This paper classifies expanding attractors and non-transitive Anosov flows on specific knot and manifold spaces.
problem Classifying expanding attractors and non-transitive Anosov flows on specific knot and manifold spaces.
method Using the derived Anosov (DA) expanding attractor and the Franks-Williams manifold, the paper proves the uniqueness of these structures.
result The DA expanding attractor and the Franks-Williams non-transitive Anosov flow are the unique structures supported by N0 and M0 respectively. The paper develops a non-transitive Cartan connection for sub-Riemannian manifolds.
problem Analyzing sub-Riemannian structures and their symmetries.
method Using Lie groupoids and non-transitive Cartan connections.
result A non-transitive Cartan connection is constructed for sub-Riemannian manifolds.
Non-transitive subgroups of the orthogonal group play an important role in the non-Euclidean geometry. If G is a closed subgroup in the orthogonal group such that the orbit of a single Euclidean unit vector does not cover the (Euclidean) unit sphere centered at the origin then there always exists a non-Euclidean Mink…
Study non-transitive pseudo-Anosov flows using group actions.
problem Characterize pseudo-Anosov flows in 3-manifolds.
method Extend pseudo-Anosov action to non-transitive flows, use group actions on orbit spaces and boundary at infinity.
result Pseudo-Anosov flows in 3-manifolds are determined by their group actions on boundary at infinity.
New model of vague knowledge without strict partitions or transitivity.
problem Standard economic models of information fail to capture real-world vague knowledge.
method Relaxing assumptions of transitivity and partition structure to formalize vague knowledge.
result Vague knowledge can distinguish some states but not partition the state space.
Let M be a most singular orbit of the isotropy representation of a simple symmetric space. Let (νi,Φi) be an irreducible factor of the normal holonomy representation (νpM,Φ(p)). We prove that there exists a basis of a section Σi⊂νi of Φi such that the corresponding shape operators have rational…
In this paper we complete the study of the normal holonomy groups of complex submanifolds (non nec. complete) of Cn or CPn. We show that irreducible but non transitive normal holonomies are exactly the Hermitian s-representations of [CD09, Table 1] (see Corollary 1.1). For each one of them we construct a non necessaril…
This paper is a continuation of Part I where the general setup was developed. Here we discuss the general equivalence problem for geometric structures and provide criteria for the equivalence, local and global, of transitive structures. Cartan's Flag Systems illustrate the theory as a major example and, finally, some a…
This paper is devoted to higher dimensional Anosov flows and consists of two parts. In the first part, we investigate fiberwise Anosov flows on affine torus bundles which fiber over 3-dimensional Anosov flows. We provide a dichotomy result for such flows --- they are either suspensions of Anosov diffeomorphisms or the …
We study the control system of a Riemannian manifold M of dimension n rolling on the sphere Sn. The controllability of this system is described in terms of the holonomy of a vector bundle connection which, we prove, is isomorphic to the Riemannian holonomy group of the cone C(M) of M. Using Berger's list, we…
This thesis extends Yang-Mills theory to Lie groupoids and algebroids, overcoming integrability and transitivity constraints.
problem Generalizing Yang-Mills theory to non-integrable and non-transitive settings.
method Introduces multiplicative Ehresmann connections and develops the theory of connections on Lie groupoids and algebroids.
result Extends Yang-Mills theory to a non-integrable and non-transitive setting, providing a pair of equations for gauge fields.
A Markov Chain approach for aligning generative models from pairwise human preferences.
problem Aligning generative models from pairwise human preferences.
method Markov Chain from Human Feedback (MCHF)
result MCHF converges geometrically fast to the stationary distribution.
This paper investigates generalisation in multi-agent games, where the generality of the agent can be evaluated by playing against opponents it hasn't seen during training. We propose two new games with concealed information and complex, non-transitive reward structure (think rock/paper/scissors). It turns out that mos…
Extends Yang-Mills theory to non-integrable Lie algebroids.
problem Developing a Yang-Mills theory for non-integrable Lie algebroids.
method Generalized Yang-Mills theory to Lie algebroids, introducing multiplicative Ehresmann connections.
result Derived self-dual solutions (instantons) in 4 and 5 dimensions.
Functional dimension varies in ReLU networks, with implications for symmetry and connectivity.
problem Understanding the functional dimension of ReLU neural networks.
method Careful definition and analysis of functional dimension, study of quotient space and fibers.
result Functional dimension is inhomogeneous and can be non-constant, with implications for symmetry and connectivity.
The paper reveals a spinning top geometry in real-world games.
problem Understanding the structure of real-world games.
method Developed a spinning top geometric model and used Nash clustering.
result Real-world games exhibit a spinning top structure with transitive and non-transitive dimensions.
Learning product representations that reflect complementary relationship plays a central role in e-commerce recommender system. In the absence of the product relationships graph, which existing methods rely on, there is a need to detect the complementary relationships directly from noisy and sparse customer purchase ac…
The interaction between transitivity and sparsity, two common features in empirical networks, implies that there are local regions of large sparse networks that are dense. We call this the blessing of transitivity and it has consequences for both modeling and inference. Extant research suggests that statistical inferen…