arXiv research

Clarifies when certain stochastic PDEs have affine state processes.

Investigates existence of affine models for Lévy-driven term structures.

Clarifies when certain stochastic PDEs have affine solutions.

We show that every Kaehler affine curvature model can be realized geometrically.

The study of which mapping class group elements can be realized as affine automorphisms of dilation surfaces.

Proposes an alternative method for HJM models' existence.

Investigates real-world interest rate dynamics using affine models.

The abstract discusses convergent realizations of Lie subalgebras in control theory.

We study geometric realization questions of curvature in the affine, Riemannian, almost Hermitian, almost para Hermitian, almost hyper Hermitian, almost hyper para Hermitian, Hermitian, and para Hermitian settings. We also express questions in Ivanov-Petrova geometry, Osserman geometry, and curvature homogeneity in ter…

We show that a large class of non-metric, non-symplectic affine holonomies can be realized, uniformly and without case by case considerations, by Weyl connections associated to the natural AHS-structures on certain generalized flag manifolds.

The affine Grassmannian is a noncompact smooth manifold that parameterizes all affine subspaces of a fixed dimension. It is a natural generalization of Euclidean space, points being zero-dimensional affine subspaces. We will realize the affine Grassmannian as a matrix manifold and extend Riemannian optimization algorit…

It is proved that the equality $Δ\ln|κ-λ|=6κ$, where $κ$ is the Gaussian curvature of a metric tensor g on a 2-dimensional manifold is a sufficient and necessary condition for local realizability of the metric as the Blaschke metric of some affine sphere.

Study on friction forces for nonholonomic systems using affine connections.

Realized GARCH model explains VIX and VRP dynamics.

We examine the local geometry of affine surfaces which are locally symmetric. There are 6 non-isomorphic local geometries. We realize these examples as Type A, Type B, and Type C geometries using a result of Opozda and classify the relevant geometries up to linear isomorphism. We examine the geodesic structures in this…

There are exactly two different types of bi-dimensional improper affine spheres: the non-convex ones can be modeled by the center-chord transform of a pair of planar curves while the convex ones can be modeled by a holomorphic map. In this paper, we show that both constructions can be generalized to arbitrary even dime…

Study shows bi-Hamiltonian structure for polygon evolutions in centro-affine space.

The paper proves there are many Lagrangian fillings for Legendrian links of affine type.

The study limits the cohomological dimension of certain affine manifolds with partially hyperbolic holonomy groups.

BN refines local partition geometry in piecewise-affine networks during training.

This paper studies the geometry of immersions into statistical manifolds. A necessary and sufficient condition is obtained for statistical manifold structures to be dual to each other for a non-degenerate equiaffine immersion. Then we obtain conditions for realizing an n-dimensional statistical manifold in an (n+1)-dim…

Smooth knots can be embedded into a specific Menger continuum.

The paper proves mirror symmetry for del Pezzo surfaces and computes related structures.

We develop a relative version of Kostant's harmonic theory and use this to prove a relative version of Kostant's theorem on Lie algebra (co)homology. These are associated to two nested parabolic subalgebras in a semisimple Lie algebra. We show how relative homology groups can be used to realize representations with low…

Proves algebraic cones for LCK manifolds with potential.

Given an $n$-dimensional manifold $N$ with an affine connection $D$, we show that the associated Patterson-Walker metric $g$ on $T^*N$ admits a global and explicit Fefferman-Graham ambient metric. This provides a new and large class of conformal structures which are generically not conformally Einstein but for which th…

The paper studies Anosov holonomy groups in complete affine manifolds.

The local structure of half conformally flat gradient Ricci almost solitons is investigated, showing that they are locally conformally flat in a neighborhood of any point where the gradient of the potential function is non-null. In opposition, if the gradient of the potential function is null, then the soliton is a ste…

Characterizes group connections on group bundles.

The paper designs neural networks with assurance for controlling nonlinear systems.

One can realize higher laminations as positive configurations of points in the affine building. The duality pairings of Fock and Goncharov give pairings between higher laminations for two Langlands dual groups $G$ and $G^{\vee}$. These pairings are a generalization of the intersection pairing between measured laminatio…

Polytopes $A(P)$ for posets $P$ compactify spaces of order-preserving maps.

The problem of immersing a simply connected surface with a prescribed shape operator is discussed. From classical and more recent work, it is known that, aside from some special degenerate cases, such as when the shape operator can be realized by a surface with one family of principal curves being geodesic, the space o…

Algorithm constructs surfaces with specific Veech groups in lattice strata.

In this paper we prove that the space of differential invariants for curves with arc-length parameter in the light cone of Lorentzian $\mathbb{R}^4$, invariants under the centro-affine action of the Lorentzian group, is Poisson equivalent to the space of conformal differential invariants for curves in the Möbius sphere…

We provide a coordinate-free version of the local classification, due to A. G. Walker [Quart. J. Math. Oxford (2) 1, 69 (1950)], of null parallel distributions on pseudo-Riemannian manifolds. The underlying manifold is realized, locally, as the total space of a fibre bundle, each fibre of which is an affine principal b…

Holomorphic map connects Hitchin components to character varieties.

String theory connects lattice models, links, and geometric Langlands.

We examine questions of geometric realizability for algebraic structures which arise naturally in affine and Riemannian geometry. Suppose given an algebraic curvature operator R at a point P of a manifold M and suppose given a real analytic (resp. C-k for finite k at least 2) pseudo-Riemannian metric on M defined near …

For a Lie groupoid $\mathcal{G}$ with Lie algebroid $A$, we realize the symplectic leaves of the Lie-Poisson structure on $A^*$ as orbits of the affine coadjoint action of the Lie groupoid $\mathcal{J}\mathcal{G}\ltimes T^*M$ on $A^*$, which coincide with the groupoid orbits of the symplectic groupoid $T^*\mathcal{G}$ …

Newton's method solves variational problems on manifolds.

Unified framework models multiple financial and insurance term structures.

Introduces formal frames for manifolds and their properties.

We focus on the problem of estimating and quantifying uncertainties on the excursion set of a function under a limited evaluation budget. We adopt a Bayesian approach where the objective function is assumed to be a realization of a Gaussian random field. In this setting, the posterior distribution on the objective func…

New framework for equivariant neural networks using Lie group decompositions.

Researchers determine all possible representations of monodromy for Schwarzian equations on punctured surfaces.

The number $|K|$ of non-isotopic framed knots that correspond to a given unframed knot $K\subset S^3$ is infinite. This follows from the existence of the self-linking number $\slk$ of a zerohomologous framed knot. We use the approach of Vassiliev-Goussarov invariants to construct ``affine self-linking numbers'' that ar…

The paper explores the geometric structure of cost functions in multiple dimensions.