Study the relationship between orbit braid group and equivariant mapping class group on surfaces.
problem Understanding the relationship between mapping class groups and braid groups with group actions.
method Using the fibration F0GMightarrowF(M/G,n) and exact sequence. result The conclusion is closely connected with the braid group of the quotient space.
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.
Theory connects Torelli subgroup homology to Sp(2g,ℤ)-modules.
problem Homology of Torelli subgroup of mapping class group.
method Equivariant group presentations and homology theory.
result Second homology group of Torelli subgroup is finitely generated as Sp(2g,ℤ)-module.
Machine learning uses invariant theory to restrict function classes.
problem Creating function classes that respect physical law constraints.
method Using equivariant machine learning and Malgrance's method to parameterize functions.
result Explicitly parameterizes equivariant functions between linear spaces.
Novikov theorem extended to rational Pontryagin classes for cyclic group C4.
problem Classifying stable Cp-smoothings of high-dimensional manifolds. method Computing equivariant homotopy groups and applying to C4. result Novikov's theorem extended to rational Pontryagin classes for C4. 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…
This paper classifies equivariant principal bundles over a 2-sphere using isotropy representations.
problem Classifying equivariant principal bundles over the 2-sphere.
method Using isotropy representations to classify bundles over the 2-sphere.
result Equivariant principal bundles over the 2-sphere can be classified by a Γ-fixed set of homotopy classes of maps and first Chern class.
Nontrivial boundary Dehn twist found on K3#K3 manifold.
problem Proving nontriviality of a Dehn twist on a specific 4-manifold.
method Algebraic criterion and equivariant topological K-theory to show non-isotopy.
result Boundary Dehn twist is nontrivial in the smooth mapping class group.
Stable approach solves equivariant Hopf theorem for G-manifolds.
problem Describe homotopy classes of G-equivariant maps into a G-sphere.
method Equivariant stable homotopy theory with semi-free G-universe.
result Degrees of maps are characterized by congruences.
Unified method for CNNs to approximate equivariant maps across various groups.
problem Limited universal approximation theorems for CNNs with specific groups and settings.
method Unified approach to derive universal approximation theorems for equivariant maps by CNNs in diverse settings.
result Ability to handle non-linear equivariant maps between infinite-dimensional spaces for non-compact groups.
The Ptolemy groupoid is a combinatorial groupoid generated by elementary moves on marked trivalent fatgraphs with three types of relations. Through the fatgraph decomposition of Teichmüller space, the Ptolemy groupoid is a mapping class group equivariant subgroupoid of the fundamental path groupoid of Teichmüller space…
A new method constructs odd characteristic classes in cyclic homology.
problem Constructing odd characteristic classes in equivariant loop space homology.
method Constructing a Chern character in the odd part of the equivariant (entire) cyclic Chen-normalized bar complex.
result Induces a well-defined group homomorphism from K−1 theory to the odd homology group. Unified theorem for deep and shallow joint-equivariant machines.
problem Universal approximation of joint-equivariant machines.
method Constructive universal approximation theorem based on ridgelet transform.
result Unified approximation of deep and shallow networks.
The paper characterizes equivariant immersions in hyperbolic space.
problem Characterizing equivariant immersions in hyperbolic space.
method Analyzes the Gauss map of equivariant immersions in hyperbolic space.
result Provides two characterizations of equivariant immersions: one in terms of the Maslov class and another for compact M in terms of Hamiltonian symplectomorphisms. A projection maps geodesic currents to Teichmüller space.
problem Mapping geodesic currents to Teichmüller space.
method Equivariant, length-minimizing projection from filling currents to Teichmüller space.
result The projection is well-behaved and maps geodesic currents to Teichmüller space.
Real Heegaard Floer theory shown to be natural and invariant.
problem Defining and proving naturality of real Heegaard Floer homology.
method Defined and proved naturality of real Heegaard Floer homology and other related theories.
result Real Heegaard Floer homology is shown to be natural and admits an action of the equivariant mapping class group.
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…
A spine is constructed for a non-orientable surface's decorated Teichmüller space.
problem Constructing a spine for a non-orientable surface's decorated Teichmüller space.
method Building on Harer's work, constructing a spine and computing its dimension, showing equivariance with the pure mapping class group.
result A spine is constructed with minimal dimension for a punctured non-orientable surface.
Schmutz Schaller and Thurston's approaches are dual.
problem Mapping class group-equivariant deformation retractions of Teichmüller space.
method Comparing Schmutz Schaller's and Thurston's methods.
result Schmutz Schaller and Thurston's approaches are dual.
Survey discusses new ideas in geometric group theory and their applications.
problem Understanding geodesic metric spaces and their equivariant wall structures.
method Introduces and highlights the impact of injective metric spaces and cubical approximation theorem.
result Rich equivariant wall structures in various geodesic metric spaces.
We show how characteristic classes determine equivariant prequantization bundles over the space of connections on a principal bundle. These bundles are shown to generalize the Chern-Simons line bundles to arbitrary dimensions. Our result applies to arbitrary bundles, and it is studied the action of both the gauge group…
Suppose one is given a discrete group G, a cocompact proper G-manifold M, and a G-self-map f of M. Then we introduce the equivariant Lefschetz class of f, which is globally defined in terms of cellular chain complexes, and the local equivariant Lefschetz class of f, which is locally defined in terms of fixed point data…
Study surjectivity of Kirwan map for generalized hyperkähler reduction.
problem Establishing surjectivity of Kirwan map for a specific class of Hamiltonian manifolds.
method Defined a close analogue of hyperkähler reduction for manifolds with equivariant functions under semi-linear G-actions. result Surjectivity of Kirwan map proved for the defined class of manifolds.
We construct a covering of Culler-Vogtmann Outer space by the Teichmuller spaces of punctured surfaces. By considering the equivariant homology for the action of Out(F_n) on this covering, we construct a spectral sequence converging to the homology of Out(F_n) that has E^1 terms given by the homology of mapping class g…
The main goal of this paper is a detailed study of asymptotic cones of the mapping class groups. In particular, we prove that every asymptotic cone of a mapping class group has a bi-Lipschitz equivariant embedding into a product of real trees, sending limits of hierarchy paths onto geodesics, and with image a median su…
Let G be a compact connected Lie group acting on a stable complex manifold M with equivariant vector bundle E. Besides, suppose φ is an equivariant map from M to the Lie algebra g. We can define some equivalence relation on the triples (M,E,φ) such that the set of equivalence classes form an …
Generalizes CNNs on homogeneous spaces like Euclidean and spherical surfaces.
problem Classifying and understanding equivariant CNNs on homogeneous spaces.
method Develops a theory for equivariant maps between field spaces of given types.
result Equivariant kernels correspond to the most general kind of equivariant linear maps.
We construct a new equivariant cohomology theory for a certain class of differential vertex algebras, which we call the chiral equivariant cohomology. A principal example of a differential vertex algebra in this class is the chiral de Rham complex of Malikov-Schechtman-Vaintrob of a manifold with a group action. The ma…
Equivariant neural networks use symmetry to interpret complex data.
problem Interpreting and understanding the behavior of equivariant neural networks.
method Decompose layers into simple representations and analyze nonlinear activation functions.
result Equivariant neural networks can be interpreted using a filtration generalizing Fourier series.
Constructs measurable equivariant maps for higher rank Lie groups.
problem Measurable cocycles with values into higher rank Lie groups.
method Extends Connell-Farb's construction to measurable cocycles.
result Constructs measurable equivariant maps with uniformly bounded Jacobian.
Generalizes CNNs for Lie group equivariance across various data types.
problem Equivariance to transformations like rotations for non-image data.
method Constructs equivariant convolutional layers for Lie groups.
result Models conserve linear and angular momentum in Hamiltonian systems.
Group equivariant neural networks simplify complex tasks with group representation theory.
problem Challenging tasks requiring input transformations like rotations.
method Group representation theory, non-commutative harmonic analysis, differential geometry.
result A neural network is group equivariant if and only if it has a convolutional structure.
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 …
Study shows infinite order in mapping class groups for certain 3D shapes.
problem Understanding the mapping class groups of certain 3D shapes.
method Analogues of Seiberg-Witten-Floer homology for 3-manifolds.
result Monodromy diffeomorphisms have infinite order in smooth mapping class groups.
EbC learns equivariant embeddings from unlabeled group actions.
problem Learning equivariant embeddings from unlabeled group actions.
method Equivariance by Contrast (EbC) method to learn equivariant embeddings from observation pairs (y,g⋅y). result High-fidelity equivariance in latent space for diverse groups.
Conditions for pre-quantizability of G-invariant forms are derived using moment maps.
problem Conditions for equivariant prequantizability of G-invariant forms.
method Conditions derived using moment maps and obstructions computed.
result Necessary and sufficient conditions for equivariant pre-quantizability are computed.
Investigates harmonic self-maps' stability on cohomogeneity one manifolds.
problem Stability of harmonic self-maps on cohomogeneity one manifolds.
method Systematic study of Jacobi equation for harmonic self-maps.
result Explicit solutions for specific cases show identity map's stability.
If K is a compact Lie group and g≥2 an integer, the space K2g is endowed with the structure of a Hamiltonian space with a Lie group valued moment map Φ. Let β be in the centre of K. The reduction Φ−1(β)/K is homeomorphic to a moduli space of flat connections. When K is simply connected, a dire…
New framework for equivariant neural networks using Lie group decompositions.
problem Limitations of existing equivariant neural network methods for Lie groups.
method Lie group structure and geometry, decomposition into subgroups and submanifolds.
result Equivariant neural networks for affine transformations outperform previous methods.
This work introduces a method for almost equivariance in neural networks using Lie algebra convolutions.
problem Real-world data often does not conform to strict group equivariances, leading to underperformance in models.
method Definition and practical implementation of almost equivariance through Lie algebra convolutions.
result Demonstrated the validity of the approach through benchmarking against fully equivariant settings.
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…
New algorithm speeds up group equivariant neural networks computations.
problem Challenging computations in group equivariant neural networks.
method Diagrammatic framework based on category theory for matrix multiplication.
result Exponential improvement in time complexity for matrix multiplication.
We give a classifying theory for LG-bundles, where LG is the loop group of a compact Lie group G, and present a calculation for the string class of the universal LG-bundle. We show that this class is in fact an equivariant cohomology class and give an equivariant differential form representing it. We then use t…
Let Y be a CW-complex with a single 0-cell, K its Kan group, a model for the loop space of Y, and let G be a compact, connected Lie group. We give an explicit finite dimensional construction of generators of the equivariant cohomology of the geometric realization of the cosimplicial manifold $\roman{Hom}(K,G)$ …
The main result of this paper is non-vanishing of the image of the index map from the G-equivariant K-homology of a proper G-compact G-manifold X to the K-theory of the C∗-algebra of the group G. Under the assumption that the Kronecker pairing of a K-homology class with a low-dimensional cohomology…
Improved sample efficiency in semantic segmentation with rotation equivariant CNNs.
problem Efficiently segmenting images with rotation and reflection symmetries.
method Introduced rotation-equivariant CNNs with new equivariant convolutions and transposed convolutions.
result Significant gains in sample efficiency and robustness to symmetry transformations.
The first author's geometric Hopf invariant of a stable map F:Σ∞X→Σ∞Y is a stable Z2-equivariant map h(F):Σ∞X→Σ∞(Y∧Y) constructed by an explicit difference construction applied to (F∧F)ΔX−ΔYF. The stable Z2-equivariant homotopy c…
Normal forms for equivariant maps in infinite dimensions established.
problem Establishing normal forms for equivariant maps in infinite-dimensional manifolds.
method Inspired by Lyapunov-Schmidt reduction and Kuranishi method, uses Slice Theorem for Fréchet manifolds.
result Abstract moduli spaces of equivariant maps are locally modeled on quotient by a compact group.