We construct Hermitian representations of Lie algebroids and associated unitary representations of Lie groupoids by a geometric quantization procedure. For this purpose we introduce a new notion of Hamiltonian Lie algebroid actions. The first step of our procedure consists of the construction of a prequantization line …
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In this paper, we prove a functorial aspect of the formal geometric quantization procedure of non-compact spin-c manifolds.
This is an expanded version of a series of two lectures given at the IMA summer program "Symmetries and Overdetermined Systems of Partial Differential Equations". The main part of the article describes the Riemannian version of the prolongation procedure for certain overdetermined system obtained recently in joint work…
The paper presents an extension of the geometric quantization procedure to integrable, big-isotropic structures. We obtain a generalization of the cohomology integrality condition, we discuss geometric structures on the total space of the corresponding principal circle bundle and we extend the notion of a polarization.
Geometric approach improves motion alignment accuracy and efficiency.
problem Temporal alignment of human motion data for various applications.
method Geometric point of view, principal fiber bundle, reparameterization invariant projection, dynamic programming, keyframe correspondences.
result Temporal alignment procedures are more accurate and computationally efficient.
Using geometric quantization procedure, the quantization of algebra of observables for physical system with Ricci-flat phase space is obtained. In the classical case the appointed physical system is reduced to harmonic oscillator when the one real parameter is vanished.
The paper examines short-term volatilities in equity indexes using a ranking procedure.
problem Understanding short-term behaviors of implied volatility in equity markets.
method Using a ranking procedure to model equity index dynamics, the paper investigates the short-term volatilities of derivatives written on indexes.
result The models reconcile the long memory of volatilities and power law of ATM skews in equity markets.
The renormalized volume of hyperbolic manifolds is a quantity motivated by the AdS/CFT correspondence of string theory and computed via a certain regularization procedure. The main aim of the present paper is to elucidate its geometrical meaning. We use another regularization procedure based on surfaces equidistant to …
Connecting ideas of geometric formulation of quantum mechanics with new results in symplectic geometry a new approach to geometrical quantization procedure is proposed. As a first result we verify that the correspondence between "classical" Poisson bracket and "quantum" one takes place.
The first part of this paper discusses general procedures for finding numerical approximations to distinguished Kahler metrics, such as Calabi-Yau metrics, on complex projective manifolds. These procedures are closely related to ideas from Geometric Invariant Theory, and to the asymptotics of high powers of positive li…
The paper constructs minimal coherent filling pairs on surfaces.
problem Finding minimal intersecting coherent filling pairs on surfaces.
method Geometric procedure starting from a torus filling pair.
result Construction of minimal intersecting coherent filling pairs on for .
Cactus Doodlesmath.GT
Cactus doodles are geometric objects derived from cactus groups.
problem Understanding geometric structures related to algebraic groups.
method Defining cactus doodles via local moves on plane curves and relating them to cactus groups.
result Established properties of cactus doodles and their connection to cactus groups.
Geometrically proves Lie algebras are identified by their Iwasawa subalgebras.
problem Determining semi-simple Lie algebras from their subalgebras.
method Geometry of Einstein solvmanifolds and algebraic procedure.
result Semi-simple Lie algebras are uniquely determined by their Iwasawa subalgebras.
CP uses geometric information to improve conformal prediction robustness.
problem CP fails under geometric data shifts, losing coverage guarantees.
method Integrates geometric pose information into CP via canonicalization.
result Integrating geometric information with CP ensures robustness under geometric shifts.
The formulation of Geometric Quantization contains several axioms and assumptions. We show that for real polarizations we can generalize the standard geometric quantization procedure by introducing an arbitrary connection on the polarization bundle. The existence of reducible quantum structures leads to considering the…
Direct proof of hyperbolicity for a family of knots.
problem Proving the hyperbolicity of a family of knots.
method Using a theorem of Casson and Rivin, constructing geometric triangulations.
result Direct proof of hyperbolicity for -pretzel knots and links with .
We propose a geometric algorithm for topic learning and inference that is built on the convex geometry of topics arising from the Latent Dirichlet Allocation (LDA) model and its nonparametric extensions. To this end we study the optimization of a geometric loss function, which is a surrogate to the LDA's likelihood. Ou…
The recently introduced dropout training criterion for neural networks has been the subject of much attention due to its simplicity and remarkable effectiveness as a regularizer, as well as its interpretation as a training procedure for an exponentially large ensemble of networks that share parameters. In this work we …
Regularisation method studies Lie algebroids via foliated structures.
problem Geometric structures on Lie algebroids.
method Regularisation procedure applied to various Lie algebroids.
result Proof of Weinstein conjecture for specific Lie algebroids.
Geometric interpretation improves VAE performance and robustness.
problem Improving Variational Autoencoder performance and robustness.
method Introducing a geometric perspective on VAEs, sampling from the Riemannian latent space.
result Improved generation and interpolations with competitive or better performance on benchmark datasets.
The construction of a linear connection on a pullback bundle from a connection on a vector bundle is explained in terms of fiberwise linear approximation. This procedure clarifies the geometric meaning of the linearized connection as well as the associated parallel transport and curvature.
To any prime alternating link, we associate a collection of hyperbolic right-angled ideal polyhedra by relating geometric, topological and combinatorial methods to decompose the link complement. The sum of the hyperbolic volumes of these polyhedra is a new geometric link invariant, which we call the right-angled volume…
We present a geometric proof of the averaging theorem for perturbed dynamical systems on a Riemannian manifold, in the case where the flow of the unperturbed vector field is periodic and the -action associated to this vector field is not necessarily trivial. We generalize the averaging procedure \cite{A…
We study local complexity measures for stochastic convex optimization problems, providing a local minimax theory analogous to that of Hájek and Le Cam for classical statistical problems. We give complementary optimality results, developing fully online methods that adaptively achieve optimal convergence guarantees. Our…
In this paper, we establish an intrinsic Gauss-Bonnet-Chern formula for Finsler manifolds by using the Mathai-Quillen's superconnection formalism, in which no extra vector field is involved. Furthermore, we prove a more general Lichnerowicz formula in this direction through a geometric localization procedure.
The loss functions of deep neural networks are complex and their geometric properties are not well understood. We show that the optima of these complex loss functions are in fact connected by simple curves over which training and test accuracy are nearly constant. We introduce a training procedure to discover these hig…
Robust Score Matchingstat.ML
Robust score matching improves parameter estimation in contaminated data.
problem Parameter estimation in data contaminated by outliers.
method Geometric median of means to develop a robust score matching procedure.
result Consistent parameter estimates in contaminated data settings.
We provide a topological procedure to obtain geometric realizations of both classical and `exotic' -manifolds, such as spheres, bundles over spheres and Kervaire manifolds. As an application, we apply the process known as Cheeger deformations to produce new metrics of both positive Ricci and almost non-negative curv…
In a previous paper, we have introduced a new unified description of the main equations of the gravitational and of the electromagnetic field, in terms of tidal tensors and connections on the tangent bundle TM of the space-time manifold. In the present work, we relate these equations to variational procedures on the ta…
New method recovers graph latent positions under edge differential privacy.
problem Recovering latent graph information from privatized graphs.
method Applying geometric insights to adjust statistical inference for privatized graphs.
result Achieves consistent recovery of latent positions under local edge differential privacy constraints.
The paper introduces a method to decorrelate circular coordinates using lattice reduction.
problem Geometric correlation between circle-valued maps when multiple cohomology classes are used.
method Systematic procedure using the Lenstra--Lenstra--Lovász algorithm for constructing low energy torus-valued maps.
result A method to obtain less correlated maps from cohomology classes using integer linear combinations.
We present a generalization of the Simultaneous Long-Short (SLS) trading strategy described in recent control literature wherein we allow for different parameters across the short and long sides of the controller; we refer to this new strategy as Generalized SLS (GSLS). Furthermore, we investigate the conditions under …
We prove that twisted correction terms in Heegaard Floer homology provide lower bounds on the Thurston norm of certain cohomology classes determined by the strong concordance class of a 2-component link in . We then specialise this procedure to knots in , and obtain a lower bound on their geomet…
Classifies solutions in multisymplectic field theories using geometric gauge freedom.
problem Classifying solutions in multisymplectic field theories.
method Using the kernel of a premultisymplectic form and equivalence relations.
result Equivalence relations and reduction procedures for sections.
We propose a dimensional reduction procedure in the Stolz--Teichner framework of supersymmetric Euclidean field theories (EFTs) that is well-suited in the presence of a finite gauge group or, more generally, for field theories over an orbifold. As an illustration, we give a geometric interpretation of the Chern charact…
This is my Habilitation a Diriger des Recherches (French thing). In this paper I summarize the work I have done to realize the program of Witten called -non abelian localization-. This work deals first with problems of localization in equivariant cohomology. The second part of this paper concerns the Guillemin-Sternber…
As a continuation of previous papers, we study the concept of a Lie algebroid structure on an affine bundle by means of the canonical immersion of the affine bundle into its bidual. We pay particular attention to the prolongation and various lifting procedures, and to the geometrical construction of Lagrangian-type dyn…
Geometric model of unbounded sl3 laminations with tropical coordinates.
problem Modeling unbounded laminations in cluster varieties.
method Introducing tropical cluster coordinates and geometric gluing procedures.
result Established a geometric gluing procedure for unbounded sl3 laminations.
A general study of symmetries in optimal control theory is given, starting from the presymplectic description of this kind of system. Then, Noether's theorem, as well as the corresponding reduction procedure (based on the application of the Marsden-Weinstein theorem adapted to the presymplectic case) are stated both in…
The visual systems of many mammals, including humans, is able to integrate the geometric information of visual stimuli and to perform cognitive tasks already at the first stages of the cortical processing. This is thought to be the result of a combination of mechanisms, which include feature extraction at single cell l…
Imputation method respects manifold structure for missing data.
problem Missing data imputation in high-dimensional data.
method Model-based imputation using mixture variational autoencoders and sampling-importance-resampling (SIR).
result Competitive performance and uncertainty quantification in imputations.
Polynomial braid combingmath.GT
Braid combing is a procedure defined by Emil Artin to solve the word problem in braid groups for the first time. It is well-known to have exponential complexity. In this paper, we use the theory of straight line programs to give a polynomial algorithm which performs braid combing. This procedure can be applied to braid…
Generalised contact structures are studied from the point of view of reduced generalised complex structures, naturally incorporating non-coorientable structures as non-trivial fibering. The infinitesimal symmetries are described in detail, with a geometric description given in terms of gerbes. As an application of the …
Gauging procedure constructs lagrangians for carrollian gravity.
problem Constructing lagrangians for carrollian gravity.
method Gauging procedure applied to Klein pairs corresponding to homogeneous spaces.
result Generalizes first-order lagrangians for four-dimensional maximally symmetric carrollian spaces.
Geometric approach clusters intersecting manifolds with high probability.
problem Clustering intersecting d-dimensional manifolds.
method Compute locality graph on d-simplices using dihedral angles, then compute LAPD to separate manifold components.
result The method separates manifold components with high probability under random sampling.
New theory for Hamiltonian actions on special geometric structures.
problem Hamiltonian actions on cosymplectic groupoids.
method Developed a moment map theory for 0-shifted cosymplectic structures.
result Established a version of the Kirwan convexity theorem.
Potential functions can be used as generating potentials of relevant geometric structures for a Riemannian manifold such as the Riemannian metric and affine connections. We study wether this procedure can also be applied to tensors of rank four and find a negative answer. We study this from the perspective of solving t…
On the geometric and Riemannian structure of the spaces of group equivariant non-expansive operatorsmath.DG
The paper studies geometric properties of group equivariant operators and their Riemannian structure.
problem Understanding the geometric structure of group equivariant operators.
method Endowing the space of group equivariant non-expansive operators with a Riemannian manifold structure and using gradient descent methods.
result Gradient descent methods can be applied to minimize cost functions on the space of group equivariant non-expansive operators.