all infinitesimal symmetries of special 2-flags.
result Existence of a continuous numerical modulus of special 2-flags in length seven.
New neural networks learn graph symmetries.
problem Learning from graph data without considering vertex relations.
method Constructs equivariant neural networks to Aut(G) group.
result Characterizes learnable, linear, Aut(G)-equivariant functions.
The paper generalizes TQFTs to fermionic systems and classifies SPTs and SETs.
problem Classifying fermionic SPTs and SETs with finite group symmetries.
method Formulating fermionic TQFTs, gauging SPTs, using bordism groups, and constructing anomalous boundary states.
result Explicit classification of fermionic SPTs and SETs, including new anomalous boundary states.
We examine groups whose resonance varieties, characteristic varieties and Sigma-invariants have a natural arithmetic group symmetry, and we explore implications on various finiteness properties of subgroups. We compute resonance varieties, characteristic varieties and Alexander polynomials of Torelli groups, and we sho…
Study shows most hyperbolic knot complements lack hidden symmetries.
problem Proving most knot complements lack hidden symmetries.
method Using rational functions on varieties associated to a link.
result Most knot complements lack hidden symmetries.
For the class of systems of PDEs, for which infinitesimal translations (with respect to some (in)dependent variables) possess specific finite-dimensional invariant subspaces of the space of generalized symmetries of the system considered. We establish when there exist generalized symmetries from these subspaces, which …
We find a remarkable subalgebra of higher symmetries of the elliptic Euler-Darboux equation. To this aim we map such equation into its hyperbolic analogue already studied by Shemarulin. Taking into consideration how symmetries and recursion operators transform by this complex contact transformation, we explicitly give …
The symmetries of paths in a manifold M are classified with respect to a given pointwise proper action of a Lie group G on M. Here, paths are embeddings of a compact interval into M. There are at least two types of symmetries: Firstly, paths that are parts of an integral curve of a fundamental vector field on $…
The paper bounds the maximum group order for graph embeddings in spheres.
problem Bounding the maximum order of a group acting on a graph embedded in a sphere.
method Proved a polynomial bound on group order based on genus and dimension.
result The degree d/2 is the best possible bound in even dimensions.
New theorems on compactness and finiteness for specific types of self-shrinkers.
problem Characterizing rotationally symmetric self-shrinkers with constraints.
method Compactness and finiteness theorems for self-shrinkers with specific symmetries and constraints.
result Existence of entropy minimizing self-shrinkers diffeomorphic to S1imesSn−1 for each n≥2. In this paper we extend the Tanaka finiteness theorem and inequality for the number of symmetries to arbitrary distributions (differential systems) and provide several applications.
Minimal surfaces with symmetries exist under certain group actions.
problem Existence of conformal minimal immersions with group symmetries.
method Identifying conditions for G-equivariant conformal minimal immersions. result Existence of such immersions under specific group actions.
New framework uses symmetry-based matrices for efficient, flexible NNs.
problem Designing neural networks with relaxed equivariance.
method Symmetry-based structured matrices, Group Matrices (GMs).
result GMs enable competitive performance with fewer parameters.
Defines pre-Kähler structures and their properties.
problem Geometry of Levi degenerate CR hypersurfaces.
method Introduces pre-Kähler structures, holomorphic degeneration, and finite-nondegeneracy.
result Symmetry algebra is finite-dimensional if and only if structure is finitely nondegenerate.
Classifies 2D complex superflows with finite symmetry groups.
problem Classifying 2D complex superflows with finite symmetry groups.
method Analyzes projective flows and rational vector fields.
result Identifies all 2D complex superflows with finite symmetry groups of U(2). We prove that for every closed, connected, orientable, irreducible 3-manifold, there exists an alternating group A_n which is not the topological symmetry group of any graph embedded in the manifold. We also show that for every finite group G, there is an embedding Γ of some graph in a hyperbolic rational homology 3-sp…
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. Study of 4D flows with nilpotent symmetry, showing immortal solutions and blowdown limits.
problem Understanding 4D generalized Ricci flows with nilpotent symmetry.
method Immortal solutions, type III curvature and diameter estimates, new energy monotonicity.
result Blowdown limits lie in a finite-dimensional family of solutions.
Constructs a stable finite-time blowup solution for a specific harmonic map heat flow problem.
problem Energy-supercritical harmonic map heat flow with 1-corotational symmetry in 7 dimensions.
method Constructs a stable finite time blowup solution under corotational symmetry.
result Constructs a stable finite time blowup solution with concentration of the universal profile.
Study verifies Joyce's conjectures for circle-invariant Lagrangian surfaces.
problem Verifying Joyce's conjectures for specific Lagrangian surfaces.
method Continuation of Lagrangian mean curvature flow through finite time neck pinches.
result Flow converges to a chain of special Lagrangians, verifying conjectures.
Study Picard groups of curves with symmetry, focusing on abelian groups and hyperelliptic curves.
problem Understanding the Picard groups of moduli spaces of curves with symmetry.
method Theory of symmetric mapping class groups, finitely generated Picard groups computation.
result Finitely generated Picard groups for moduli spaces of curves with abelian automorphisms.
Using the adjoint action of the infinitesimal translations (with respect to some (in)dependant variables) on specific finite-dimensional subspaces of the space of generalized symmetries of some system of partial differential equations, we explicitly determine the dependance of coefficients of generalized symmetries fro…
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 …
We provide some insights in the study of branching problems of reductive groups, and a method of investigations into symmetry breaking operators. First, we give geometric criteria for finiteness property of linearly independent continuous (respectively, differential) operators that intertwine two induced representation…
New methods reveal symmetries in Chern-Simons theory.
problem Understanding symmetries in Chern-Simons theory.
method Introduced a special basis in the center of the universal enveloping algebra to present group factors in arbitrary representations.
result Computed Vassiliev invariants and proved the tug-the-hook symmetry of the colored HOMFLY polynomial.
Study finds maximal symmetry groups for CR structures with specific properties.
problem Determining the maximal dimension of symmetry groups for CR structures.
method Proved the sharp upper bound for the dimension of symmetry groups for homogeneous, 2-nondegenerate CR manifolds.
result The maximal dimension is n2+7 for n≥3. We construct examples of S1-manifolds with finite second homotopy group and non-vanishing A^-genus. This is related to the classification of positive quaternionic Kaehler manifolds.
The paper proves limitations on hyperbolic knot complements with hidden symmetries.
problem Identifying hyperbolic knot complements with hidden symmetries.
method Obstructions to infinitely many fillings producing knot complements with hidden symmetries.
result The figure-eight knot complement is the only hyperbolic knot complement with hidden symmetries.
We prove that certain dual pairs of Calabi-Yau manifolds have matching orbifold Euler characteristics.
problem Constructing mirror symmetric Calabi-Yau manifolds with specific symmetry groups.
method Generalized Berglund-Hübsch-Henningson construction to include permutations of variables.
result Reduced orbifold Euler characteristics of dual pairs coincide up to sign under cyclic permutation groups satisfying parity condition.
We study "warped Berger" solutions $\big(\mc S^1\times\mc S^3,G(t)\big)$ of Ricci flow: generalized warped products with the metric induced on each fiber {s}×SU(2) a left-invariant Berger metric. We prove that this structure is preserved by the flow, that these solutions develop finite-time neckpinch …
Study of minimal surfaces in 4D with specific ends.
problem Characterize minimal surfaces in R4 with specific ends. method Modification of Costa and Hoffman-Meeks method, generalized Weierstrass representation.
result Minimal surfaces with specific ends are J-holomorphic under certain conditions. New equations use Pin(2) symmetry to study spinor and connection solutions.
problem Study of spinor and connection solutions on 4-manifolds.
method Define and analyze Seiberg-Witten-like equations with Rarita-Schwinger operators.
result Moduli space of solutions is non-compact and non-empty under certain conditions.
Geometrically represents the Jacobian for a mechanical system with symmetry.
problem Path integral reduction for a mechanical system with symmetry.
method Geometric representation using scalar curvature and adapted coordinates.
result Obtained geometric representation of the Jacobian.
Combining M-algebra and hyperbolic involutory algebra extends exceptional tangent spaces to 11 dimensions.
problem Combining symmetries in M-theory to extend exceptional tangent spaces.
method Combining known results to show hyperbolic involutory algebra acts on M-algebra through brane-rotating symmetry.
result Extends the hierarchy of exceptional tangent spaces from n ≤ 7 to n = 11.
We briefly review the hierarchy for the hyper-Kähler equations and define a notion of symmetry for solutions of this hierarchy. A four-dimensional hyper-Kähler metric admits a hidden symmetry if it embeds into a hierarchy with a symmetry. It is shown that a hyper-Kähler metric admits a hidden symmetry if it admits a ce…