arXiv research

Classifies non-linear Fredholm maps linking to stable homotopy groups of spheres.

Investigates linearity of group amalgams and new examples of non-linear groups.

The paper generalizes equivariant neural networks on homogeneous spaces to the non-linear setting.

In the two parts of this paper we solve a problem of De Rham, proving that Reidemeister torsion invariants determine topological equivalence of linear G-representations, for G a finite cyclic group. Methods in controlled K-theory and surgery theory are developed to establish, and effectively calculate, a necessary and …

Non linear sigma models are quantum field theories describing, in the large deviations sense, random fluctuations of harmonic maps between a Riemann surface and a Riemannian manifold. Via their formal renormalization group analysis, they provide a framework for possible generalizations of the Hamilton-Perelman Ricci fl…

Non-linear Hopf manifolds can be embedded into linear ones and admit LCK metrics.

Generalizing the canonical symplectization of contact manifolds, we construct an infinite dimensional non-linear Stiefel manifold of weighted embeddings into a contact manifold. This space carries a symplectic structure such that the contact group and the group of reparametrizations act in a Hamiltonian fashion with eq…

Defines spectral selectors on lens spaces for contactomorphisms.

We present a geometric construction of central extensions of covering groups of the group of volume preserving diffeomorphisms, integrating central extensions of the Lie algebra of divergence free vector fields defined by Lichnerowicz cocycles. Certain covering spaces of non-linear Grassmannians can be realized as preq…

SAFLe solves federated learning's trade-off between non-linearity and scalability.

For a given manifold $M$ we consider the non-linear Grassmann manifold $Gr_n(M)$ of $n$-dimensional submanifolds in $M$. A closed $(n+2)$-form on $M$ gives rise to a closed 2-form on $Gr_n(M)$. If the original form was integral, the 2-form will be the curvature of a principal $S^1$-bundle over $Gr_n(M)$. Using this $S^…

The price of financial assets are, since Bachelier, considered to be described by a (discrete or continuous) time sequence of random variables, i.e a stochastic process. Sharp scaling exponents or unifractal behavior of such processes has been reported in several works. In this letter we investigate the question of sca…

Improved disability insurance model with collective health claims.

Let G be a rank two finite group, and let $\cH$ denote the family of rank one p-subgroups of G, at all primes where G has p-rank two. We show that a rank two finite group G which satisfies certain group-theoretic conditions admits a finite G-CW-complex X with isotropy in $\cH$, whose fixed sets are homotopy spheres. Ou…

We discuss from a geometric point of view the connection between the renormalization group flow for non--linear sigma models and the Ricci flow. This offers new perspectives in providing a geometrical landscape for 2D quantum field theories. In particular we argue that the structure of Ricci flow singularities suggests…

Maps in Carnot groups are equivalent to solutions of a PDE system.

Essential curves in certain graph manifolds imply non-linear groups.

In this paper, we investigate the non-linear Black--Scholes equation: $u_t+ax^2u_{xx}+bx^3u_{xx}^2+c(xu_x-u)=0,\quad a,b>0,\ c\geq0.$ and show that the one can be reduced to the equation $u_t+(u_{xx}+u_x)^2=0$ by an appropriate point transformation of variables. For the resulting equation, we study the group-theore…

We discuss in rather general terms quantum field theories dealing with spaces of maps between Riemannian manifolds. In particular we explore the well--known connection between the renormalization group flow for non--linear sigma models and the Ricci flow.

New proof of generalized Chow-Rashevskii theorem for non-linear systems.

Estimates signals from a continuous dictionary with sparse mixtures using optimization.

We give dimension-free regularity conditions for a class of possibly degenerate sub-elliptic equations in the Heisenberg group exhibiting super-quadratic growth in the horizontal gradient; this solves an issue raised by Manfredi & Mingione (Math. Ann. 2007) where only dimension dependent bounds for the growth exponent …

For a closed symplectic manifold $(M,ω)$ with compatible Riemannian metric $g$ we study the Sobolev $H^1$ geometry of the group of all $H^s$ diffeomorphisms on $M$ which preserve the symplectic structure. We show that, for sufficiently large $s$, the $H^1$ metric admits globally defined geodesics and the corresponding …

In this paper we investigate the properties of a semi-linear problem on a spin manifold involving the Dirac operator, through the construction of Rabinowitz-Floer homology groups. We give several existence results for sub-critical and critical non-linearities as application of the computation of the different homologie…

In this paper we examine the Riemannian geometry of the group of contactomorphisms of a compact contact manifold. We compute the sectional curvature of $\mathcal{D}_θ(M)$ in the sections containing the Reeb field and show that it is non-negative. We also solve explicitly the Jacobi equation along the geodesic correspon…

SL(N,C) is the phase space of the Poisson SU(N). We calculate explicitly the symplectic structure of SL(N,C), define an analogue of the Hamiltonian of the free motion on SU(N) and solve the corresponding equations of motion. Velocity is related to the momentum by a non-linear Legendre transformation.

Study the geometry of hydrodynamics equations using diffeomorphism groups.

Unified method for CNNs to approximate equivariant maps across various groups.

Geometry arising from two diffusion operators (smooth semi-elliptic, second order differential operators) on different spaces but intertwined by a smooth map is described. Particular cases arise from Riemannian submersions when the operators are Laplace-Beltrami operators, from equivariant operators on the total space …

To every Darboux integrable system there is an associated Lie group $G$ which is a fundamental invariant of the system and which we call the Vessiot group. This article shows that solving the Cauchy problem for a Darboux integrable partial differential equation can be reduced to solving an equation of Lie type for the …

Recent developments in the understanding of $N=2$ supersymmetric Yang-Mills theory in four dimensions suggest a new point of view about Donaldson theory of four manifolds: instead of defining four-manifold invariants by counting $SU(2)$ instantons, one can define equivalent four-manifold invariants by counting solution…

In this note, we study non-linear gauge theories for principal bundles, where the structure group is replaced by a Lie groupoid. We follow the approach of Moerdijk-Mrcun and establish its relation with the existing physics literature. In particular, we derive a new formula for the gauge transformation which closely res…

We construct a Weil-Petersson geodesic completion of Teichmuller space through the formalism of Coxeter complex with the Teichmuller space as its non-linear non-homogeneous fundamental domain. We show that the metric and geodesic completions both satisfy a finite rank property, demonstrating a similarity with the non-c…

The geometric non-linear Schrodinger equation (GNLS) on the complex Grassmannian manifold M is the Hamiltonian equation for the energy functional on C(R,M) with respect to the symplectic form induced from the Kahler form on M. It has a Lax pair that is gauge equivalent to the Lax pair of the matrix non-linear Schroding…

Researchers can retrieve Yang-Mills-Higgs fields from Minkowski space measurements.

The paper studies the holonomy of spherically symmetric Finsler metrics.

We develop the theory of equivariant harmonic self-maps of compact cohomogeneity one manifolds and construct new harmonic self-maps of the compact Lie groups SO(4L+2), L >= 1, with degree -3, of SO(8), SO(14) and SO(26) with degree -5 each, of SO(10) with degree -7, and of SO(14) with degree -11 by exhibiting linear so…

The $L^2$-metric or Fubini-Study metric on the non-linear Grassmannian of all submanifolds of type $M$ in a Riemannian manifold $(N,g)$ induces geodesic distance 0. We discuss another metric which involves the mean curvature and shows that its geodesic distance is a good topological metric. The vanishing phenomenon for…

Two Fisher information matrix estimators are analyzed for neural networks, focusing on their variances and trade-offs.

In genomic analysis, biomarker discovery, image recognition, and other systems involving machine learning, input variables can often be organized into different groups by their source or semantic category. Eliminating some groups of variables can expedite the process of data acquisition and avoid over-fitting. Research…

Paper uses non-linear dimension reduction for better economic forecasting.

We show how the tangent functor extends from ordinary smooth maps to "microformal morphisms" (also called "thick morphisms") of supermanifolds. Microformal morphisms generalize ordinary maps and correspond to formal canonical relations between the cotangent bundles specified by generating functions depending on positio…

Study shows solutions to certain equations form smooth manifolds.

Optimal algorithms for non-linear ridge bandits reduce burn-in cost.

Uniform bounds derived for fully non-linear equations.

TaCo prevents non-linear classifiers from detecting sensitive attributes.

Proves energy estimates for tensorial wave equations, decoupling components for stability proof.

We present a PDE-based framework that generalizes Group equivariant Convolutional Neural Networks (G-CNNs). In this framework, a network layer is seen as a set of PDE-solvers where geometrically meaningful PDE-coefficients become the layer's trainable weights. Formulating our PDEs on homogeneous spaces allows these net…