138,233 research outputs found
Spectral Curves, Opers and Integrable Systems
We establish a general link between integrable systems in algebraic geometry
(expressed as Jacobian flows on spectral curves) and soliton equations
(expressed as evolution equations on flat connections). Our main result is a
natural isomorphism between a moduli space of spectral data and a moduli space
of differential data, each equipped with an infinite collection of commuting
flows. The spectral data are principal G-bundles on an algebraic curve,
equipped with an abelian reduction near one point. The flows on the spectral
side come from the action of a Heisenberg subgroup of the loop group. The
differential data are flat connections known as opers. The flows on the
differential side come from a generalized Drinfeld-Sokolov hierarchy. Our
isomorphism between the two sides provides a geometric description of the
entire phase space of the Drinfeld-Sokolov hierarchy. It extends the Krichever
construction of special algebro-geometric solutions of the n-th KdV hierarchy,
corresponding to G=SL(n).
An interesting feature is the appearance of formal spectral curves, replacing
the projective spectral curves of the classical approach. The geometry of these
(usually singular) curves reflects the fine structure of loop groups, in
particular the detailed classification of their Cartan subgroups. To each such
curve corresponds a homogeneous space of the loop group and a soliton system.
Moreover the flows of the system have interpretations in terms of Jacobians of
formal curves.Comment: 64 pages, Latex, final version to appear in Publications IHE
Multipartite quantum correlations: symplectic and algebraic geometry approach
We review a geometric approach to classification and examination of quantum
correlations in composite systems. Since quantum information tasks are usually
achieved by manipulating spin and alike systems or, in general, systems with a
finite number of energy levels, classification problems are usually treated in
frames of linear algebra. We proposed to shift the attention to a geometric
description. Treating consistently quantum states as points of a projective
space rather than as vectors in a Hilbert space we were able to apply powerful
methods of differential, symplectic and algebraic geometry to attack the
problem of equivalence of states with respect to the strength of correlations,
or, in other words, to classify them from this point of view. Such
classifications are interpreted as identification of states with `the same
correlations properties' i.e. ones that can be used for the same information
purposes, or, from yet another point of view, states that can be mutually
transformed one to another by specific, experimentally accessible operations.
It is clear that the latter characterization answers the fundamental question
`what can be transformed into what \textit{via} available means?'. Exactly such
an interpretations, i.e, in terms of mutual transformability can be clearly
formulated in terms of actions of specific groups on the space of states and is
the starting point for the proposed methods.Comment: 29 pages, 9 figures, 2 tables, final form submitted to the journa
Classification of first order sesquilinear forms
A natural way to obtain a system of partial differential equations on a
manifold is to vary a suitably defined sesquilinear form. The sesquilinear
forms we study are Hermitian forms acting on sections of the trivial
-bundle over a smooth -dimensional manifold without boundary.
More specifically, we are concerned with first order sesquilinear forms,
namely, those generating first order systems. Our goal is to classify such
forms up to gauge equivalence. We achieve this
classification in the special case of and by means of geometric and
topological invariants (e.g. Lorentzian metric, spin/spin structure,
electromagnetic covector potential) naturally contained within the sesquilinear
form - a purely analytic object. Essential to our approach is the interplay of
techniques from analysis, geometry, and topology.Comment: Minor edit
The differential geometric structure in supervised learning of classifiers
In this thesis, we study the overfitting problem in supervised learning of classifiers from a geometric perspective. As with many inverse problems, learning a classification function from a given set of example-label pairs is an ill-posed problem, i.e., there exist infinitely many classification functions that can correctly predict the class labels for all training examples. Among them, according to Occam's razor, simpler functions are favored since they are less overfitted to training examples and are therefore expected to perform better on unseen examples. The standard technique to enforce Occam's razor is to introduce a regularization scheme, which penalizes some type of complexity of the learned classification function. Some widely used regularization techniques are functional norm-based (Tikhonov) techniques, ensemble-based techniques, early stopping techniques, etc. However, there is important geometric information in the learned classification function that is closely related to overfitting, and has been overlooked by previous methods. In this thesis, we study the complexity of a classification function from a new geometric perspective. In particular, we investigate the differential geometric structure in the submanifold corresponding to the estimator of the class probability P(y|x), based on the observation that overfitting produces rapid local oscillations and hence large mean curvature of this submanifold. We also show that our geometric perspective of supervised learning is naturally related to an elastic model in physics, where our complexity measure is a high dimensional extension of the surface energy in physics. This study leads to a new geometric regularization approach for supervised learning of classifiers. In our approach, the learning process can be viewed as a submanifold fitting problem that is solved by a mean curvature flow method. In particular, our approach finds the submanifold by iteratively fitting the training examples in a curvature or volume decreasing manner.
Our technique is unified for both binary and multiclass classification, and can be applied to regularize any classification function that satisfies two requirements: firstly, an estimator of the class probability can be obtained; secondly, first and second derivatives of the class probability estimator can be calculated. For applications, where we apply our regularization technique to standard loss functions for classification, our RBF-based implementation compares favorably to widely used regularization methods for both binary and multiclass classification. We also design a specific algorithm to incorporate our regularization technique into the standard forward-backward training of deep neural networks.
For theoretical analysis, we establish Bayes consistency for a specific loss function under some mild initialization assumptions. We also discuss the extension of our approach to situations where the input space is a submanifold, rather than a Euclidean space.2018-11-30T00:00:00
Moduli Spaces of Flat GSp-Bundles
A classical problem in the theory of differential equations is the classification of first-order singular differential operators up to gauge equivalence. A related algebro-geometric problem involves the construction of moduli spaces of meromorphic connections. In 2001, P. Boalch constructed well-behaved moduli spaces in the case that each of the singularities are diagonalizable. In a recent series of papers, C. Bremer and D. Sage developed a new approach to the study of the local behavior of meromorphic connections using a geometric variant of fundamental strata, a tool originally introduced by C. Bushnell for the study of p-adic representation theory. Not only does this approach allow for the generalization of diagonalizable singularities, but it is adaptable to the study of flat G-bundles for G a reductive group. In this dissertation, the objects of study are irregular singular flat GSp-bundles. The main results of this dissertation are two-fold. First, the local theory of fundamental strata for GSp-bundles is made explicit; in particular, the fundamental strata necessary for the construction of well-behaved moduli spaces are shown to be associated to uniform symplectic lattice chain filtrations. Second, a construction of moduli spaces of flat GSp-bundles is given which has many of the geometric features that have been important in the work of P. Boalch and others
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