arXiv research

The study shows strong formality in certain complex manifolds.

Strong formal properties for toric and homogeneous Kähler manifolds.

This paper explores formal verification for autonomous systems, identifying limitations and proposing improvements.

Formalizes weak and strong verification for LLMs, controlling errors without assumptions.

Study on properties of special Kähler metrics and their interplay.

Study geometric formal metrics and Massey products on Kähler manifolds with torsion.

The paper defines strong emergence in field theories and proves it exists between certain theories.

Motivated by deformation quantization, we introduced in an earlier work the notion of formal Morita equivalence in the category of $^*$-algebras over a ring $\ring C$ which is the quadratic extension by $\im$ of an ordered ring $\ring R$. The goal of the present paper is twofold. First, we clarify the relationship betw…

Self-supervised skip-tree training improves mathematical reasoning in language models.

Motivated by understanding the limiting case of a certain systolic inequality we study compact Riemannian manifolds having all harmonic 1-forms of constant length. We give complete characterizations as far as Kähler and hyperbolic geometries are concerned. In the second part of the paper, we give algebraic and topologi…

Paper analyzes weak-to-strong generalization in CNNs, identifying data-scarce and data-abundant regimes.

Given for instance a finite volume negatively curved Riemannian manifold $M$, we give a precise relation between the logarithmic growth rates of the excursions into cusps neighborhoods of the strong unstable leaves of negatively recurrent unit vectors of $M$ and their linear divergence rates under the geodesic flow. As…

The paper proves an index theorem for loop spaces of compact manifolds.

Geodesic currents in strongly hyperbolic spaces are dense.

We call a knot in the 3-sphere $SU(2)$-simple if all representations of the fundamental group of its complement which map a meridian to a trace-free element in $SU(2)$ are binary dihedral. This is a generalisation of being a 2-bridge knot. Pretzel knots with bridge number $\geq 3$ are not $SU(2)$-simple. We provide an …

New method protects whistleblowers from retaliation by ensuring their reports remain private.

We find a minimal differential graded (dg) operad whose generic representations in $R^n$ are in one-to-one correspondence with formal germs of those endomorphisms of the tangent bundle to $R^n$ which satisfy the Nijenhuis integrability condition. This operad is of a surprisingly simple origin -- it is the cobar constru…

We define parametrized cobordism categories and study their formal properties as bivariant theories. Bivariant transformations to a strongly excisive bivariant theory give rise to characteristic classes of smooth bundles with strong additivity properties. In the case of cobordisms between manifolds with boundary, we pr…

Given an n-manifold M and an n-category C, we define a chain complex (the "blob complex") B_*(M;C). The blob complex can be thought of as a derived category analogue of the Hilbert space of a TQFT, and as a generalization of Hochschild homology to n-categories and n-manifolds. It enjoys a number of nice formal properti…

The optimal approach is to theorize after examining data, not before.

Study shows global invertibility in nonlinear elasticity with vanishing self-repulsion term.

I show that the solution of a standard clearing model commonly used in contagion analyses for financial systems can be expressed as a specific form of a generalized Katz centrality measure under conditions that correspond to a system-wide shock. This result provides a formal explanation for earlier empirical results wh…

Formal verification of neural networks is essential for their deployment in safety-critical areas. Many available formal verification methods have been shown to be instances of a unified Branch and Bound (BaB) formulation. We propose a novel framework for designing an effective branching strategy for BaB. Specifically,…

The Minimum Description Length (MDL) principle selects the model that has the shortest code for data plus model. We show that for a countable class of models, MDL predictions are close to the true distribution in a strong sense. The result is completely general. No independence, ergodicity, stationarity, identifiabilit…

We exploit the symmetry concepts developed in the companion review of this article to introduce a stochastic version of link reversal symmetry, which leads to an improved understanding of the reciprocity of directed networks. We apply our formalism to the international trade network and show that a strong embedding in …

This paper solves steady-state planning for multichain MDPs.

New optimal transport method handles mass creation and destruction.

Proves bounds on copyright risk for generative models.

The rise of connected personal devices together with privacy concerns call for machine learning algorithms capable of leveraging the data of a large number of agents to learn personalized models under strong privacy requirements. In this paper, we introduce an efficient algorithm to address the above problem in a fully…

This is a sequel of \cite{Wang}, which provides a general formalism for this paper. We mainly investigate thoroughly a subclass of toric generalized K$\ddot{a}$hler manifolds of symplectic type introduced by Boulanger in \cite{Bou}. We find torus actions on such manifolds are all \emph{strong Hamiltonian} in the sense …

In this paper, we study deformations of coisotropic submanifolds in a locally conformal symplectic manifold. Firstly, we derive the equation that governs $C^\infty$ deformations of coisotropic submanifolds and define the corresponding $C^\infty$-moduli space of coisotropic submanifolds modulo the Hamiltonian isotopies.…

Establishes Poincaré's lemma for formal manifolds.

GP3 framework efficiently analyzes Gaussian processes on GPUs.

A scalable method for heavy-tailed data using robust aggregation.

New stability criteria for Fano varieties using generalized b-divisors.

Structured subsampling improves privacy in deep time series forecasting.

Foundations laid for formal manifolds in differential geometry.

Explores local structure of morphisms and formal submanifolds in formal manifolds theory.

Study non-formal pseudo-differential operators over formal ones.

The isotropy action on certain symmetric spaces is shown to be equivariantly formal.

The paper develops a method to map knots in a cylinder to virtual-flat knots.

For G a complex reductive group and X a smooth projective or convex quasi-projective polarized G-variety we construct a formal map in quantum K-theory from the equivariant quantum K-theory $QK^G(X)$ to the quantum K-theory of the git quotient $QK(X//G)$ assuming the quotient $X//G$ is a smooth Deligne-Mumford stack wit…

This paper is the first attempt to formalize a new field of economics; studding the Intangibles Goods available on the Internet. We are taking advantage of the digital world's specific rules, in particular the zero marginal cost, to propose a theory of trading & sharing unified. A function based money is created as a w…

A metric is formal if all products of harmonic forms are again harmonic. The existence of a formal metric implies Sullivan formality of the manifold, and hence formal metrics can exist only in presence of a very restricted topology. We show that a warped product metric is formal if and only if the warping function is c…

Differential privacy is a strong notion for privacy that can be used to prove formal guarantees, in terms of a privacy budget, $ε$, about how much information is leaked by a mechanism. However, implementations of privacy-preserving machine learning often select large values of $ε$ in order to get acceptable utility of …

SAL framework uses unlabeled data to improve OOD detection.

TGLS generates text by optimizing search results and learning from them.

Formal manifolds with non-negative Ricci curvature have formal covers.