Construct minimal Lagrangian surfaces in complex projective plane via loop group method.
problem Construct minimal Lagrangian immersions from arbitrary Riemann surfaces into complex projective plane.
method Loop group method, perturbed equivariant minimal Lagrangian surfaces, Delaunay cylinders approximation.
result Construct a class of minimal Lagrangian cylinders approximating Delaunay cylinders.
We present a version of the equivariant gradient degree defined for equivariant gradient perturbations of an equivariant unbounded self-adjoint operator with purely discrete spectrum in Hilbert space. Two possible applications are discussed.
We use bifurcation theory to determine the existence of infinitely many new examples of triply periodic minimal surfaces in R3. These new examples form branches issuing from the H-family, the rPD-family, the tP-family, and the tD-family, that converge to some degenerate embedding of the families. As to nonde…
New invariants for 3-manifolds derived from equivariant Cerf theory.
problem Existence of perturbative SU(n) Casson invariants on integer homology spheres. method Equivariant Cerf theory for Morse functions, adapted to infinite-dimensional setting.
result Existence and explicit formula for SU(4) Casson invariants. New robustness measure accounts for task-specific symmetries.
problem Traditional robustness measures fail for tasks with inherent symmetries.
method Sound notion of adversarial robustness for equivariant tasks, using randomized smoothing and graph edit distance certificates.
result Provable robustness can be achieved for various tasks with inherent symmetries.
New framework for conformal equivariant cycles in KK-theory.
problem Tackles conformal equivariance in unbounded KK-theory.
method Extends unbounded Kasparov theory with novel perturbation theory.
result Defines new unbounded representatives of Kasparov classes.
Study stabilizes Morse-Bott cohomology for equivariant manifolds.
problem Equivariant cohomology of manifolds with group actions.
method Stabilization technique to construct Morse-Bott functions.
result Realization of equivariant transversality and orientability.
Analyzes string topology operations using Chen's integrals and homotopy transfer.
problem Relating string topology to perturbative Chern-Simons theory.
method Develops integrals over configuration spaces and applies homotopy transfer.
result Intertwines involutive Lie bialgebra structures on homology.
Bound on equivariant index for min-max surfaces.
problem Bounding the index of equivariant min-max surfaces.
method Equivariant min-max procedure with group action.
result Equivariant index bound by number of parameters.
Study on embedding surfaces into 3-manifolds, focusing on equivariant cases.
problem Embedding hyperbolic surfaces into hyperbolic 3-manifolds with specific symmetries.
method Examined orientation-preserving and orientation-reversing actions on surfaces, including nonorientable ones.
result Found conditions for equivariant embeddings of hyperbolic surfaces into hyperbolic 3-manifolds.
The study constructs equivariant harmonic maps into symmetric spaces with applications to Willmore surfaces.
problem Constructing harmonic maps into symmetric spaces.
method Equivariant primitive harmonic maps construction.
result Examples of S1-equivariant Willmore Moebius strips in S3. Study automorphism equivariant Hitchin index for Riemann surfaces.
problem Define and study an index for Riemann surfaces under automorphisms.
method Define automorphism equivariant Hitchin index and prove a formula in terms of cohomological pairings.
result Prove a formula for the automorphism equivariant Hitchin index.
We apply Lescop's construction of Z-equivariant perturbative invariant of knots and 3-manifolds to the explicit equivariant propagator of "AL-paths" given in arXiv:1403.8030. We obtain an invariant Z^n of certain equivalence classes of fiberwise Morse functions on a 3-manifold fibered over S1, whi…
The purpose of this paper is to study finite dimensional equivariant moduli problems from the viewpoint of stratification theory. We show that there exists a stratified obstruction system for a finite dimensional equivariant moduli problem. In addition, we define a coindex for a G-vector bundle which is determined by t…
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.
Getzler-Jones-Petrack introduced A∞ structures on the equivariant complex for manifold M with smooth S1 action, motivated by geometry of loop spaces. Applying Witten's deformation by Morse functions followed by homological perturbation we obtained a new set of A∞ structures. We extend and …
Classifies Cp-surfaces using equivariant surgery methods.
problem Classifying closed, connected 2-manifolds with cyclic group actions.
method Equivariant surgery methods
result Provides representatives of each isomorphism class using surgery operations.
Nonorientable spanning surfaces of periodic knots can have arbitrarily high first Betti number.
problem Periodic knots do not always have nonorientable spanning surfaces of high genus.
method Examples and calculations of nonorientable spanning surfaces of periodic knots.
result The first Betti number of nonorientable spanning surfaces can be arbitrarily large.
In this paper we continue our study of equivariant minimal Lagrangian surfaces in CP2, characterizing the rotationally equivariant cases and providing explicit formulae for relevant geometric quantities of translationally equivariant minimal Lagrangian surfaces in terms of Weierstrass elliptic functions.
In this paper, for a compact Lie group action,we prove the anomaly formula and the functoriality of the equivariant Bismut-Cheeger eta forms with perturbation operators when the equivariant family index vanishes. In order to prove them, we extend the Melrose-Piazza spectral section and its main properties to the equiva…
New minimal surfaces found in ball with boundary constraints.
problem Finding minimal surfaces with boundary conditions.
method Equivariant differential geometry approach.
result A family of free boundary minimal surfaces in the unit ball.
The study examines how equivariance in networks affects generalization error using PAC-Bayesian bounds.
problem Understanding how equivariance in networks impacts generalization error.
method Utilized PAC-Bayesian analysis for equivariant networks, deriving norm-based bounds for generalization error.
result The bound indicates that using larger group size in the model improves generalization error.
This paper explores the trade-off between spatial and adversarial robustness in neural networks.
problem Understanding the trade-off between spatial and adversarial robustness in neural networks.
method Quantitative analysis and empirical testing with curriculum learning.
result Spatial robustness and adversarial robustness are quantitatively related and can be improved simultaneously.
Paper shows mapping class group-equivariant Teichmüller space deformation to Thurston spine.
problem Mapping Teichmüller space to Thurston spine.
method Equivariant deformation retraction of Teichmüller space onto a cell complex.
result Thurston spine contains points corresponding to hyperbolic surfaces with shortest geodesics forming polygons.
GCNNs gain rotation invariance with more training augmentation, making SVD-Universal more effective.
problem Improving robustness of GCNNs to adversarial attacks.
method SVD-Universal technique applied to GCNNs trained with larger rotations.
result SVD-Universal becomes more effective as GCNNs gain rotation invariance.
Study connections on complex Riemann surfaces for Lie algebroid structures.
problem Investigating connections on holomorphic Lie algebroid structures on Riemann surfaces.
method Analyzing equivariant holomorphic Lie algebroid connections on holomorphic principal bundles over compact Riemann surfaces.
result Every holomorphic principal G-bundle admits an equivariant holomorphic Lie algebroid connection under certain conditions.
In this note, we present a new look at translationally equivariant minimal Lagrangian surfaces in the complex projective plane via the loop group method.
New invariant distinguishes non-orientable surfaces.
problem Distinguishing non-orientable surfaces bounded by the same knot.
method Mixed invariant from Lee and Bar-Natan deformations of Khovanov homology.
result Distinguishes exotic non-orientable surfaces.
The paper defines new knot genera and finds bounds for stabilization distances.
problem Finding bounds for stabilization distances of symmetric surfaces.
method Defining new knot genera and using them to find bounds.
result Constructs unknotted symmetric 2-spheres without symmetric 3-ball bounds.
For an equivariant Morse stratification which contains a unique open stratum, we introduce the notion of equivariant antiperfection, which means the difference of the equivariant Morse series and the equivariant Poincare series achieves the maximal possible value (instead of the minimal possible value 0 in the equivari…
The paper calculates the equivariant genus for a specific type of knot.
problem Calculating the equivariant genus of marked strongly invertible knots associated with 2-bridge knots.
method Analyzing invariant Seifert surfaces for marked strongly invertible knots.
result The paper completely determines the equivariant genus for every marked strongly invertible knot with K a 2-bridge knot. Selects points from Jordan domains on Riemannian surfaces.
problem Selecting points from Jordan domains in Riemannian surfaces.
method Fiber bundle theory and conformal mappings.
result Space of Jordan domains retracts onto round disks.
The paper proves a generalized Bour's theorem for invariant surfaces.
problem Proving a generalized Bour's theorem for invariant surfaces.
method Equivariant geometry techniques applied to three-manifolds.
result A generalized Bour's theorem holds for invariant surfaces.
Constructs perturbations of a minimal surface with triple junctions.
problem Minimal surfaces with triple junctions in curved spaces.
method Constructs stationary perturbations with given boundary conditions.
result Constructs minimal surfaces with triple junctions in R2imesS1. Proves conditions for minimal surfaces in complex hyperbolic space.
problem Conditions for finite energy equivariant minimal surfaces in complex hyperbolic space.
method Analyzes peripheral holonomy and uses Higgs bundles.
result Explicit parametrization and construction of minimal surfaces.
Local index density of perturbed de Rham complex is invariant under certain conditions.
problem Invariance of local index density for perturbed de Rham complex.
method Invariance theory applied to perturbed Laplacian and local index density.
result Local index density is invariant under perturbation by closed 1-forms.
We construct the odd symplectic structure and the equivariant even (pre)symplectic one from it on the space of differential forms on the Riemann manifold. The Poincare -- Cartan like invariants of the second structure define the equivariant generalizations of the Euler classes on the surfaces.
We prove the automorphic property of the invariant of K3 surfaces with involution, which we obtained using equivariant analytic torsion, in the case where the dimension of the moduli space is less than or equal to 2.
Proves representability of complex semigroup systems.
problem Representability of systems of proportionally modular numerical semigroups.
method Canonical equivariant resolution of weighted homogeneous surface singularities.
result Every system of proportionally modular numerical semigroups is representable.
Unstable minimal surfaces in n-space link to hyperbolic products.
problem Characterizing unstable minimal surfaces in Rn and their product counterparts. method Lifting to R-trees, deforming to hyperbolic products, and proving instability equivalence. result Unstable minimal surfaces in Rn imply unstable surfaces in product hyperbolic spaces. We consider an equivariant analogue of a conjecture of Borcherds. Let Y be a real K3 surface without real points. Let g be a Ricci-flat Kaehler metric on Y invariant under the complex conjugation. We shall prove that the equivariant determinant of the Laplacian of (Y,g) with respect to the complex conjugation…
Geodesic surfaces embed into hyperbolic 3-manifolds for all finite group actions.
problem Embedding geodesic surfaces into hyperbolic 3-manifolds.
method Analyzing finite group actions on surfaces and proving geodesic embeddings for all irreducible cases.
result All quasiplatonic surfaces embed geodesically into hyperbolic 3-manifolds.
For a 3-manifold M with b1(M)=1 fibered over S1 and the fiberwise gradient ξ of a fiberwise Morse function on M, we introduce the notion of amidakuji-like path (AL-path) on M. An AL-path is a piecewise smooth path on M consisting of edges each of which is either a part of a critical locus of ξ or a fl…
Constructs a unique surface in a ball with specific properties.
problem Creating a minimal surface with specific topological and geometric constraints.
method Variational methods, equivariant min-max theory, nontrivial sweepout.
result First genus one critical catenoid in a unit ball.
Defines a new homomorphism for strongly invertible knots, proving equivariant algebraic concordance.
problem Equivariant algebraic concordance of strongly invertible knots.
method Defining a homomorphism Φ from equivariant concordance group to a new equivariant algebraic concordance group, proving it lifts known homomorphisms and provides new obstructions. result Obtains a new obstruction to equivariant sliceness and novel lower bounds on equivariant slice genus.
New invariants refine link homology, showing large genus differences.
problem Understanding genus differences in equivariant cobordisms.
method Refined Bar-Natan homology for involutive links, constructing new numerical invariants.
result Difference between equivariant and isotopy-equivariant slice genera can be arbitrarily large.
Study on invariant Seifert surfaces for strongly invertible knots, showing large gaps in genus.
problem Understanding gaps in genus between strongly invertible knots and their invariant Seifert surfaces.
method Analysis of invariant Seifert surfaces and proof of genus gaps, with variants of Edmonds' theorem.
result Gap between equivariant genus and usual genus can be arbitrarily large for strongly invertible knots.
Characterizes area-minimizing maps for surfaces of genus ≥ 2.
problem Equivariant area-minimizing maps on surface covers.
method Classifies minimal surfaces in Hilbert spheres with constant negative Gaussian curvature.
result Characterizes all equivariantly area-minimizing maps from the universal cover of a surface to a Hilbert sphere.