arXiv research

Unified method for CNNs to approximate equivariant maps across various groups.

The study constructs equivariant harmonic maps into symmetric spaces with applications to Willmore surfaces.

Equivariant neural networks use symmetry to interpret complex data.

Researchers found uncountable harmonic self-maps in complex projective spaces.

Constructs harmonic maps between special geometric shapes.

Proves unique maps from certain spaces to others.

This survey studies equivariant harmonic maps arising from Higgs bundles. We explain the non-abelian Hodge correspondence and focus on the role of equivariant harmonic maps in the correspondence. With the preparation, we review current progress towards some open problems in the study of equivariant harmonic maps.

We present a general theory of Group equivariant Convolutional Neural Networks (G-CNNs) on homogeneous spaces such as Euclidean space and the sphere. Feature maps in these networks represent fields on a homogeneous base space, and layers are equivariant maps between spaces of fields. The theory enables a systematic cla…

We prove that a primitive harmonic map is equivariant if and only if it admits a holomorphic potential of degree one. We investigate when the equivariant harmonic map is periodic, and as an application discuss constant mean curvature cylinders with screw motion symmetries.

Investigates harmonic self-maps' stability on cohomogeneity one manifolds.

Harmonic maps depend analytically on representations.

The construction of topological index maps for equivariant families of Dirac operators requires factoring a general smooth map through maps of a very simple type: zero sections of vector bundles, open embeddings, and vector bundle projections. Roughly speaking, a normally non-singular map is a map together with such a …

Normal forms for equivariant maps in infinite dimensions established.

Conditions for pre-quantizability of G-invariant forms are derived using moment maps.

Study the relationship between orbit braid group and equivariant mapping class group on surfaces.

We show that if G is a discrete subgroup of the group of the isometries of the hyperbolic k-space H^k, and if R is a representation of G into the group of the isometries of H^n, then any R-equivariant map F from H^k to H^n extends to the boundary in a weak sense in the setting of Borel measures. As a consequence of thi…

Paper shows mapping class group-equivariant Teichmüller space deformation to Thurston spine.

Generalizes Floer homotopy via Morse-Bott theory.

In this paper we define strongly projectively flatness of holomorphic maps into the complex Grassmannian manifold, which is a kind of generalization of holomorphic maps into the complex projective space and prove a rigidity of equivariant strongly projectively flat maps of compact simply connected homogeneous Kähler ma…

New method for deep learning hierarchies like sequences and graphs.

New translation equivariant neural processes improve spatio-temporal data modeling.

Machine learning uses invariant theory to restrict function classes.

We show a non-existence result for some class of equivariant maps between sphere bundles over tori. The notion of equivariant KO-degree is used in the proof. As an application to Seiberg-Witten theory, for a connected closed oriented spin 4-manifold with indefinite intersection form, we have a new bound of the second B…

We give two generalizations of the Atiyah-Bott-Berline-Vergne localization theorem for the equivariant cohomology of a torus action: 1) replacing the torus action by a compact connected Lie group action, 2) replacing the manifold having a torus action by an equivariant map. This provides a systematic method for calcula…

We present a convolutional network that is equivariant to rigid body motions. The model uses scalar-, vector-, and tensor fields over 3D Euclidean space to represent data, and equivariant convolutions to map between such representations. These SE(3)-equivariant convolutions utilize kernels which are parameterized as a …

Generalizes CNNs for Lie group equivariance across various data types.

We introduce and study a new spectral sequence associated with a Poisson group action on a Poisson manifold and an equivariant momentum mapping. This spectral sequence is a Poisson analog of the Leray spectral sequence of a fibration. The spectral sequence converges to the Poisson cohomology of the manifold and has the…

We develop a Chern character map for twisted equivariant non-abelian cohomology.

Stable approach solves equivariant Hopf theorem for G-manifolds.

We present effective methods to compute equivariant harmonic maps from the universal cover of a surface into a nonpositively curved space. By discretizing the theory appropriately, we show that the energy functional is strongly convex and derive convergence of the discrete heat flow to the energy minimizer, with explic…

The paper generalizes equivariant neural networks on homogeneous spaces to the non-linear setting.

For a compact Lie group acting on a smooth manifold, we define the differential cohomology of a certain quotient stack involving principal bundles with connection. This produces differential equivariant cohomology groups that map to the Cartan-Weil equivariant forms and to Borel's equivariant integral cohomology. We sh…

We survey work by the author and Ralf Meyer on equivariant KK-theory. Duality plays a key role in our approach. We organize the survey around the objective of computing a certain homotopy-invariant of a space equipped with a proper action of a group or groupoid called the Lefschetz map. The Lefschetz map associates an …

Solves initial value problem for harmonic maps on specific manifolds.

In this paper we classify maps from a torus phase space $X$ to $\mathcal{H}_n^*$, the space of $n \times n$, non-singular hermitian operators up to equivariant homotopy. The equivariance is with respect to a time-reversal involution on $X$ and an involution on $\mathcal{H}_n^*$ defining a certain symmetry class. Furthe…

The theory of principal $G$-bundles over a Lie groupoid is an important one, unifying the various types of principal $G$-bundles, including those over manifolds, those over orbifolds, as well as equivariant principal $G$-bundles. In this paper, we study the differential geometry of these objects, including connections …

Functor connects symplectic and contact structures via cutting and blowups.

By results of Loeffler and Comezana, the Pontrjagin-Thom map from geometric G-equivariant bordism to homotopy theoretic equivariant bordism is injective for compact abelian G. If G = S^1 x ... x S^1, we prove that the associated fixed point square is a pull back square, thus confirming a recent conjecture of D. Sinha. …

The paper characterizes equivariant immersions in hyperbolic space.

Let $G$ a semisimple Lie group of non-compact type and let $\mathcal{X}_G$ be the Riemannian symmetric space associated to it. Suppose $\mathcal{X}_G$ has dimension $n$ and it has no factor isometric to either $\mathbb{H}^2$ or $\text{SL}(3,\mathbb{R})/\text{SO}(3)$. Given a closed $n$-dimensional Riemannian manifold $…

Equivariance is a nice property to have as it produces much more parameter efficient neural architectures and preserves the structure of the input through the feature mapping. Even though some combinations of transformations might never appear (e.g. an upright face with a horizontal nose), current equivariant architect…

We propose a semantic segmentation model that exploits rotation and reflection symmetries. We demonstrate significant gains in sample efficiency due to increased weight sharing, as well as improvements in robustness to symmetry transformations. The group equivariant CNN framework is extended for segmentation by introdu…

Two constructions of Chern character for equivariant vector bundles in noncommutative geometry.

The first author's geometric Hopf invariant of a stable map $F:Σ^{\infty}X \to Σ^{\infty}Y$ is a stable ${\mathbb Z}_2$-equivariant map $h(F):Σ^{\infty}X \to Σ^{\infty}(Y \wedge Y)$ constructed by an explicit difference construction applied to $(F \wedge F)Δ_X - Δ_Y F$. The stable ${\mathbb Z}_2$-equivariant homotopy c…

In this short note we prove an equivariant version of the formality of multidiffirential operators for a proper Lie group action. More precisely, we show that the equivariant Hochschild-Kostant-Rosenberg quasi-isomorphism between the cohomology of the equivariant multidifferential operators and the complex of equivaria…

This paper contains two main results. The first is the existence of an equivariant Weil-Petersson geodesic in Teichmueller space for any choice of pseudo-Anosov mapping class. As a consequence one obtains a classification of the elements of the mapping class group as Weil-Petersson isometries which is parallel to the T…

Unified theorem for deep and shallow joint-equivariant machines.

Proves an equivariant version of Heegaard Floer link surgery formula.