arXiv research

This study explores complex structures on Lie algebras from graph perspectives.

Given a closed real analytic Riemannian manifold, we construct and study a one parameter family of adapted complex structures on the manifold of its geodesics.

The paper studies hyperkähler structures and adapted complex structures using the Monge-Ampère equation.

Study integrability of specific geometric structures on odd Courant algebroids.

Let M be a real analytic Riemannian manifold. An adapted complex structure on $TM$ is a complex structure on a neighborhood of the zero section such that the leaves of the Riemann foliation are complex submanifolds. This structure is called entire if it may be extended to the whole of $TM$. We prove here that the only …

Let $\,Ξ\,$ be the crown domain associated with a non-compact irreducible hermitian symmetric space $\,G/K$. We give an explicit description of the unique $\,G$-invariant adapted hyper-Kähler structure on $\,Ξ$,$\$i.$\,$e.$\$compatible with the adapted complex structure $\,J_{ad}\,$ and with the $\,G$-invariant Kähle…

In this paper, we give a new construction of the adapted complex structure on a neighborhood of the zero section in the tangent bundle of a compact, real-analytic Riemannian manifold. Motivated by the "complexifier" approach of T. Thiemann as well as certain formulas of V. Guillemin and M. Stenzel, we obtain the polari…

We define and study complex structures and generalizations on spaces consisting of geodesics or harmonic maps that are compatible with the symmetries of these spaces. The main results are about existence and uniqueness of such structures.

New proofs for complex Hopf manifolds using geometric structures.

A characterization of maximal domains of existence of adapted complex structures for Riemannian homogeneous manifolds under certain extensibility assumptions on their geodesic flow is given. This is applied to generalized Heisenberg groups and naturally reductive Riemannian homogeneous spaces. As an application it is s…

The paper explores linear generalised complex structures over vector bundles.

We classify complex surfaces $(M,\,J)$ admitting Engel structures $\mathcal{D}$ which are complex line bundles. Namely we prove that this happens if and only if $(M,\,J)$ has trivial Chern classes. We construct examples of such Engel structures by adapting a construction due to Geiges. We also study associated Engel de…

Harmonic almost complex structures on specific Lie groups and solvmanifolds identified.

Enhances ocean floor mapping with adaptive uncertainty estimates.

The aim of this article is to use generalized complex structures in order to extend the definition of twistor spaces given by Penrose. We will adapt the integrability result of Atiyah, Hitchin and Singer. We will deduce new correspondences betwenn differential geometry and (generalized) complex geometry. In the last se…

FVI method calculates bicausal OT with neural networks, outperforming other methods.

A Levi-Malcev type decomposition for $2$-step solvable Lie algebras with a complex structure

A generalized complex structure is called stable if its defining anticanonical section vanishes transversally, on a codimension-two submanifold. Alternatively, it is a zero elliptic residue symplectic structure in the elliptic tangent bundle associated to this submanifold. We develop Gompf-Thurston symplectic technique…

Improved optimal regularity for harmonic almost complex structures.

A novel unsupervised domain adaptation method using hierarchical optimal transport.

Shallow diffusion models learn hidden low-dimensional structures effectively.

New bounds adaptively control spectral complexity of trained Transformers.

A beautiful solution to the problem of isometric immersions in $\mathbb{R}^n$ using spinors was found by Bayard, Lawn and Roth. However to use spinors one must assume that the manifold carries a $\mbox{Spin}$-structure and, especially for complex manifolds where is more natural to consider $\mbox{Spin}^{\mathbb{C}}$-st…

In this paper, we adapt part of Weinberger, Xie and Yu's breakthrough work, to define additive higher rho invariant for topological structure group by differential geometric version of signature operators, or in other words, unbounded Hilbert-Poincaré complexes.

Paper simplifies complex AI exploration by predicting future rewards.

Optimal CATE estimation with structured contrast functions using KRR.

We introduce interactive structure discovery, a generic framework that encompasses many interactive learning settings, including active learning, top-k item identification, interactive drug discovery, and others. We adapt a recently developed active learning algorithm of Tosh and Dasgupta (2017) for interactive structu…

Structure plays a key role in learning performance. In centralized computational systems, hyperparameter optimization and regularization techniques such as dropout are computational means to enhance learning performance by adjusting the deep hierarchical structure. However, in decentralized deep learning by the Interne…

The Serre construction of rank two holomorphic bundles with a section is adapted to construct generalized holomorphic bundles on a generalized complex 4-manifold from the data of a set of points on an elliptic curve. The motivation is the special case of rank two Poisson modules on a complex surface with a holomorphic …

The agent-based model of stock price dynamics on a directed evolving complex network is suggested and studied by direct simulation. The stationary regime is maintained as a result of the balance between the extremal dynamics, adaptivity of strategic variables and reconnection rules. The inherent structure of node agent…

Banyaga has shown that the group of symplectomorphisms Symp(N) of a compact symplectic manifold (N,w) determines the symplectic structure. This motivates the study of the homotopy properties of Symp(N). Gromov has shown that the group of symplectomorphisms of N is homotopic to SO(3)\times SO(3) when N is the product of…

DIVA generates diverse tasks for complex simulators, enabling adaptive agent training.

We introduce algorithms for online, full-information prediction that are competitive with contextual tree experts of unknown complexity, in both probabilistic and adversarial settings. We show that by incorporating a probabilistic framework of structural risk minimization into existing adaptive algorithms, we can robus…

A new method reduces bias in adaptive Lasso estimates.

Data-driven method for error estimation without needing class complexity.

UDN adapts depth to data complexity, outperforming standard neural networks.

SASE improves attributed graph clustering for large graphs with linear time and space complexity.

Adapts MBDOE for real-time parameter estimation in complex systems.

We give necessary and sufficient conditions for the real distributions defined by a metallic pseudo-Riemannian structure to be integrable and geodesically invariant, in terms of associated tensor fields to the metallic structures and of adapted connections. In the integrable case, we prove a Chen-type inequality for th…

Paper describes integrable structure of Hitchin moduli spaces.

New method uses limited labeled data and multiple starts to adapt models across domains.

SPARTAN learns sparse interaction graphs between objects in scenes.

Over the last decade, both the neural network and kernel adaptive filter have successfully been used for nonlinear signal processing. However, they suffer from high computational cost caused by their complex/growing network structures. In this paper, we propose two random Euler filters for complex-valued nonlinear filt…

Adaptive algorithm improves nonlinear data assimilation for non-Gaussian systems.

We analyze the computational limits of LoRA for transformer models using fine-grained complexity theory.

PrAda-GAN improves synthetic data generation under differential privacy.

We study existence of complex structures on semidirect products $\g \oplus_ρ \v$ where $\g$ is a real Lie algebra and $ρ$ is a representation of $\g$ on $\v$. Our first examples, the Euclidean algebra $\e(3)$ and the Poincaré algebra $ \e(2,1)$, carry complex structures obtained by deformation of a regular complex stru…

AdaRL improves robust RL by adaptively adjusting policy complexity.