New method uses Coulomb gases for Monte Carlo integration with reduced errors.
problem Reducing integration errors in numerical algorithms.
method Using Gibbs measures with a large deviations approach.
result Preserves large deviation principle for improved integration.
CoulGAT interprets GAT models by analyzing node interactions.
problem Understanding and interpreting the complex interactions within graph attention networks.
method Developed a CoulGAT framework to analyze and interpret GAT model layers and datasets.
result Extracted node-node and node-feature interactions to define a standard model for graph structure.
We study the non-asymptotic behavior of a Coulomb gas on a compact Riemannian manifold. This gas is a symmetric n-particle Gibbs measure associated to the two-body interaction energy given by the Green function. We encode such a particle system by using an empirical measure. Our main result is a concentration inequalit…
Compactification of AdS5 allows studying meson behavior in QCD.
problem Understanding meson behavior in Quantum Chromodynamics (QCD).
method Deforming AdS5 metric to model Coulomb interaction between charges.
result Proposed conformal deformation provides a quantum mechanical description of mesons.
Quantized Coulomb branches linked to skein algebras.
problem Understanding the relationship between quantized Coulomb branches and skein algebras.
method Association of quantized Coulomb branches to surfaces, description of relationship for specific surfaces, formulation of a conjecture.
result A conjecture linking quantized Coulomb branches and skein algebras.
We study quantized Coulomb branches of quiver gauge theories of Jordan type. We prove that the quantized Coulomb branch is isomorphic to the spherical graded Cherednik algebra in the unframed case, and is isomorphic to the spherical cyclotomic rational Cherednik algebra in the framed case. We also prove that the quanti…
Study of Hilbert schemes and Coulomb branches of hypertoric varieties.
problem Understanding the geometry and topology of Coulomb branches of hypertoric varieties.
method Investigation of transverse equivariant Hilbert schemes and Hamiltonian reductions, proposing new metrics.
result Coulomb branches of hypertoric varieties can be constructed as Hilbert schemes or Hamiltonian reductions.
Paper analyzes convergence rates of mean-field SVGD method.
problem Establishing quantitative rates of convergence for mean-field SVGD.
method Quantitative analysis of mean-field SVGD dynamics on torus.
result Explicit polynomial convergence rates in L2-norm for Riesz-type kernels.
We calculate the free energy of Coulomb gas systems on Riemann surfaces.
problem Analyzing the free energy of Coulomb gas systems on Riemann surfaces.
method Using bosonization formula and analytic torsion, we derive the asymptotic expansion of the partition function.
result We prove the geometric version of the Zabrodin-Wiegmann conjecture in the determinantal case.
Study quantized SL2-character variety of a once-punctured torus, finding three Coulomb branch isomorphisms.
problem Understanding the structure of quantized SL2-character variety of a once-punctured torus.
method Analyzing the quantized algebra and its subalgebras isomorphic to Coulomb branches.
result Three Z2-invariant subalgebras of the quantized algebra are isomorphic to Coulomb branches. Learning of low dimensional structure in multidimensional data is a canonical problem in machine learning. One common approach is to suppose that the observed data are close to a lower-dimensional smooth manifold. There are a rich variety of manifold learning methods available, which allow mapping of data points to the…
We establish existence and regularity results for normal Coulomb frames in the normal bundle of two-dimensional surfaces of disc-type embedded in Euclidean spaces of higher dimensions.
Floer homology connects to quiver Hecke algebras in Coulomb branches.
problem Understanding the representation theory of quiver Hecke algebras.
method Using Lagrangian Floer homology in multiplicative Coulomb branches.
result Cylindrical KLRW algebras are realized by Floer homology.
This is an introduction to a provisional mathematical definition of Coulomb branches of 3-dimensional N=4 supersymmetric gauge theories, studied in arXiv:1503.03676, arXiv:1601.03586
Paper proves existence of minimal doublings on surfaces with specific properties.
problem Existence of minimal doublings on surfaces with given properties.
method Variational approach to finding nondegenerate critical points of a Coulomb-type energy.
result Proves existence of minimal doublings for surfaces of index one in a generic 3-manifold.
Proposes a Coulomb-like model for international trade flows, fitting real-world data.
problem Describing and predicting international trade flows between countries.
method Formulated a coulomb force model where GDP represents charge and distance is influenced by various factors.
result Developed a trade strength distribution equation that fits real-world data well.
Generative adversarial networks (GANs) evolved into one of the most successful unsupervised techniques for generating realistic images. Even though it has recently been shown that GAN training converges, GAN models often end up in local Nash equilibria that are associated with mode collapse or otherwise fail to model t…
Consider the 3-dimensional N=4 supersymmetric gauge theory associated with a compact Lie group Gc and its quaternionic representation M. Physicists study its Coulomb branch, which is a noncompact hyper-Kähler manifold with an SU(2)-action, possibly with singularities. We give a math…
2d GLSM connects Berry connections to Coulomb branch via difference equations.
problem Connecting Berry connections to Coulomb branch via difference equations.
method Boundary 2d GLSM, 3d A-twisted gauge theory, spectral data of monopoles.
result Coulomb branch algebra actions derived from 2d GLSM.
Method identifies causal interactions between time series using extreme eigenvalue variability.
problem Detecting causal interactions between time series.
method Largest eigenvalue of lagged correlation matrices, measuring causal interactions through variability.
result The method outperforms traditional Granger causality tests in detecting structural changes.
The paper connects G2-manifolds to Coulomb and Higgs phases of gauge theories.
problem Exploring the physical interpretation of special singularities in G2-holonomy manifolds. method Analyzing desingularizations of orbifold singularities and relating them to gauge theories.
result Shows an isomorphism between moduli spaces of Ricci flat metrics and flat ADE-connections.
This is a companion paper of arXiv:1601.03586. We study Coulomb branches of unframed and framed quiver gauge theories of type ADE. In the unframed case they are isomorphic to the moduli space of based rational maps from CP1 to the flag variety. In the framed case they are slices in the affine Grassmannia…
Tripod spiders' energy control analyzed for Hooke and Coulomb potentials.
problem Control of tripod spiders' energy configurations.
method Morse theory for Hooke potential, stationary charges for Coulomb energy.
result For positive charges in a regular triangle, the domain of robust control is non-void.
This is an introduction to a provisional mathematical definition of Coulomb branches of 3-dimensional N=4 supersymmetric gauge theories, studied in arXiv:1503.03676, arXiv:1601.03586. This is an expanded version of an article arXiv:1612.09014 appeared in the 61st DAISUUGAKU symposium proceeding (2016), wri…
This is the third companion paper of arXiv:1601.03586. When a gauge theory has a flavor symmetry group, we construct a partial resolution of the Coulomb branch as a variant of the definition. We identify the partial resolution with a partial resolution of a generalized slice in the affine Grassmannian, Hilbert scheme o…
We introduce a novel class of localized atomic environment representations, based upon the Coulomb matrix. By combining these functions with the Gaussian approximation potential approach, we present LC-GAP, a new system for generating atomic potentials through machine learning (ML). Tests on the QM7, QM7b and GDB9 biom…
New gauge condition fixes L2 metric divergence in hyperbolic monopole spaces.
problem Divergence of L2 metric on hyperbolic monopole moduli spaces. method Alternative gauge-fixing condition inspired by supersymmetry.
result Resulting geometry is hyperbolic hyperkähler, analogous to Euclidean monopole spaces.
Smooth Yang-Mills fields proved in supercritical dimensions.
problem Regularity of weak Yang-Mills connections in high dimensions.
method ε-regularity theorem, Coulomb gauges construction.
result Stationary Yang-Mills fields are smooth away from a small singular set.
In this paper we consider the existence and regularity problem for Coulomb frames in the normal bundle of two-dimensional surfaces with higher codimension in Euclidean spaces. While the case of two codimensions can be approached directly by potential theory, more sophisticated methods have to be applied for codimension…
Study analyzes low-energy behavior of Schrödinger operators with Coulomb potentials.
problem Analyzing the limiting resolvent of Schrödinger operators at low energies.
method Using Vasy's second microlocal approach (Lagrangian approach), uniformly analyzing the resolvent from E=0. result Obtained oscillatory asymptotics for the resolvent output at low energy, differing from short-range cases.
New theorems in 2D and 4D for metrics with curvature or singularity.
problem Proving new versions of Huber theorem in dimensions 2 and 4.
method Using Coulomb frames and Bach tensor conditions to construct conformal metrics.
result Constructs conformal metrics with regularity across singularities in 4D.
Extends Coulomb gauge existence to non-associative gauge theory.
problem Existence of configurations with divergence-free torsion in non-associative gauge theory.
method Analyzes non-associative gauge theory based on smooth loops and their tangent algebras, proving existence of configurations with divergence-free torsion.
result Proves existence of configurations with divergence-free torsion given a sufficiently small torsion in a Sobolev norm.
Consider the 3-dimensional N=4 supersymmetric gauge theory associated with a compact Lie group G and its quaternionic representation M. Physicists study its Coulomb branch, which is a noncompact hyper-Kähler manifold, such as instanton moduli spaces on R4, SU(2)-monopole moduli spa…
The study shows how to regularize weakly harmonic maps using Sobolev norms and Coulomb frames.
problem Regularity of weakly harmonic maps between Riemannian manifolds.
method New structure equations and Coulomb-frame methods combined with Hardy-BMO duality.
result Sufficient conditions on Sobolev norms ensure full regularity of weakly harmonic maps.
Study on Wasserstein gradient flow for MMD between Coulomb measures.
problem Analyzing the long-time behavior of MMD between probability and target measures using Coulomb kernels.
method Existence of global weak solutions, ultracontractive estimate, regularity analysis, exponential decay proof, defective Polyak-Lojasiewicz inequality.
result Exponential decay of squared MMD toward a uniformly positive target measure on flat torus.
This is a supplement to [arXiv:1503.03676], where an approach towards a mathematically rigorous definition of the Coulomb branch of a 3-dimensional N=4 SUSY gauge theory was proposed. We ask questions on their expected properties, especially in relation to the corresponding Higgs branch, partly motivated b…
The paper recasts Penrose-Sparling's non-Hausdorff twistor space using noncommutative geometry.
problem Reinterpreting Penrose-Sparling's non-Hausdorff twistor space.
method Introduces noncommutative geometry techniques to reinterpret the space, using explicit etale gluing groupoid and convolution algebra.
result The source-adapted cyclic pairing recovers the Coulomb charge, demonstrating the effectiveness of the new algebraic model.
Narasimhan and Ramadas showed that the Gribov ambiguity was maximal for the product SU(2) bundle over S^3. Specifically they showed that the holonomy group of the Coulomb connection is dense in the connected component of the identity of the gauge group. Instead of base manifold S^3, we consider here a base manifold wit…
Emulator speeds up landslide run-out modeling sensitivity analysis.
problem Computational challenges in assessing landslide run-out model sensitivity.
method Gaussian process emulation integrated into r.avaflow.
result Strong interactions detected between friction coefficients and release volume.
Monoidal categorifies genus zero skein algebra using K-theory.
problem Relating skein algebra to K-theory.
method Using quantized K-theoretic Coulomb branch and Grothendieck ring of equivariant coherent sheaves.
result Monoidal categorification of the skein algebra.
Differential Cohomotopy theory predicts brane interactions via chord diagrams.
problem Quantization of brane charges and moduli spaces.
method Differential refinement of Cohomotopy theory, configuration spaces, chord diagrams.
result Higher observables on brane moduli spaces are given by weight systems on chord diagrams.
This is the second companion paper of arXiv:1601.03586. We consider the morphism from the variety of triples introduced in arXiv:1601.03586 to the affine Grassmannian. The direct image of the dualizing complex is a ring object in the equivariant derived category on the affine Grassmannian (equivariant derived Satake ca…
Let A be a W1,2-connection on a principle SU(2)-bundle P over a compact 4-manifold M whose curvature FA satisfies ∥FA∥L2(M)≤Λ. Our main result is the existence of a global section σ:M→P with finite singularities on M such that the connection form σ∗A satisfies the Coulomb…
New approach reduces particle simulation complexity to linear time and space.
problem Challenges in learning dynamics from particle interactions, especially N-body problems.
method Transforms fully-connected interaction graphs into hierarchical ones, reducing complexity.
result Linear time and space complexity for large-scale simulations, retaining high accuracy.
Study on Yang-Mills fields blow-up in 4D, proving certain configurations impossible.
problem Prohibiting specific configurations of Yang-Mills fields in 4D.
method Expanding connection forms on long cylinders, proving equations relating bubble and limit connections.
result Proves certain configurations of Yang-Mills fields in 4D are impossible.
Study regularity of Schrödinger eigenfunctions with Coulomb-type potentials.
problem Regularity of eigenfunctions for Schrödinger operators with singular potentials.
method Blow-ups of manifolds with corners and Lie manifolds.
result Proves regularity estimates in weighted Sobolev spaces for eigenfunctions.
New G2-holonomy manifolds from 5d N=1 theories domain walls.
problem Geometrizing domain walls in 5d N=1 theories.
method Constructing 7-manifolds by fibering a Calabi-Yau over a real line.
result 7-manifolds with G2-holonomy from domain walls in 5d theories. Study topological twists of massive SQCD with gauge group SU(2) and 3 or fewer fundamental hypermultiplets.
problem Evaluate topological partition functions of massive SQCD with arbitrary gauge bundles and hypermultiplet masses.
method Develop techniques for evaluating low-energy path integrals on the Coulomb branch.
result Formulate theories for arbitrary gauge bundles on compact four-manifolds.