arXiv research

Study Poisson algebras for Hamiltonian systems linearization.

Investigates linearity of group amalgams and new examples of non-linear groups.

CoLA automates efficient numerical linear algebra for complex matrix structures.

LightOn OPUs accelerate randomized numerical linear algebra, reducing computational costs.

Unified determinants via a single equation.

Investigates integrable systems with linear periodic integral for e(3) Lie algebra.

We show that formal isomorphism of intransitive linear Lie equations along transversal to the orbits can be extended to neighborhoods of these transversal. In analytic cases, the word formal is dropped from theorems. Also, we associate an intransitive Lie algebra with each intransitive linear Lie equation, and from the…

New basis and Schur-Weyl duality for loop Hecke algebra defined.

Efficiently solves inverse PDE problems with Gaussian processes.

This chapter is based on lectures on Randomized Numerical Linear Algebra from the 2016 Park City Mathematics Institute summer school on The Mathematics of Data.

Theory of ends of spaces using linear algebra.

New examples show algebraically slice knots with specific genus bounds.

These are lecture notes that are based on the lectures from a class I taught on the topic of Randomized Linear Algebra (RLA) at UC Berkeley during the Fall 2013 semester.

In planar algebras, we show how to project certain simple "quadratic" tangles onto the linear space spanned by "linear" and "constant" tangles. We obtain some corollaries about the principal graphs and annular structure of subfactors.

Study on Hermitian curvature flow on special linear groups, disproving a conjecture and finding non-algebraic solitons.

Study Poisson cohomology and linearize Lie algebra structures.

We develop the theory of linear algebra over a (Z_2)^n-commutative algebra (n in N), which includes the well-known super linear algebra as a special case (n=1). Examples of such graded-commutative algebras are the Clifford algebras, in particular the quaternion algebra H. Following a cohomological approach, we introduc…

In this note we prove that an analytic symplectic action of a semisimple Lie algebra can be locally linearized in Darboux coordinates. This result yields simultaneous analytic linearization for Hamiltonian vector fields in a neighbourhood of a common zero. We also provide an example of smooth non-linearizable Hamiltoni…

Necessary and sufficient conditions for the exponentiation of finite-dimensional real Lie algebras of linear operators on complete Hausdorff locally convex spaces are obtained, focused on the equicontinuous case - in particular, necessary conditions for exponentiation to compact Lie groups are established. Applications…

The notion of Poisson manifold with compatible pseudo-metric was introduced by the author in [1]. In this paper, we introduce a new class of Lie algebras which we call a pseudo-Rieamannian Lie algebras. The two notions are strongly related: we prove that a linear Poisson structure on the dual of a Lie algebra has a com…

By using the viewpoint of modern computational algebraic geometry, we explore properties of the optimization landscapes of the deep linear neural network models. After clarifying on the various definitions of "flat" minima, we show that the geometrically flat minima, which are merely artifacts of residual continuous sy…

A simplified proof for satellite knot signatures.

Classifies Lie symmetry algebras for 2D quasilinear equations, linking symmetry to linearity.

New hyperbolic knots not concordant to algebraic ones found.

Computes the component group of arbitrary real algebraic groups.

We classify all of the 4-dimensional linear Poisson structures of which the corresponding Lie algebras can be considered as the extension by a derivation of 3-dimensional unimodular Lie algebras. The affine Poisson structures on R^3 are totally classified.

Unified geometric framework for quantum states using dual number algebras.

This research introduces Lie brackets on spaces of biderivations in Lie algebras.

Linear algebra algorithms are used widely in a variety of domains, e.g machine learning, numerical physics and video games graphics. For all these applications, loop-level parallelism is required to achieve high performance. However, finding the optimal way to schedule the workload between threads is a non-trivial prob…

New proof of SL(n) skein algebra for twice punctured sphere, showing it's a polynomial algebra.

Introduces mobility algebra for modeling geodesics on n-spheres.

A linear Lie rack structure on a finite dimensional vector space $V$ is a Lie rack operation $(x,y)\mapsto x\rhd y$ pointed at the origin and such that for any $x$, the left translation $\mathrm{L}_x:y\mapsto \mathrm{L}_x(y)= x\rhd y$ is linear. A linear Lie rack operation $\rhd$ is called analytic if for any $x,y\in V…

The paper analyzes privacy leakage in federated learning using linear algebra and optimization theory.

New characterization of vector bundles using Lie algebras of symbols.

The paper characterizes vector bundles and differential operators using Lie algebras and their symbols.

We study phase retrieval from magnitude measurements of an unknown signal as an algebraic estimation problem. Indeed, phase retrieval from rank-one and more general linear measurements can be treated in an algebraic way. It is verified that a certain number of generic rank-one or generic linear measurements are suffici…

Study algebraic invariants from lightning self-attention models.

Linear algebra approach for parallel deep learning models.

This paper offers a new algebraic perspective of GCCA using subspace intersection.

We relate canonical algebraic curvature tensors that are built from a self-adjoint ($R^S_A$) or skew adjoint ($R^Λ_A$) linear operator A. Several authors have proven that any algebraic curvature tensor $R$ may be expressed as a sum of $R^S_A$, or as a sum of $R^Λ_A$. This motivates our interest in relating them as well…

We develop an approach to construct Poisson algebras for non-linear scalar field theories that is based on the Cahiers topos model for synthetic differential geometry. In this framework the solution space of the field equation carries a natural smooth structure and, following Zuckerman's ideas, we can endow it with a p…

Irreducible skew-Berger algebras $\g\subset\gl(n,\Co)$, i.e. algebras spanned by the images of the linear maps $R:\odot^2\Co^n\to\g$ satisfying the Bianchi identity, are classified. These Lie algebras can be interpreted as irreducible complex Berger superalgebras contained in $\gl(0|n,\Co)$.

Given a front projection of a Legendrian knot $K$ in $\mathbb{R}^{3}$ which has been cut into several pieces along vertical lines, we assign a differential graded algebra to each piece and prove a van Kampen theorem describing the Chekanov-Eliashberg invariant of $K$ as a pushout of these algebras. We then use this the…

Second part of proving linearization theorem for sl2(C).

Algebraic treatment of connection reduction over a special disc.

Results on characterization of manifolds in terms of certain Lie algebras growing on them, especially Lie algebras of differential operators, are reviewed and extended. In particular, we prove that a smooth (real-analytic, Stein) manifold is characterized by the corresponding Lie algebra of linear differential operator…

In order to understand the structure of the cohomologies involved in the study of projectively equivariant quantizations, we introduce a notion of affine representation of a Lie algebra.We show how it is related to linear representations and 1-cohomology classes of the algebra. We classify the affine representations of…

New algebraic structure derived from Kähler manifolds.