Related papers: Finite matrix multiplication algorithms from infin…
We present an exposition of our ongoing project in a new area of applicable mathematics: practical computation with finitely generated linear groups over infinite fields. Methodology and algorithms available for practical computation in…
In the context of group-theoretic fast matrix multiplication the TPP capacity is used to bound the exponent $\omega$ of matrix multiplication. We prove a new and sharper upper bound for the TPP subgroup capacity of a finite group
Let $n$ be a positive integer and $\mathcal M$ a set of rational $n \times n$-matrices such that $\mathcal M$ generates a finite multiplicative semigroup. We show that any matrix in the semigroup is a product of matrices in $\mathcal M$…
We describe an algorithm that computes the index of a finitely generated subgroup in a finitely $L$-presented group provided that this index is finite. This algorithm shows that the subgroup membership problem for finite index subgroups in…
In this work the algorithms of fast multiplication of matrices are considered. To any algorithm there associated a certain group of automorphisms. These automorphism groups are found for some well-known algorithms, including algorithms of…
We study the capability of the Fast Fourier Transform (FFT) to accelerate exact and approximate matrix multiplication without using Strassen-like divide-and-conquer. We present a simple exact algorithm running in $O(n^{2.89})$ time, which…
Using a ``3 by 3 matrix trick'' we previously showed that multiplication in a C*-algebra A, an algebraic structure, is determined by the geometry of the C*-algebra of the 3 by 3 matrices with entries from A. As an application of this…
We generalize the classical construction principles of infinite-dimensional real (and complex) Lie groups to the case of Lie groups over non-discrete topological fields. In particular, we discuss linear Lie groups, mapping groups, test…
Feit and Fine derived a generating function for the number of ordered pairs of commuting n by n matrices over the finite field F_q. This has been reproved and studied by Bryan and Morrison from the viewpoint of motivic Donaldson-Thomas…
We present a new algorithm to decide finiteness of matrix groups defined over a field of positive characteristic. Together with previous work for groups in zero characteristic, this provides the first complete solution of the finiteness…
We introduce a topological property for finitely generated groups called stackable that implies the existence of an inductive procedure for constructing van Kampen diagrams with respect to a particular finite presentation. We also define…
We give an algorithm to determine finitely many generators for a subgroup of finite index in the unit group of an integral group ring $\mathbb{Z} G$ of a finite nilpotent group $G$, this provided the rational group algebra $\mathbb{Q} G$…
Cohn and Umans proposed a framework for developing fast matrix multiplication algorithms based on the embedding computation in certain groups algebras. In subsequent work with Kleinberg and Szegedy, they connected this to the search for…
In comparison to graphs, combinatorial methods for the isomorphism problem of finite groups are less developed than algebraic ones. To be able to investigate the descriptive complexity of finite groups and the group isomorphism problem, we…
Lie group theory states that knowledge of a $m$-parameters solvable group of symmetries of a system of ordinary differential equations allows to reduce by $m$ the number of equations. We apply this principle by finding some \emph{affine…
In this article we survey recent progress in the algorithmic theory of matrix semigroups. The main objective in this area of study is to construct algorithms that decide various properties of finitely generated subsemigroups of an infinite…
We use an extension of the diagrammatic rules in random matrix theory to evaluate spectral properties of finite and infinite products of large complex matrices and large hermitian matrices. The infinite product case allows us to define a…
We survey recent progress in computing with finitely generated linear groups over infinite fields, describing the mathematical background of a methodology applied to design practical algorithms for these groups. Implementations of the…
Matrix multiplication is a fundamental computation in many scientific disciplines. In this paper, we show that novel fast matrix multiplication algorithms can significantly outperform vendor implementations of the classical algorithm and…
We introduce a combinatorial property for finitely generated groups called stackable that implies the existence of an inductive procedure for constructing van Kampen diagrams with respect to a canonical finite presentation. We also define…