arXiv research

Analyzes Berry phases and connection matrices on Siegel-Jacobi spaces.

Study local topological constraints on Berry curvature in spin-orbit coupled Bose-Einstein condensates.

We explain how the kind of ``parallel transport'' of a wavefunction used in discussing the Berry or Geometrical phase induces the conventional parallel transport of certain real vectors. These real vectors are associated with operators whose commutators yield diagonal operators; or in Lie algebras those operators whose…

Geometric observables detect financial regime shifts with high accuracy.

Study of rotation angles in a rotating disc model.

We analyze 2-dimensional Ginzburg-Landau vortices at critical coupling, and establish asymptotic formulas for the tangent vectors of the vortex moduli space using theorems of Taubes and Bradlow. We then compute the corresponding Berry curvature and holonomy in the large volume limit.

In this paper, we discuss some aspects of the averaging method for Poisson connections on foliated manifolds with symmetry generalizing the previous results on the Hannay-Berry connections on fibrations due to \cite{Mn-88,MaMoRa-90} which play an important role in the normal form theory for Hamiltonian systems of adiab…

Study Berry connections for 2d GLSMs, linking to cohomology theories.

2d GLSM connects Berry connections to Coulomb branch via difference equations.

Researchers extend topological classification to knotted semimetals in 3D.

Tying knots and linking microscopic loops of polymers, macromolecules, or defect lines in complex materials is a challenging task for material scientists. We demonstrate the knotting of microscopic topological defect lines in chiral nematic liquid crystal colloids into knots and links of arbitrary complexity by using l…

New statistical test for change-point detection using relative entropy.

A treatment of the spin-statistics relation in nonrelativistic quantum mechanics due to Berry and Robbins [Proc. R. Soc. Lond. A (1997) 453, 1771-1790] is generalised within a group-theoretical framework. The construction of Berry and Robbins is re-formulated in terms of certain locally flat vector bundles over n-parti…

We study supersymmetric deformations of N = 4 quantum mechanics with a Kahler target space admitting a holomorphic isometry. We show that the twisted mass deformation generalises to a deformation constructed from matrix-valued functions of the moment map, which obey the Nahm equations. We also explain how N = 4 supersy…

A new geometric framework resolves singularities in anomalous transport.

Paper improves confidence intervals for LSA with multiplier bootstrap.

Paper develops efficient incomplete U-statistics for degenerate cases.

Deep Gaussian Processes with polynomial kernels can collapse rapidly without proper hyperparameter tuning.

We give sufficient conditions for the existence of a Dirac structure on the total space of a Poisson fiber bundle endowed with a compatible connection. We also show that Cartan and Cartan-Hannay-Berry connections give rise to coupling Dirac structures.

Paper derives convergence rates and confidence intervals for LSA with Markovian noise.

New bounds for SGD in high dimensions improve inference efficiency.

Study on variance of Laplace eigenfunctions on manifolds.

In the space $\mathbb U^4$ of cubic forms of surfaces, regarded as a $G$-space and endowed with a natural invariant metric, the ratio of the volumes of those representing umbilic points with negative to those with positive indexes is evaluated in terms of the asymmetry of the metric, defined here. A connection of this …

Study of 2d gauged linear sigma models to derive difference equations and spectral data.

Kernel smoothing on unknown manifolds with bounds and asymptotic normality.

Paper stabilizes bandit learning with regularization, improving inference under adaptive sampling.

In this paper we show how to place Michael Berry's discovery of knotted zeros in the quantum states of hydrogen in the context of general knot theory and in the context of our formulations for quantum knots. Berry gave a time independent wave function for hydrogen, as a map from three space to the complex plane and suc…

Paper develops robust methods for large-scale testing without tuning parameters.

We give a comprehensive theoretical characterization of a nonparametric estimator for the $L_2^2$ divergence between two continuous distributions. We first bound the rate of convergence of our estimator, showing that it is $\sqrt{n}$-consistent provided the densities are sufficiently smooth. In this smooth regime, we t…

Mondrian random forests improve statistical inference for regression.

Smartphone app counts grapes for accurate yield estimation.

We study a simple model of bicycle motion: a segment of fixed length in multi-dimensional Euclidean space, moving so that the velocity of the rear end is always aligned with the segment. If the front track is prescribed, the trajectory of the rear wheel is uniquely determined via a certain first order differential equa…

Study on volumes of random inscribed polytopes in projective geometries.

QCML uses quantum geometry to represent data.

We give a positive answer to the Berry-Robbins problem for any compact Lie group G, i.e. we show the existence of a smooth W-equivariant map from the space of regular triples in a Cartan subalgebra to the flag manifold G/T. This map is constructed via solutions to Nahm's equations and it is compatible with the SO(3) ac…

Random forests remain among the most popular off-the-shelf supervised learning algorithms. Despite their well-documented empirical success, however, until recently, few theoretical results were available to describe their performance and behavior. In this work we push beyond recent work on consistency and asymptotic no…

The paper provides Gaussian approximations for decentralized Federated Learning.

We apply the geometric-topology surgery theory on spacetime manifolds to study the constraints of quantum statistics data in 2+1 and 3+1 spacetime dimensions. First, we introduce the fusion data for worldline and worldsheet operators capable creating anyon excitations of particles and strings, well-defined in gapped st…

Solves Dirichlet problem for Lagrangian phase equation with critical and supercritical phase.

Berry et al. (1997) initiated the development of the infinite arms bandit problem. They derived a regret lower bound of all allocation strategies for Bernoulli rewards with uniform priors, and proposed strategies based on success runs. Bonald and Proutière (2013) proposed a two-target algorithm that achieves the regret…

Motivated by a question of Rubel, we consider the problem of characterizing which noncompact hypersurfaces in $\RR^n$ can be regular level sets of a harmonic function modulo a $C^\infty$ diffeomorphism, as well as certain generalizations to other PDEs. We prove a versatile sufficient condition that shows, in particular…

We consider the problem of providing nonparametric confidence guarantees for undirected graphs under weak assumptions. In particular, we do not assume sparsity, incoherence or Normality. We allow the dimension $D$ to increase with the sample size $n$. First, we prove lower bounds that show that if we want accurate infe…

Novel bounds improve TD learning consistency in RL.

A major issue in harmonic analysis is to capture the phase dependence of frequency representations, which carries important signal properties. It seems that convolutional neural networks have found a way. Over time-series and images, convolutional networks often learn a first layer of filters which are well localized i…

Paper develops Gaussian approximations and bootstrap for federated LSA with trade-off bounds.

New algorithms handle phase retrieval with rank d measurements, revealing phase transitions.

New estimators improve efficiency in two-phase designs with coarsened data.

Paper improves CLT and bootstrap approximations for LSA with decreasing step size.