arXiv subjects
Publications and source records attributed to Alexandre Borovik.
We briefly explain how to implement the morphisms in our paper ``Natural representations of black box groups encrypting $SL_2(\mathbb{F})$" and provide some examples.
We give a review of one of the lines in development of the theory of groups of finite Morley rank. These groups naturally appear in model theory as model-theoretic analogues of Galois groups, therefore their actions and their role as permutation groups is of primary interest. We restrict our story to the study of connected groups of finite Morley rank $G$ acting in a definably primitive way on a set $X$ and containing a definable abelian normal subgroup $V$ which acts on $X$ regularly -- the so-called \emph{primitive groups of affine type}. For reasons explained in the paper, this case plays a central role in the theory.
In this work, we complete the classification of generically multiply transitive actions of groups on solvable groups in the finite Morley rank setting. We prove that if $G$ is a connected group of finite Morley rank acting definably, faithfully and generically $m$-transitively on a connected solvable group $V$ of finite Morley rank where $\operatorname{rk}(V)\leqslant m$, then $\operatorname{rk}(V)=m$, $V$ is a vector space of dimension $m$ over an algebraically closed field $F$, $G\cong \operatorname{GL}_m(F)$, and the action is equivalent to the natural action of $\operatorname{GL}_m(F)$ on $F^m$. This generalises our previous work arXiv:2107.09997. As an application of our result, we classify definably primitive groups of finite Morley rank and affine type acting on a set $X$ with a generic transitivity degree of $\operatorname{rk}(X)+1$.
This paper provides an example of structural proxies for black box rings encrypting rings of 2 by 2 matrices of finite fields of odd order.
University level mathematics in a number of countries is under pressure to `decolonise the curriculum'. This paper considers, as a test case, a possible `decolonisation' of linear algebra. This is a representative case, since linear algebra is one of the core courses of undergraduate mathematics and a mathematical discipline with a millennia long historic tradition. This paper is written for my colleagues, university mathematicians. In my opinion, it could help them to determine their position and calmly stick to it without entering into an unnecessary debate with promoters of `decolonisation of the curricula'. The paper offers a simple and honest defence against `decolonisation' pressures: tell students the real (pre)history of a particular mathematical discipline. Let us call this activity `historical enrichment'. It would be useful if all attempts at `decolonization' (and vice versa, `historical enrichment') were known to a wider circle of the mathematical community. Publicity and an open discussion are the best way to resist outside pressures to engage in virtue signalling at the expense of historical and mathematical truth. The international mathematical community should defend academic freedom and insist on our right to formulate our curricula and evaluate the history of mathematics and judge mathematicians of the past according to criteria developed within the profession, and ignore any kind of political fads and pressure
The words ``Programming is the second literacy'' were coined more than 40 years ago but never came to life. This paper is one in the series of papers aimed at the analysis of mathematical requirements for a merge of school mathematics with computer science and computer programming. First indications are this demands development of quite serious mathematical tools most of which, hopefully, will be hidden "under the hood'' of software systems used in the process, but many will feature prominently in the Domain Specific Language needed for support of mathematical exchanges between Learner, Teacher, and Computer. We focus on "hardcore" mathematical aspects of this development.
Leibniz scholarship is currently an area of lively debate. We respond to some recent criticisms by Archibald et al.
In the history of infinitesimal calculus, we trace innovation from Leibniz to Cauchy and reaction from Berkeley to Mansion and beyond. We explore 19th century infinitesimal lores, including the approaches of Simeon-Denis Poisson, Gaspard-Gustave de Coriolis, and Jean-Nicolas Noel. We examine contrasting historiographic approaches to such lores, in the work of Laugwitz, Schubring, Spalt, and others, and address a recent critique by Archibald et al. We argue that the element of contingency in this history is more prominent than many modern historians seem willing to acknowledge.
In the Soviet Union a reform movement in mathematics education was triggered by Andrey Kolmogorov in the 1970s, and followed by a counter-reform. This movement was rooted in the very different socioeconomic conditions of that time and place, and followed a strategy with very significant contrasts to similar programs in the USA, England, or France. This provides an interesting case study which may illuminate the way such movements arise and succeed or fail, and, at the social level, certain fundamental commonalities of constraints as well as significant differences according to local conditions. We shall show that the principal reasons of the failure of the Kolmogorov reform were political: (1) The reform ignored the reality of the socio-economic conditions of the country; (2) The human factor was ignored, and very little attention was given to professional development and retraining of, and methodological help to, the whole army of teachers; (3) An attempt to transfer mathematical content and methods from the highly successful advanced extension stream for mathematically strong and highly engaged children to mainstream education was an especially grievous error.
Mathematics enters the period of change unprecedented in its history, perhaps even a revolution: a switch to use of computers as assistants and checkers in production of proofs. This requires rethinking traditional approaches to mathematics education which is struggling through a crisis of its own, socio-economic and political by its nature. The mathematical community faces Pandora's box of problems, which, surprisingly, are not usually discussed in any connected form. The present paper attempts to address this issue in a bit more joint and cohesive way.
We investigate the configuration where a group of finite Morley rank acts definably and generically $m$-transitively on an elementary abelian $p$-group of Morley rank $n$, where $p$ is an odd prime, and $m\geqslant n$. We conclude that $m=n$, and the action is equivalent to the natural action of $\operatorname{GL}_n(F)$ on $F^n$ for some algebraically closed field $F$. This strengthens our earlier result in arXiv:1802.05222, and partially answers two problems posed in [9].
The classical platonist / formalist dilemma in philosophy of mathematics can be expressed in lay terms as a deceptively naive question: \emph{Is new mathematics discovered or invented? Using examples from my own mathematical work during the Coronavirus lockdown, I argue that there is also a third way: new mathematics can also be inherited. And entering into possession, making it your own, could be great fun.
This paper discusses, from a mathematician's point of view, the thesis formulated by Israel Gelfand, one of the greatest mathematicians of the 20th century, and one of the pioneers of mathematical biology: "There is only one thing which is more unreasonable than the unreasonable effectiveness of mathematics in physics, and this is the unreasonable ineffectiveness of mathematics in biology."
This paper develops some general results about actions of finite groups on (infinite) abelian groups in the finite Morley rank category. They are linked to a range of problems on groups of finite Morley rank discussed in [16]. Crucially, these results are needed for the forthcoming work by Ay\c{s}e Berkman and myself [5] where we remove the `sharpness' assumption from [4]. Also, they yield a proof of the long standing conjecture of linearity of irreducible definable actions of simple algebraic groups on elementary abelian $p$-groups of finite Morley rank [16, Conjecture 12].
This paper is a compressed summary of some principal definitions and concepts in the approach to the black box algebra being developed by the authors. We suggest that black box algebra could be useful in cryptanalysis of homomorphic encryption schemes, and that homomorphic encryption is an area of research where cryptography and black box algebra may benefit from exchange of ideas.
Given a global exponent $E$ for a black box group $\mathsf{Y}$ encrypting ${\rm SL}_2(\mathbb{F})$, where $\mathbb{F}$ is an unknown finite field of unknown odd characteristic, we construct, in probabilistic time polynomial in $\log E$, the isomorphisms \[ \mathsf{Y} \longleftrightarrow {\rm SL}_2(\mathsf{K}), \] where $\mathsf{K}$ is a black box field encrypting $\mathbb{F}$. Our algorithm makes no reference to any additional oracles. We also give similar algorithms for black box groups encrypting ${\rm PGL}_2(\mathbb{F})$, ${\rm PSL}_2(\mathbb{F})$.
The present survey aims at being a list of Conjectures and Problems in an area of model-theoretic algebra wide open for research, not a list of known results. To keep the text compact, it focuses on structures of finite Morley rank, although the same questions can be asked about other classes of objects, for example, groups definable in $ω$-stable and $o$-minimal theories. In many cases, answers are not known even in the classical category of algebraic groups over algebraically closed fields.
We describe structure of locally finite groups of finite centraliser dimension.