arXiv research

Study of control problems on Carnot groups with SO(3) symmetry using geometric algebra.

Geometrically deforms $L_\infty$ algebras to Lie algebroids, revealing new invariants.

Paper connects geometric structures to algebra in high dimensions.

In this paper we study the difference between algebraic and geometric solutions of the hyperbolic Dehn filling equations for ideally triangulated 3-manifolds. We show that any geometric solution is an algebraic one, and we prove the uniqueness of the geometric solutions. Then we do explicit calculations for three inter…

Geometric deformations preserve post-Lie algebra structure in regularity structures.

The study examines extensions of Lie algebras with specific geometric structures.

Geometric AD framework simplifies derivative computation in JAX.

Geometric Algebra Transformer (GATr) handles various geometric data types efficiently.

Generative model designs highly designable proteins using geometric algebra.

Study uses geometric algebra to analyze credit cycles, revealing dangerous feedback loops.

Abstract: Geometrically reformulates estimation theory for finite-dimensional C*-algebras.

This dissertation explores Clifford bundles and spinor fields in geometric and algebraic contexts.

We study the 8 natural GL equivariant geometric realization questions for the space of generalized algebraic curvature tensors. All but one of them is solvable; a non-zero projectively flat Ricci antisymmetric generalized algebraic curvature is not geometrically realizable by a projectively flat Ricci antisymmetric tor…

Unified geometric framework for quantum states using dual number algebras.

Geometric structures over algebras describe geodesics and spaces.

A novel geometric algebra-based KG embedding framework improves link prediction.

An analogue of geometric quantization of Poisson algebras obtained by algebraic reduction of symmetries is developed. Interpretation of the obtained results and their application to the problem of commutativity of quantization and reduction are given

Randomized Geometric Algebra for Convex Neural Networks Optimizes Transfer Learning.

This is the first paper in a series of eight where in the first three we develop a systematic approach to the geometric algebras of multivectors and extensors, followed by five papers where those algebraic concepts are used in a novel presentation of several topics of the differential geometry of (smooth) manifolds of …

Geometric models for Lie algebras from simple singularities.

Study quandle modules over geometric quandles and their relation to Lie-Yamaguti representations.

Study connects spectral and algebraic torsion in geometric contexts.

This is the first paper in a series (of four) designed to show how to use geometric algebras of multivectors and extensors to a novel presentation of some topics of differential geometry which are important for a deeper understanding of geometrical theories of the gravitational field. In this first paper we introduce t…

Some natural hidden symmetries in the Verma modules over the Virasoro algebra are constructed in terms of geometric quantization. Their differential geometric meaning is established and their expression via $q_R$-conformal symmetries in the Verma modules over the Lie algebra $sl(2,C)$ is found. The analysis and the unr…

Maps geometric deformations to algebraic classes in Lie groupoids and algebroids.

These lectures given in Montreal in Summer 1997 are mainly based on, and form a condensed survey of, the book by N. Chriss and V. Ginzburg: `Representation Theory and Complex Geometry', Birkhauser 1997. Various algebras arising naturally in Representation Theory such as the group algebra of a Weyl group, the universal …

Geometrically proves Lie algebras are identified by their Iwasawa subalgebras.

We show that every Kaehler algebraic curvature tensor is geometrically realizable by a Kaehler manifold of constant scalar curvature. We also show that every para-Kaehler algebraic curvature tensor is geometrically realizable by a para-Kaehler manifold of constant scalar curvature

Researchers address the generation of differential invariants for geometric structures.

L-GATr transforms high-energy physics data using geometric algebra and Lorentz symmetry.

Study of Type IIA flow on symplectic Lie algebras for geometric structures.

The approach we present is a modification of the Morse theory for unital C*-algebras. We provide tools for the geometric interpretation of noncommutative CW complexes. These objects were introduced and studied in [2],[7] and [14]. Some examples to illustrate these geometric information in practice are given. A classifi…

Motivated by Kohno's result on the holonomy Lie algebra of a hyperplane arrangement, we define the holonomy Lie algebra of a finite geometric lattice in a combinatorial way. For a solvable pair of lattices, we show that the holonomy Lie algebra is an almost-direct product of the holonomy Lie algebra of the sublattice a…

In this paper, we investigate the relationship between algebraic soliton metrics and soliton metrics for geometric evolution equations on Lie groups. After discussing the general relationship between algebraic soliton metrics and soliton metrics, we investigate the cross curvature flow and the second order renormalizat…

The paper extends a geometric model using singular curves.

Geometrically convex return risk measures on AM-algebras

Transformed geometry into algebra to prove Pick's theorem efficiently.

We prove that the set of non-degenerate second order maximally superintegrable systems in the complex Euclidean plane carries a natural structure of a projective variety, equipped with a linear isometry group action. This is done by deriving the corresponding system of homogeneous algebraic equations. We then solve the…

We show that the algebraic intersection number of Scott and Swarup for splittings of free groups coincides with the geometric intersection number for the sphere complex of the connected sum of copies of $S^2\times S^1$.

New algebraic framework for Jacobi manifolds connects geometric mechanics and dimensional analysis.

Novel analysis of neural networks using geometric algebra and convex optimization.

Study algebraic K-theory of 3-manifold groups using Farrell-Jones isomorphism and geometrization.

We work in both the complex and in the para-complex categories and examine (para)-Kähler Weyl structures in both the geometric and in the algebraic settings. The higher dimensional setting is quite restrictive. We show that any (para)-Kaehler Weyl algebraic curvature tensor is in fact Riemannian in dimension at least 6…

Equivalence proven between algebraic stability and geometric stability.

This paper gives an algebraic conjecture which is shown to be equivalent to Thurston's Geometrization Conjecture for closed, orientable 3-manifolds. It generalizes the Stallings-Jaco theorem which established a similar result for the Poincare Conjecture. The paper also gives two other algebraic conjectures; one is equi…

We characterize unimodular solvable Lie algebras with Vaisman structures in terms of Kähler flat Lie algebras equipped with a suitable derivation. Using this characterization we obtain algebraic restrictions for the existence of Vaisman structures and we establish some relations with other geometric notions, such as Sa…

Based on representation theory of Clifford algebra, Ferus, Karcher and Münzner constructed a series of isoparametric foliations. In this paper, we will survey recent studies on isoparametric hypersurfaces of OT-FKM type and investigate related geometric constructions with mean curvature flow.

Study group actions on hyperbolic spaces to find algebraic and geometric properties.