Study hidden symmetry in global analysis of spaces.
problem Relationship between representations of groups and global analysis.
method Extension of G'-action to G, analysis of branching laws.
result Trick transfers finite to infinite-dimensional results.
The paper examines the geometry of a curve's centre symmetry set.
problem Global geometrical properties of a curve's centre symmetry set.
method Study of the envelope of affine chords.
result Number of singularities and asymptotes of the centre symmetry set.
We consider a generalization of the notion of a natural mechanical system to the case of additional forces of gyroscopic type. Such forces appear, for example, as a result of global reduction of a natural system with symmetry. We study symmetries in the systems with gyroscopic forces to find out when these systems admi…
Study shows no global symmetries for Kähler metrics on compact manifolds.
problem Identifying symmetries in the space of Kähler metrics on compact manifolds.
method Proving isometries are induced by biholomorphisms and anti-biholomorphisms, analyzing Mabuchi's metric and its completion.
result No global symmetries exist for Kähler metrics on compact manifolds, including for Mabuchi's metric.
Defines risk-free portfolios and risk-free rate using gauge symmetries.
problem Identifying a consistent definition of risk-free rate in economics.
method Introduces three gauge invariant differential operators to define risk-free portfolios and identifies the risk-free rate as the return of an infinitely diversified portfolio.
result Identifies the risk-free rate as the return of an infinitely diversified portfolio and connects it to global price rescaling as a gauge symmetry.
We prove several global existence theorems for spacetimes with toroidal or hyperbolic symmetry with respect to a geometrically defined time. More specifically, we prove that generically, the maximal Cauchy development of T2-symmetric initial data with positive cosmological constant Λ>0, in the vacuum or with Vlaso…
Theory explains symmetry and saddle points in nonconvex optimization landscapes.
problem Understanding the optimization landscape of nonconvex matrix factorization problems.
method Characterizing stationary points and null spaces via invariant groups.
result Identifies infinitely many nonisolated strict saddle points and global minima.
L-CNNs maintain gauge symmetry on non-Abelian lattice theories.
problem Applying convolutional neural networks to non-Abelian lattice gauge theories while preserving gauge symmetry.
method Developed a geometric formulation of L-CNNs that are equivariant under global symmetries and gauge transformations.
result Convolutional operations in L-CNNs are a specific case of gauge-equivariant neural networks on SU(N) principal bundles. Willmore flow converges globally for surfaces with rotational symmetry below a specific energy threshold.
problem Global existence and convergence of Willmore flow with Dirichlet boundary conditions.
method Considered surfaces with rotational symmetry, proved global existence and convergence for initial data below a sharp energy threshold.
result Sharp threshold for global existence and convergence of Willmore flow depends on boundary conditions.
We investigate the initial value problem for the Einstein-Euler equations of general relativity under the assumption of Gowdy symmetry on T3, and we construct matter spacetimes with low regularity. These spacetimes admit, both, impulsive gravitational waves in the metric (for instance, Dirac mass curvature singularitie…
The paper creates symmetrical discrete minimal nets using Schwarz reflection.
problem Creating symmetrical discrete minimal nets.
method Extending Schwarz reflection principle to discrete minimal surfaces.
result Global examples of discrete minimal nets with high symmetry.
Method implements symmetries in TQFT for finite groups.
problem Implementing symmetries in TQFT for finite groups.
method Cohomology classes and bordisms with a (d−q−2)-skeleton. result Generalizes existing methods to arbitrary dimensions and q.
A new algebraic method extracts symmetry anomalies from 5D SCFTs.
problem Extracting global symmetry anomalies from 5D superconformal field theories.
method Path algebra of branes probing Calabi-Yau cones provides a complementary approach.
result Combinatorial approach to symmetry anomalies in 5D SCFTs.
Local normal forms for symmetrical contact structures on 3-manifolds.
problem Understanding symmetrical contact structures on 3-manifolds.
method Determining local normal forms for pairs of transverse contact distributions with symmetries.
result Orientable Anosov flows can be globally represented by intersecting contact distributions with maximal symmetries.
Under weak regularity assumptions, only, we develop a fully geometric theory of vacuum Einstein spacetimes with T2 symmetry, establish the global well-posedness of the initial value problem for Einstein's field equations, and investigate the global causal structure of the constructed spacetimes. Our weak regularity ass…
Singular fiber resolution does not describe the spontaneous breaking of gauge symmetry in F-theory, as the corresponding branch of the moduli space does not exist in the theory. Accordingly, even non-abelian gauge theories have not been fully understood in global F-theory compactifications. We present a systematic disc…
Using the symmetry group theory of second order PDEs, one finds the symmetry group associated to Tzitzeica surfaces partial differential equation. One studies the inverse problem and one shows that the Tzitzeica surfaces PDE is an Euler-Lagrange equation. One determines the variational symmetry group of the associated …
A BV algebra is a formal framework within which the BV quantization algorithm is implemented. In addition to the gauge symmetry, encoded in the BV master equation, the master action often exhibits further global symmetries, which may be in turn gauged. We show how to carry this out in a BV algebraic set up. Depending o…
Survey of tractable nonconvex problems using symmetry.
problem Nonlinear models with symmetries create complex, nonconvex objective landscapes.
method Analysis of geometric structure and symmetry roles.
result Efficient methods can find global minimizers due to symmetry.
Study of a series of Lorentzian structures on SL(2,R) with SO(1,1) symmetry.
problem Global optimality of extremal trajectories in a series of Lorentzian structures.
method Analysis of a one-parametric series of left-invariant Lorentzian structures on SL(2,R) with SO(1,1) symmetry.
result Properties of the Lorentzian structures deform to those of the sub-Lorentzian structure in a limit case.
Conserved quantities on multisymplectic manifolds extend symplectic geometry results.
problem Defining and analyzing conserved quantities on multisymplectic manifolds.
method Defining globally conserved quantities as Lie derivatives of exact forms, extending classical results in symplectic geometry.
result A family of globally conserved quantities can be obtained under certain symmetries and homotopy co-momentum maps.
New method constructs geometric flat outputs for robotic systems using symmetry.
problem Finding flat outputs for arbitrary robotic systems remains an open question.
method Employing symmetry directly to construct a flat output.
result Demonstrated geometric flat outputs for various robotic systems.
Summarizes quantum field theories with discrete symmetry, classifying representations and anomalies.
problem Classifying representations and anomalies in quantum field theories with discrete symmetry.
method Classification of representations and anomalies using the ring of profinite integers.
result Rich and complex classification of representations and anomalies.
Characterizes conjugacy classes of pairs in SU(3,1) using symmetry.
problem Classifying conjugacy classes of pairs of matrices in SU(3,1).
method Minimal global coordinate system on SL(4,C)-character variety, symmetry analysis.
result 22 coordinates subject to 5 relations determine conjugacy classes.
We prove a geometrically meaningful stochastic representation of the derivative of the heat semigroup on sub-Riemannian manifolds with tranverse symmetries. This representation is obtained from the study of Bochner-Weitzenbock type formulas for sub-Laplacians on 1-forms. As a consequence, we prove new hypoelliptic heat…
Unique steady and expanding solitons with spherical links identified.
problem Characterizing steady and expanding Ricci solitons with specific asymptotic symmetries.
method Symmetry principle applied to asymptotically cylindrical and conical GRSs, proving uniqueness for Bryant solitons.
result Bryant steady and expanding solitons are the unique asymptotically cylindrical and conical GRSs with spherical links under certain conditions.
The paper uncovers symmetries in large language models through layer-peeled optimization.
problem Understanding geometric structure in large language model weights and context embeddings.
method Constrained layer-peeled optimization program to analyze symmetries in next-token distributions.
result Symmetries in target next-token distributions are transferred to optimal model weights and context embeddings.
Study describes global existence and convergence of flows on surfaces and fibrations.
problem Global existence and convergence of flows on surfaces and fibrations.
method Complete description of Ricci-Yang-Mills flow and pluriclosed flow on Tk bundles over Riemann surfaces. result Equivalence of solutions to generalized Ricci flow and pluriclosed flow with symmetry.
Study of symmetries in 2D Yang-Mills theory, including orbifolds and higher forms.
problem Understanding symmetries and anomalies in 2D Yang-Mills theory.
method Combining continuum methods, topological defects, and higher gauge theory.
result Unified description of higher and lower form gauge fields, identifying spontaneous symmetry breaking.
6D manifolds with special symmetries have limited automorphism groups.
problem Understanding the maximum symmetry groups of 6D manifolds.
method Analyzing almost product structures and their automorphism groups.
result The automorphism group of 6D manifolds with non-degenerate Nijenhuis tensor has a limited dimension, with the maximum being 14.
New theorem links symmetries to first integrals in plasma physics.
problem Understanding the relationship between symmetries and first integrals in divergence-free fields.
method Developed a Noether-type Theorem reformulation for three-dimensional divergence-free vector fields.
result Converse of the Noether-type Theorem holds on the toroidal region, proving the existence of flux coordinates.
We classify higher-SPTs and their anomalies via cobordism theory.
problem Understanding higher symmetries and anomalies in quantum field theories.
method Developed a generalized cobordism theory using advanced mathematical tools.
result Classified higher-SPTs and their boundary anomalies.
Study very stable Higgs bundles on Riemann surfaces, linking to multiplicity and mirror symmetry.
problem Existence and properties of very stable Higgs bundles.
method Bialynicki-Birula theory, C∗-actions, Hecke transformations, Fourier-Mukai transforms. result Precise formula for multiplicity of very stable components of global nilpotent cone.
Study of partially-massless higher-spin algebras in (A)dSd+1.
problem Understanding global symmetries of partially-massless higher-spin fields.
method Defined and quotiented ideals in the universal enveloping algebra of (A)dSd+1 isometries, derived bilinear forms, and used Howe duality. result Explicit calculation of bilinear forms revealing additional ideals for non-negative integer values of ℓ−d/2. This research bridges Killing vectors and Lie algebras through induced vector fields.
problem Understanding the relationship between Killing vector fields and Lie algebras.
method Defining and exploring induced vector fields to connect Killing vector fields with isometry Lie groups.
result Established a new connection between Killing vector fields and Lie algebras.
Vacuum spacetimes with conformal symmetry are rigid and split.
problem Understanding the rigidity of vacuum spacetimes under conformal symmetry.
method Analyzing globally hyperbolic vacuum spacetimes with timelike conformal Killing fields.
result Rigidity result for vacuum spacetimes with compact Cauchy surfaces and timelike conformal Killing fields.
Latent MoS learns multiple symmetries for efficient dynamic learning.
problem Efficiently learning dynamics from limited system measurements.
method Latent Mixture of Symmetries (Latent MoS) with hierarchical architecture.
result Latent MoS outperforms baselines in interpolation and extrapolation tasks.
New insights into 4d YM and 5d topological field theories with higher symmetries.
problem Exploring new topological field theories with higher symmetries.
method Dynamic gauging of 1-form symmetry, higher anomalies, and lattice simplicial complex regularizations.
result Discovery of new higher-form gauge fields and exotic anyonic statistics.
Characterizes conical angles for metrics with dihedral symmetry.
problem Understanding metrics with specific symmetry properties.
method Using recent results on local invariants of quadratic differentials.
result Complete characterization of conical angles for dihedral spherical metrics.
Our research proves neural collapse in deep ResNets and transformers is globally optimal.
problem Understanding neural collapse in deep learning models.
method Analysis of deep regularized transformers and ResNets trained with cross entropy or mean squared error loss.
result Global optima of deep regularized transformers and ResNets are approximately collapsed, becoming more prominent as depth increases.
Gauge equivariant CNNs improve image segmentation and climate pattern analysis.
problem Improving image segmentation and climate pattern analysis on non-Euclidean surfaces.
method Developed gauge equivariant convolutional neural networks (gCNNs) for icosahedral surfaces.
result Significant improvements in image segmentation and climate pattern analysis.
New mesh network preserves symmetries in deep learning.
problem No existing mesh processing architecture is equivariant to all symmetries.
method Equivariant attention-based mesh network using relative tangential features.
result The network achieves improved performance and is equivariant to various transformations.
Study on naked singularities without symmetry, forming incomplete future null infinity and singular inner Cauchy horizon.
problem Formation of naked singularities in Einstein-scalar field system without symmetry assumptions.
method Employing four-type differences and scale-invariant weighted norms to control geometry.
result Global naked singularity structure with incomplete future null infinity and singular inner Cauchy horizon.
Deep networks exhibit permutation saddles and valleys between equivalent minima.
problem Understanding the structure of loss landscapes in deep neural networks.
method Geometric approach to constructing paths between equivalent minima and saddle points.
result Existence of permutation saddles and valleys in deep neural networks.
The possibility of the global Lagrangian reduction of a mechanical system with symmetry is shown to be connected with the characteristic class of a principal fiber bundle of the configuration space over the factor manifold. It is proved that the reduced system is globally Lagrangian if and only if the product of the mo…
Study quantum aspects of 1-form symmetries using BV-BRST cohomology.
problem Quantum aspects of gauging continuous 1-form global symmetries.
method BV-BRST quantization of a U(1) 2-form gauge field, Lie 2-algebroid construction, Čech-de Rham bicomplex.
result Anomaly descent for U(1) 1-form symmetries is naturally set up in the Čech-de Rham bicomplex.
Paper proves trapped surface formation for EMCSF system without symmetry assumptions.
problem Formation of trapped surfaces for the Einstein--Maxwell--charged scalar field system.
method Scale-critical trapped surface formation result established from past null infinity.
result Focusing of gravitational waves, concentration of electromagnetic fields, or condensation of scalar fields can lead to trapped surface formation.
Proves transversality for local Morse homology with symmetries.
problem Defining local Morse chain complexes with finite cyclic group symmetry.
method Special regularized distance functions and inductive perturbation process.
result Global existence theorem for symmetric Morse-Smale pairs.