Study of invariants on manifolds with boundary involving equivariant spectral flow and η-invariants.
problem Equivariant invariants on manifolds with boundary.
method Analysis of Dirac operators, winding numbers, spectral flow, Maslov indices, and η-invariants.
result Established relation between equivariant η-invariants and Maslov triple indices.
Develops equivariant Seiberg-Witten-Floer cohomology for 3-spheres.
problem Finite group actions on rational homology 3-spheres.
method Equivariant version of Seiberg-Witten-Floer stable homotopy type.
result Definition of d-invariants with Froyshov-type inequality. 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 of Seiberg-Witten invariants for 4-manifolds with group actions.
problem Understanding Seiberg-Witten invariants for manifolds with group actions.
method Introduced equivariant Seiberg-Witten invariants, studied their properties and established relations.
result Established localisation formulas and gluing formulas for the invariants.
Program connects birational invariants with G-equivariant ones using Gromov-Witten theory.
problem Establishing a connection between birational invariants and G-equivariant ones.
method Gromov-Witten theory, Chen-Ruan cohomology, and equivariant atoms.
result New interpretations and applications of classical invariants.
We show that the S^1-equivariant Yamabe invariant of the 3-sphere, endowed with the Hopf action, is equal to the (non-equivariant) Yamabe invariant of the 3-sphere. More generally, we establish a topological upper bound for the S^1-equivariant Yamabe invariant of any closed oriented 3-manifold endowed with an S^1-actio…
Study on 2-bridge knots, proving equivariant concordance order is infinite.
problem Equivariant concordance of 2-bridge knots.
method Formula for butterfly polynomial, two proofs of non-equivariant sliceness, new invariant for strongly invertible knots.
result Equivariant concordance order of 2-bridge knots is infinite.
Improved upper bound for equivariant Yamabe invariant in 3D.
problem Upper bound for G-equivariant Yamabe invariant in 3-manifolds. method Used topological assumptions to show an upper bound.
result Improved Hebey-Vaugon conjecture in dimension 3.
Develops equivariant grid homology for strongly invertible knots.
problem Invariants of strongly invertible knots.
method Equivariant grid diagrams and mapping cones.
result Equivariant unknotting numbers and genus bounds.
In this paper, we develop a theory about the relationship between G-invariant/equivariant functions and deep neural networks for finite group G. Especially, for a given G-invariant/equivariant function, we construct its universal approximator by deep neural network whose layers equip G-actions and each affine t…
Our work improves VAE latent space clustering by enforcing invariant and equivariant learning.
problem Current VAEs fail to learn invariant and equivariant clusters in latent space.
method We use a mixture model pdf like Gaussian mixtures to enforce deep, group-invariant learning and separate semantic and equivariant variables.
result Our model effectively learns to disentangle invariant and equivariant representations, improving learning rate and image recognition.
Numerous invariant (or equivariant) neural networks have succeeded in handling invariant data such as point clouds and graphs. However, a generalization theory for the neural networks has not been well developed, because several essential factors for the theory, such as network size and margin distribution, are not dee…
Study knot Floer homology to create concordance invariants and slice genus bounds.
problem Developing concordance invariants using knot Floer homology.
method Using knot Floer homology, define and analyze equivariant concordance invariants.
result Showed a family of strongly invertible slice knots with arbitrarily large equivariant slice genus.
The paper explores how equivariant models' biases affect latent representations for better performance.
problem The impact of inductive biases on latent representations in equivariant models.
method Demonstrates the importance of accounting for inductive biases in latent representations of equivariant models.
result Effective invariant projections can be used to retain information in latent representations, improving downstream tasks.
New invariants from Seiberg-Witten theory for 3-spheres with involution.
problem Equivariant Seiberg-Witten Floer theory of rational homology 3-spheres.
method Coupling involution to Seiberg-Witten theory, constructing delta-invariants.
result New Floer-theoretic invariants with properties and applications.
Group invariant and equivariant Multilayer Perceptrons (MLP), also known as Equivariant Networks, have achieved remarkable success in learning on a variety of data structures, such as sequences, images, sets, and graphs. Using tools from group theory, this paper proves the universality of a broad class of equivariant M…
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.
Study of Khovanov homology invariants from U(1)imesU(1)-equivariant algebra.
problem Understanding concordance invariants from Khovanov homology.
method Analysis of U(1)imesU(1)-equivariant Khovanov homology using algebraic filtrations. result Extracted two families of concordance invariants.
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.
2-knots with S4 symmetry are classified up to equivariant concordance.
problem Classifying 2-knots with S4 symmetry up to equivariant concordance. method Constructing a new invariant called periodic, based on the Arf invariant.
result The smooth equivariant concordance group of 2-knots in S4 is isomorphic to Z/2Z for all d≥2. We prove inverse spectral results for differential operators on manifolds and orbifolds invariant under a torus action. These inverse spectral results involve the asymptotic equivariant spectrum, which is the spectrum itself together with "very large" weights of the torus action on eigenspaces. More precisely, we show …
The study shows symmetry improves machine learning generalization.
problem Improving machine learning generalization through symmetry.
method Using an averaging operator to prove equivariance reduces test risk.
result Equivariant predictors reduce test risk compared to non-equivariant ones.
Frame Averaging makes neural networks invariant or equivariant to new symmetries.
problem Designing neural networks that respect symmetries while being expressive and efficient.
method Introduces Frame Averaging (FA) as a systematic framework to adapt architectures to become invariant or equivariant to new symmetries.
result Frame Averaging guarantees exact invariance or equivariance while being simpler to compute than full group averaging.
L-CNNs preserve gauge symmetry in lattice simulations.
problem Breaking gauge symmetry in neural network models.
method Lattice gauge equivariant convolutional neural networks (L-CNNs).
result L-CNNs represent gauge invariant functions on the lattice.
Study symmetries in equivariant Khovanov homology.
problem Understand symmetries in equivariant Khovanov homology.
method Construction of an involution, integral lifting, splitting of theories, and relation to Rasmussen's invariant.
result Established symmetries in equivariant Khovanov homology.
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. The Bauer-Furuta invariants of smooth 4-manifolds are investigated from a functorial point of view. This leads to a definition of equivariant Bauer-Furuta invariants for compact Lie group actions. These are studied in Galois covering situations. We show that the ordinary invariants of all quotients are determined by th…
STAR improves equivariant and invariant representation learning by routing projection heads.
problem Redundant feature learning in equivariant and invariant representation learning.
method Soft Task-Aware Routing (STAR) for projection heads specialization.
result Lower canonical correlations between invariant and equivariant embeddings.
Invariant and equivariant networks have been successfully used for learning images, sets, point clouds, and graphs. A basic challenge in developing such networks is finding the maximal collection of invariant and equivariant linear layers. Although this question is answered for the first three examples (for popular tra…
IsoGCNs learn invariant and equivariant graph features for efficient simulations.
problem Learning isometric transformation invariant and equivariant features in graphs for simulations.
method Transformation invariant and equivariant Graph Convolutional Networks (IsoGCNs).
result IsoGCNs outperform state-of-the-art methods on geometrical and physical simulation tasks.
Unified framework for image restoration using equivariant denoisers.
problem Restoring images with realistic priors and invariant transformations.
method Unified framework named ERED based on equivariant denoisers and stochastic optimization.
result Unified framework ERED converges and improves image restoration.
Paper proves non-zero generalization boost for equivariant models.
problem Improving generalization in machine learning models.
method Analyzes simplest case of linear models, focusing on invariant/equivariant properties.
result First provably non-zero improvement in generalization for invariant/equivariant models.
Paper proves finiteness and Morse index estimates for equivariant min-max hypersurfaces.
problem Existence and finiteness of G-invariant minimal hypersurfaces. method Equivariant min-max theory, compactness theorem, bumpy metrics theorem.
result Generalization of Morse index estimates to equivariant setting.
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.
The study proves a generic multiplicity one theorem for G-invariant minimal hypersurfaces.
problem Proving a generic multiplicity one theorem for G-invariant minimal hypersurfaces. method Equivariant min-max theory and analysis of G-homology classes. result Shows a generic multiplicity one theorem for G-invariant minimal hypersurfaces. Lie groupoid equivariant neural networks are a new type of neural network.
problem Designing neural networks that respect the structure of Lie groupoids.
method Introducing Lie groupoid equivariant convolutions and layers, and showing their equivalence to Lie algebroid-equivariant networks.
result Lie groupoid equivariant neural networks are equivalent to certain Lie algebroid-equivariant networks.
We define the equivariant holonomy of an invariant connection on a principal U(1)-bundle. The properties of the ordinary holonomy are generalized to the equivariant setting. In particular, equivariant U(1)-bundles with connection are shown to be classified by its equivariant holonomy modulo isomorphisms. We also show t…
We derive a formula for the eta invariants of equivariant Dirac operators on quotients of compact Lie groups, and for their infinitesimally equivariant extension. As an example, we give some computations for spheres.
Graph Neural Networks (GNN) come in many flavors, but should always be either invariant (permutation of the nodes of the input graph does not affect the output) or equivariant (permutation of the input permutes the output). In this paper, we consider a specific class of invariant and equivariant networks, for which we …
EquivCNP learns group symmetries for conditional data.
problem Learning conditional models with data symmetries.
method Group equivariant decomposition and Lie group convolutional layers.
result EquivCNP achieves comparable performance and zero-shot generalization.
We use virtual neighborhood technique to establish GW-invariants, Quantum cohomology, equivariant GW-invariants, equivariant quantum cohomology and Floer cohomology for general symplectic manifold. We also establish GW-invariants for a family of symplectic manifolds. As a consequence, we prove Arnold conjecture for non…
New approach to Z-stability and critical metrics on Kähler manifolds.
problem Determining Z-stability and existence of Z-critical metrics on Kähler manifolds. method Equivariant localisation applied to integrals over test configurations.
result Existence of Z-critical metrics is equivalent to Z-stability. Paper studies spectral invariants and monopole Floer homology for rational homology three-spheres.
problem Tackles the existence of positive scalar curvature metrics on ribbon homology cobordisms.
method Defines an R-filtration on the equivariant complex of monopole Floer homology via Chern-Simons-Dirac functional, leading to a spectral invariant.
result Shows that the spectral invariant provides an obstruction to the existence of positive scalar curvature metrics on ribbon homology cobordisms.
Sign equivariant networks improve model expressiveness for spectral geometric learning.
problem Limited expressiveness of sign invariant models for tasks like graph link prediction.
method Developed sign equivariant neural network architectures based on new analytic sign equivariant polynomials.
result Sign equivariant models achieve theoretical benefits in spectral geometric learning tasks.
Constructs equivariant spectral flow for Dirac-type operators on manifolds.
problem Calculating spectral flow for Dirac-type operators on manifolds with group actions.
method Equivariant spectral flow construction for paths of Dirac-type operators on manifolds.
result Relates delocalised η-invariants and ρ-invariants for different positive scalar curvature metrics.
We compute K-theory invariants of algebras of pseudodifferential operators on manifolds with corners and prove an equivariant index theorem for operators invariant with respect to an action of Rk. We discuss the relation between our results and the η-invariant.
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…
New invariants prove exotic slice disks for knots.
problem Proving exotic slice disks for knots.
method Involutive Khovanov homology and derived invariants.
result Reprove exotic slice disks for knots Jn.