The Toda hierarchy and the KdV hierarchy

We show that spin generalization of elliptic Calogero-Moser system, elliptic extension of Gaudin model and their cousins are the degenerations of Hitchin systems. Applications to the constructions of integrals of motion, angle-action variables and quantum systems are discussed. The constructions of classical systems are motivated by Conformal Field Theory, and their quantum counterparts can be thought of as being the degenerations of the critical level Knizhnik-Zamolodchikov-Bernard equations.     

The relation between the Toda hierarchy and the KdV hierarchy

Under three relations connecting the field variables of Toda flows and that of KdV flows, we present three new sequences of combinations of the equations in the Toda hierarchy which have the KdV hierarchy as a continuous limit. The relation between the Poisson structures of the KdV hierarchy and the Toda hierarchy in the continuous limit is also studied.     

A hierarchy hypothesis

Why attraction and repulsion between likes should not enjoy equal status in nature is considered. By postulating a hierarchy of isolated systems of finite radii whose associated charges form a geometric series with enormous imaginary common ratio, and by identifying a universe (the content of an infinite cosmos within a Hubble radius of an observer), an electron, and a neutrino as three consecutive members of the hierarchy (in fact the only three observable because of the uncertainty principle), it is possible to treat gravitational and electromagnetic phenomena as perfectly analogous and complementary for the overall structure of the cosmos. An isolated system behaves, from an external viewpoint, as an elementary particle, and from an internal viewpoint, as a universe. Remarkable relationships between physical constants emerge.     

Algebraic study on the super-KP hierarchy and the ortho-symplectic super-KP hierarchy

Bilinear residue formulas are established for the super-KP hierarchy and the ortho-symplectic super-KP hierarchy. Furthermore, superframes corresponding to the ortho-symplectic super-KP hierarchy are completely characterized. Soliton solutions to the super-KP hierarchy are given.     

Geometric dequantization

Dequantization is a set of rules which turn quantum mechanics (QM) into classical mechanics (CM). It is not the WKB limit of QM. In this paper we show that, by extending time to a 3-dimensional “supertime,” we can dequantize the system in the sense of turning the Feynman path integral version of QM into the functional counterpart of the Koopman-von Neumann operatorial approach to CM. Somehow this procedure is the inverse of geometric quantization and we present it in three different polarizations: the Schrödinger, the momentum and the coherent states ones.     

Geometric transitions

The purpose of this paper is to give, on one hand, a mathematical exposition of the main topological and geometrical properties of geometric transitions, on the other hand, a quick outline of their principal applications, both in mathematics and in physics.     

Geometric fermions

We study the Kähler-Dirac equation which linearizes the laplacian on the space of antisymmetric tensor fields. In flat space-time it is equivalent to the Dirac equation with internal symmetry and on the lattice it reproduces Susskind fermions. The KD equation in curved space-time differs from the Dirac equation by coupling the gravitational field to the internal symmetry generators. This new way of treating fermionic degrees of freedom may lead to a solution of the generation puzzle but is in conflict with the equivalence principle and with Lorentz invariance on the Planck-mass scale.     

The generalized KP hierarchy

The KP hierarchy has been extended to a large set of integrable hierarchies. The main idea is to utilize the additional symmetry of the KP hierarchy, although not commuting among themselves. But we found that their proper combinations do commute with the original KP flows and among themselves, so as to give rise to an enlarged integrable hierarchy, which we call the generalized KP hierarchy.     

Geometric Landau-Zener interferometry

We propose a new type of interferometry, based on geometric phases accumulated by a periodically driven two-level system undergoing multiple Landau-Zener transitions. As a specific example, we study its implementation in a superconducting charge pump. We find that interference patterns appear as a function of the pumping frequency and the phase bias, and clearly manifest themselves in the pumped charge. We also show that the effects described should persist in the presence of realistic decoherence.     

Geometric statistical inference

A reparametrization-covariant formulation of the inverse problem of probability is explicitly solved for finite sample sizes. The inferred distribution is explicitly continuous for finite sample size. A geometric solution of the statistical inference problem in higher dimensions is outlined.     

几何量子计算

实现可集成的量子计算的关键步骤是实现保真度足够高的一组普适量子逻辑门，最近几年发展的几何量子计算使用几何位相来实现量子逻辑门，其特点是利用几何位相的整体几何性质来避免某些局域的无规噪声的影响，从而实现较高保真度的量子门，文章先简要介绍常规几何量子逻辑门的概念，然后重点介绍最近提出的非常规几何量子计算：量子计算中使用的逻辑门的总位相既包含有几何位相，又包含有动力学位相，但它仅依赖于一些几何特征，而且，对于任意的量子位输入态，在量子门操作过程中积累的位相要么是零，要么是仅依赖几何特征的位相。     

Geometric intrinsic symmetries

The problem of geometric symmetries in the intrinsic frame of a many-body system (nucleus) is considered. An importance of symmetrization group notion is discussed. Ageneral structure of the intrinsic symmetry group structure is determined.     

Geometric Variational Inference

Efficiently accessing the information contained in non-linear and high dimensional probability distributions remains a core challenge in modern statistics. Traditionally, estimators that go beyond point estimates are either categorized as Variational Inference (VI) or Markov-Chain Monte-Carlo (MCMC) techniques. While MCMC methods that utilize the geometric properties of continuous probability distributions to increase their efficiency have been proposed, VI methods rarely use the geometry. This work aims to fill this gap and proposes geometric Variational Inference (geoVI), a method based on Riemannian geometry and the Fisher information metric. It is used to construct a coordinate transformation that relates the Riemannian manifold associated with the metric to Euclidean space. The distribution, expressed in the coordinate system induced by the transformation, takes a particularly simple form that allows for an accurate variational approximation by a normal distribution. Furthermore, the algorithmic structure allows for an efficient implementation of geoVI which is demonstrated on multiple examples, ranging from low-dimensional illustrative ones to non-linear, hierarchical Bayesian inverse problems in thousands of dimensions.     

The real gauge hierarchy problem

We discuss the problem of generating two vastly different mass scales through spontaneous symmetry breaking. We find that to all orders in perturbation theory this can only be achieved if there is a non-trivial relation between the couplings of the theory. While this relation may come from a higher symmetry we find that this symmetry can only be broken at the low mass scale.     

Two-reduction of the super-KP hierarchy

Recursion relations are established for the residues of fractional powers of a two-reduced super-KP operator making use of the Baker-Akhiezer function. These show the integrability of the two-reduced even (or bosonic) flows of the super-KP hierarchy. Similar recursion relations are also proven for the residues of operators associated with the odd (or fermionic) flows of the Mulase-Rabin super-KP hierarchy. Due to the presence of a spectral parameter and its fermionic partner in the Baker-Akhiezer function, these recursion relations should be relevant to any attempt to prove or disprove a recent proposal that the integrable hierarchy underlying two-dimensional quantum supergravity is the Mulase-Rabin super-KP hierarchy.