The paper classifies and proves properties of symmetry breaking operators for specific groups.
problem Classifying and understanding symmetry breaking operators for de Sitter and Lorentz groups.
method Constructing and classifying differential symmetry breaking operators, proving localness, and showing sporadic nature.
result All symmetry breaking operators are differential and sporadic, not obtainable by residue formulas.
We present Bernstein-Sato identities for scalar-, spinor- and differential form-valued distribution kernels on Euclidean space associated to conformal symmetry breaking operators. The associated Bernstein-Sato operators lead to partially new formulae for conformal symmetry breaking differential operators on functions, …
Part I. We prove a one-to-one correspondence between differential symmetry breaking operators for equivariant vector bundles over two homogeneous spaces and certain homomorphisms for representations of two Lie algebras, in connection with branching problems of the restriction of representations. We develop a new method…
The paper classifies and constructs differential symmetry breaking operators from a line bundle to a vector bundle over real projective spaces.
problem Classifying and constructing differential symmetry breaking operators.
method Utilizing factorization identities and branching laws of generalized Verma modules.
result Differential symmetry breaking operators from a line bundle to a vector bundle over real projective spaces are classified and constructed.
We prove an explicit residue formula for a meromorphic continuation of conformally covariant integral operators between differential forms on Rn and on its hyperplane. The results provide a simple and new construction of the conformally covariant differential symmetry breaking operators between differential fo…
Paper introduces a new symbol map for differential symmetry breaking operators.
problem Generalizing the symbol map to non-abelian settings.
method Introduces and studies the truncated symbol map Symb0(D). result Classified and constructed differential intertwining operators and homomorphisms.
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…
Researchers classify differential operators between 3-sphere and 2-sphere bundles.
problem Classifying differential symmetry breaking operators between 3-sphere and 2-sphere bundles.
method Constructing and classifying all differential symmetry breaking operators D_{λ,ν}^m.
result Necessary and sufficient conditions for the existence of these operators.
We study conformal symmetry breaking differential operators which map differential forms on Rn to differential forms on a codimension one subspace Rn−1. These operators are equivariant with respect to the conformal Lie algebra of the subspace Rn−1. They correspond to homomorphism…
Rapid progress has been made recently on symmetry breaking operators for real reductive groups. Based on Program A-C for branching problems (T.Kobayashi [Progr.Math.2015]), we illustrate a scheme of the classification of (local and nonlocal) symmetry breaking operators by an example of conformal representations on diff…
Geometric mechanism mimics physics' symmetry breaking.
problem Understanding spontaneous symmetry breaking in geometry.
method Analogous to physics, studying symmetry breaking in differential geometry.
result Symmetry breaking can be used to solve geometric problems.
We give a complete classification of intertwining operators (symmetry breaking operators) between spherical principal series representations of G=O(n+1,1) and G'=O(n,1). We construct three meromorphic families of the symmetry breaking operators, and find their distribution kernels and their residues at all poles explic…
Constructs moduli spaces for monopoles with arbitrary symmetry breaking.
problem Finding moduli spaces for monopoles with varying symmetry.
method Defined configuration space with asymptotic conditions, performed quotient construction, used b-calculus and scattering calculus.
result Constructs hyper-Kähler moduli spaces for monopoles with arbitrary symmetry breaking.
We give a complete classification of conformally covariant differential operators between the spaces of differential i-forms on the sphere Sn and j-forms on the totally geodesic hypersphere Sn−1 by analyzing the restriction of principal series representations of the Lie group O(n+1,1). Further, we provide…
Study symmetry breaking in quantum mechanics to understand many-body physics.
problem Understanding many-body physics from quantum mechanics.
method Analyzing potentials with unstable critical points and local minima.
result Emergence of many-body physics from spontaneous symmetry breaking.
We give a complete classification of conformally covariant differential operators between the spaces of i-forms on the sphere Sn and j-forms on the totally geodesic hypersphere Sn−1. Moreover, we find explicit formulæ for these new matrix-valued operators in the flat coordinates in terms of basic operators …
For a pseudo-Riemannian manifold X and a totally geodesic hypersurface Y, we consider the problem of constructing and classifying all linear differential operators Ei(X)→Ej(Y) between the spaces of differential forms that intertwine multiplier representations of the Lie algebra of confor…
Symmetry-breaking in three differential geometry conjectures.
problem Exploring the role of symmetry in three differential geometry conjectures.
method Examining the Carathéodory, Willmore, and Lawson Conjectures through the lens of symmetry in 3D space-forms.
result Symmetry is broken, and more general ambient metrics are considered, leading to the failure of the conjectures.
We introduce new aspects in conformal geometry of some very natural second-order differential operators. These operators are termed shift operators. In the flat space, they are intertwining operators which are closely related to symmetry breaking differential operators. In the curved case, they are closely connected wi…
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.
Generative diffusion models are analyzed for their information dynamics.
problem Lack of a unified theoretical understanding of generative diffusion models.
method Integrated perspective connecting information-theoretic, dynamical, and thermodynamic aspects.
result Generative bandwidth is directly governed by the divergence of the score function's vector field.
Investigates spontaneous symmetry breaking in non-equilibrium systems.
problem Spontaneous symmetry breaking of ergodicity in non-equilibrium systems.
method Mathematical and effective field theory approaches to investigate symmetry breaking.
result Symmetry breaking phenomena observed in stochastic processes.
Symmetry proven for positive solutions of a weighted p-Laplace operator inequality.
problem Proving symmetry of positive solutions to a specific type of inequality.
method Analyzing positive critical points of Caffarelli-Kohn-Nirenberg inequalities with a weighted p-Laplace operator.
result Complete classification and symmetry result for positive solutions in a range of parameters.
SymPE breaks symmetries in equivariant networks, improving performance across various tasks.
problem Equivariant networks cannot break symmetries, leading to poor performance in tasks with symmetrical inputs.
method Novel equivariant conditional distributions and randomized canonicalization.
result SymPE significantly improves performance of group-equivariant and graph neural networks.
New method shows random, diverse initializations are not essential for deep neural networks.
problem The necessity of random, diverse initializations in deep neural networks.
method Constructed a deep convolutional network with identical features by initializing weights to 0, enabling signal propagation and stable gradients.
result Random, diverse initializations are not necessary for training neural networks.
Noether's framework reveals symmetry-breaking in neural networks.
problem Understanding the role of symmetry breaking in neural networks.
method Developed a theoretical framework using Lagrangian mechanics.
result Identified 'kinetic symmetry breaking' and its effect on learning dynamics.
New method breaks symmetry in neural networks, improving sample efficiency.
problem Symmetry in neural networks limits their ability to learn unique features.
method Introduces 'relaxed equivariance' to overcome symmetry limitations.
result Equivariant multilayer perceptrons (E-MLPs) can now break symmetry at the sample level.
We introduce the concept of spontaneous symmetry breaking to arbitrage modeling. In the model, the arbitrage strategy is considered as being in the symmetry breaking phase and the phase transition between arbitrage mode and no-arbitrage mode is triggered by a control parameter. We estimate the control parameter for mom…
Study on spontaneous symmetry breaking in financial markets using quantum mechanics.
problem Analyzing spontaneous symmetry breaking in financial markets.
method Using Hamiltonian form of Black-Scholes and Merton-Garman equations, analyzing symmetry breaking and interpreting Nambu-Goldstone bosons.
result Interpretation of Nambu-Goldstone bosons in financial markets.
We reveal connections between RBMs and Bosons, explaining symmetry breaking in their energy landscapes.
problem Understanding the relationships among different deep generative models and their learning mechanisms.
method Introducing a reciprocal space formulation to RBMs, revealing connections to diffusion processes and Bosons.
result Symmetry breaking in RBM energy landscapes is characterized by singular values and weight matrix eigenvectors.
Metric evaluates symmetry-breaking in datasets, revealing severe biases.
problem Symmetry-breaking in datasets can hinder the performance of symmetry-aware methods.
method Developed a metric to quantify symmetry-breaking using a two-sample classifier test.
result Symmetry-breaking can prevent optimal performance of invariant methods, even when labels are invariant.
Symmetry in inverse problems leads to multiple solutions, but breaking symmetry helps deep learning.
problem Symmetry in physical systems causes multiple solutions in inverse problems, hindering deep learning.
method Careful symmetry breaking on training data helps solve inverse problems and improve deep learning performance.
result Symmetry breaking on training data significantly improves deep learning performance in inverse problems.
Explains conformal symmetry with examples in geometry and analysis.
problem None explicitly stated; focuses on introduction.
method Introduction based on examples of Yamabe operator and its applications.
result Illustrates conformal symmetry in geometry and analysis.
New theory shows how membranes can break symmetry.
problem Understanding symmetry breaking in membranes with boundaries.
method Applied bifurcation theory and reduced membrane equation.
result Existence of symmetry breaking bifurcation in membrane solutions.
Classifies patterns of symmetry breaking and vacuum degeneracy in scalar and gauge fields.
problem Understanding patterns of symmetry breaking and vacuum degeneracy in complex field systems.
method Uses mathematical classification of singular foliations to encode and classify patterns of spontaneous symmetry breaking and vacuum degeneracy.
result Mathematical classification provides a qualitative understanding of possible patterns of vacuum degeneracy.
We construct in projective differential geometry of the real dimension 2 higher symmetry algebra of the symplectic Dirac operator ${D}\kern-0.5em\raise0.22ex\hbox{/}_s$ acting on symplectic spinors. The higher symmetry differential operators correspond to the solution space of a class of projectively invariant overde…
Transformers reduce redundancy by focusing on invariant relational quantities.
problem Substantial internal redundancy in Transformer models due to coordinate-dependent representations and continuous symmetries.
method Reformulate representations, attention mechanisms, and optimization dynamics in terms of invariant relational quantities, eliminating redundant degrees of freedom by construction.
result Architectures that operate directly on relational structures, providing a principled geometric framework for reducing parameter redundancy and analyzing optimization.
The paper connects financial vacuum conditions to spontaneous symmetry breaking in quantum finance.
problem Understanding the conditions under which the martingale condition is a non-degenerate vacuum.
method Expressing financial equations in Hamiltonian form and analyzing symmetry breaking.
result Conditions for the martingale condition to be a non-degenerate vacuum are identified.
This paper analyzes how kinetic terms in stock market equations can affect symmetry breaking.
problem Spontaneous symmetry breaking in quantum finance and its impact on stock market dynamics.
method Analyzes the role of kinetic terms in the context of the martingale condition in stock market equations.
result Kinetic terms can shift the effective location of the vacuum state, affecting symmetry breaking patterns.
On locally conformally flat manifolds we describe a construction which maps generalised conformal Killing tensors to differential operators which may act on any conformally weighted tensor bundle; the operators in the range have the property that they are symmetries of any natural conformally invariant differential ope…
DNNs can learn complex functions efficiently by breaking the curse of dimensionality.
problem Learning complex functions efficiently in high-dimensional spaces.
method Combining compositionality and symmetry learning with generalization bounds.
result DNNs can learn functions with bounded F1-norm efficiently, reducing the curse of dimensionality. Symmetric critical points lead to symmetry breaking in neural networks.
problem Understanding symmetry in critical points of invariant functions.
method Analyzing the symmetry of critical points and their neighbors in invariant nonconvex functions.
result Symmetric critical points in invariant nonconvex functions are generically followed by symmetry breaking adjacent points.
Paper proposes a method to break symmetries in Bayesian matrix factorization.
problem Symmetries in posterior distribution reduce MCMC sampling efficiency.
method Modification to Gaussian prior mean and covariance to break symmetries.
result Breaking symmetries leads to lower autocorrelation and reconstruction errors.
New operators generalize Michelsohn's on almost Hermitian manifolds.
problem Generalizing differential operators to almost Hermitian manifolds.
method Introducing two differential operators on sections of the complex Clifford bundle over compact almost Hermitian manifolds.
result Surprising Kähler-like symmetries in the kernel of the Laplacians of these operators.
Based on operator identities and their formal adjoints, we derive two symmetry operators for the linearized Einstein operator on vacuum backgrounds of Petrov type D and in particular the Kerr spacetime. One of them is of differential order four and coincides with a result of Cohen and Kegeles. The other one is a new op…
We exploit the symmetry concepts developed in the companion review of this article to introduce a stochastic version of link reversal symmetry, which leads to an improved understanding of the reciprocity of directed networks. We apply our formalism to the international trade network and show that a strong embedding in …
We construct continuously parametrised families of conformally invariant boundary operators on densities. These may also be viewed as conformally covariant boundary operators on functions and generalise to higher orders the first-order conformal Robin operator and an analogous third-order operator of Chang-Qing. Our fa…
Continuous symmetries and their breaking play a prominent role in contemporary physics. Effective low-energy field theories around symmetry breaking states explain diverse phenomena such as superconductivity, magnetism, and the mass of nucleons. We show that such field theories can also be a useful tool in machine lear…