arXiv research

Develops a finite construction for self-duality and related moduli spaces over Riemann surfaces.

The study describes metrics geodesically compatible with Nijenhuis operators and their applications to integrable systems.

Study minimal networks on spheres and balls near standard metrics.

We review how a reduction procedure along a principal fibration and an unfolding procedure associated to a suitable momentum map allow to describe the Kähler geometry of a finite dimensional complex projective spaces.

Extends SGM to functional spaces for multimodal data.

The paper solves integrable systems of PDEs, including famous equations.

Investigates properties of moment maps and stratifications on Lie groups.

We prove a theorem on singular symplectic cotangent bundle reduction in the Fréchet setting and apply it to Yang-Mills-Higgs theory with special emphasis on the Higgs sector of the Glashow-Weinberg-Salam model. For the latter model we give a detailed description of the reduced phase space and show that the singular str…

These notes give an introduction to Geometric Invariant Theory and symplectic reduction, with lots of pictures and simple examples. We describe their applications to moduli of bundles and varieties, and their infinite dimensional analogues in gauge theory and the theory of special metrics on algebraic varieties. Donald…

Given a central extension of Lie groups, we study the classification problem of lifting the structure group together with a given connection. For reductive structure groups we introduce a new connective structure on the lifting gerbe associated to this problem. Our main result classifies all connections on the central …

Geometrically represents the Jacobian for a mechanical system with symmetry.

In this paper we study the reductions of evolutionary PDEs on the manifold of the stationary points of time--dependent symmetries. In particular we describe how that the finite dimensional Hamiltonian structure of the reduced system is obtained from the Hamiltonian structure of the initial PDE and we construct the time…

It is shown that the equation which describes constant mean curvature surface via the generalized Weierstrass-Enneper inducing has Hamiltonian form. Its simplest finite-dimensional reduction has two degrees of freedom, integrable and its trajectories correspond to well-known Delaunay and do Carmo-Dajzcer surfaces (i.e.…

Reduces multisymplectic Lie systems through symmetry analysis.

We study finite dimensional almost and quasi-effective prolongations of nilpotent Z-graded Lie algebras, especially focusing on those having a decomposable reductive structural subalgebra. Our assumptions generalize effectiveness and algebraicity and are appropriate to obtain Levi-Malčev and Levi-Chevalley decompositio…

Paper constructs solutions to WDVV equations for Frobenius manifolds.

Invariant reduction preserves Poisson structures in PDEs.

Develops neural networks for reductive Lie groups, enhancing symmetry respect.

Develops a correspondence between symplectic orbits and Grassmannians.

Contact Lie systems analyze integral curves of Hamiltonian vector fields.

A method for noise reduction in functional time series using FPCA.

Reduces path integrals for interacting systems using dependent coordinates.

The popular cubic smoothing spline estimate of a regression function arises as the minimizer of the penalized sum of squares $\sum_j(Y_j - μ(t_j))^2 + λ\int_a^b [μ"(t)]^2 dt$, where the data are $t_j,Y_j$, $j=1,..., n$. The minimization is taken over an infinite-dimensional function space, the space of all functions wi…

Periodic geodesics on Hilbert half-Lie groups exist whenever the fundamental group is nontrivial.

We consider a curve of Fredholm pairs of Lagrangian subspaces in a fixed Banach space with continuously varying (weak) symplectic structures. Assuming vanishing index, we obtain intrinsically a continuously varying splitting of the total Banach space into pairs of symplectic subspaces. Using such decompositions we defi…

The paper extends LDDMM framework to include Lie group actions in large deformation shape registration.

Geometrically represents path integral reduction Jacobian for interacting systems.

The ``classical BRST construction'' as developed by Batalin-Fradkin-Vilkovisky is a homological construction for the reduction of the Poisson algebra $P = C^\infty (W)$ of smooth functions on a Poisson manifold $W$ by the ideal $I$ of functions which vanish on a constraint locus. This ideal is called first class if $I$…

In this paper, we explain how generalized dynamical r-matrices can be obtained by (quasi-)Poisson reduction. New examples of Poisson structures and Poisson groupoid actions naturally appear in this setting. As an application, we use a generalized dynamical r-matrix induced by the gauge fixing procedure to give a new fi…

This paper simplifies finding least favorable priors by reducing dimensionality.

We classify the finite type (in the sense of E. Cartan theory of prolongations) subalgebras $\mathfrak{h}\subset\mathfrak{sp}(V)$, where $V$ is the symplectic 4-dimensional space, and show that they satisfy $\mathfrak{h}^{(k)}=0$ for all $k>0$. Using this result, we reduce the problem of classification of graded transi…

A new path development layer reduces dimensionality for irregular time series.

In this paper we describe all the nilradicals of parabolic subalgebras of split real simple Lie algebras admitting symplectic structures. The main tools used to obtain this list are Kostant's description of the highest weight vectors (hwv) of the cohomology of these nilradicals and some necessary conditions obtained fo…

Stokes-Dirac structures are infinite-dimensional Dirac structures defined in terms of differential forms on a smooth manifold with boundary. These Dirac structures lay down a geometric framework for the formulation of Hamiltonian systems with a nonzero boundary energy flow. Simplicial triangulation of the underlaying m…

We present a new construction for Poisson transforms between vector bundle valued differential forms on homogeneous parabolic geometries and the corresponding Riemannian symmetric space, which can be described in terms of finite dimensional representations of reductive Lie groups. In particular, we use these operators …

In this paper we adopt an alternative, analytical approach to Arnol'd problem \cite{A1} about the existence of closed and embedded $K$-magnetic geodesics in the round $2$-sphere $\mathbb S^2$, where $K: \mathbb S^2 \rightarrow \mathbb R$ is a smooth scalar function. In particular, we use Lyapunov-Schmidt finite-dimensi…

We study the problem asking if one can embed manifolds into finite dimensional Euclidean spaces by taking finite number of eigenvector fields of the connection Laplacian. This problem is essential for the dimension reduction problem in massive data analysis. Singer-Wu proposed the vector diffusion map which embeds mani…

We give a construction of a Poisson transform mapping density valued differential forms on generalized flag manifolds to differential forms on the corresponding Riemannian symmetric spaces, which can be described entirely in terms of finite dimensional representations of reductive Lie groups. Moreover, we will explicit…

Develops neural network approximations for infinite-dimensional input-output maps.

Let $G$ be a Lie group, with an invariant non-degenerate symmetric bilinear form on its Lie algebra, let $π$ be the fundamental group of an orientable (real) surface $M$ with a finite number of punctures, and let $\bold C$ be a family of conjugacy classes in $G$, one for each puncture. A finite-dimensional construction…

Study applies inverse scattering to BKM systems, linking spectra and integrable systems.

Let $X$ be a differentiable manifold endowed with a transitive action $α:A\times X\longrightarrow X$ of a Lie group $A$. Let $K$ be a Lie group. Under suitable technical assumptions, we give explicit classification theorems, in terms of explicit finite dimensional quotients, of three classes of objects: {enumerate} equ…

For each connected complex reductive group G, we find a family of new examples of complex quasi-Hamiltonian G-spaces with G-valued moment maps. These spaces arise naturally as moduli spaces of (suitably framed) meromorphic connections on principal G-bundles over a disc, and they generalise the conjugacy class example o…

We consider the energy supercritical wave maps from $\mathbb{R}^d$ into the $d$-sphere $\mathbb{S}^d$ with $d \geq 7$. Under an additional assumption of 1-corotational symmetry, the problem reduces to the one dimensional semilinear wave equation $$\partial_t^2 u = \partial^2_r u + \frac{(d-1)}{r}\partial_r u - \frac{(d…

Study on Higgs bundles and hyperpolygon spaces using Hitchin metrics.

Survey of recent results on homogeneous finite-dimensional spaces.

New method reconstructs Black-Scholes option prices from current profiles.

We construct from first principles the operator 'A-hat' that annihilates the partition functions (or wavefunctions) of three-dimensional Chern-Simons theory with gauge groups SU(2), SL(2,R), or SL(2,C) on a knot complement M. The operator 'A-hat' is a quantization of the knot complement's classical A-polynomial A(l,m).…