Alexandre Borovik

The book itself, "mathematics under the Microscope: Notes on Cognitive Aspects of Mathematical Practice", is available from http://www.ams.org/bookstore-getitem/item=mbk-71

So far, there are no corrections -- but a plenty of additions and comments.

I started to record systematically and analyse logical difficulties experienced by professional mathematicians at the earliest stages of their learning of elementary mathematics. I concentrate my study on recollections of my fellow mathematicians and on interviews with so-called “mathematically able” children because only they posses an adequate language which allows them to describe their personal experiences. My approach is justified by the success of Vadim Krutetskii's (1976) classical study of “mathematically able” children: it provides remarkable insights into mathematical thinking, and adult mathematicians instantly recognise themselves in Krutetskii's young subjects.

I describe hidden structures of elementary mathematics which may intrigue and – like shadows in the night – sometimes scare an inquisitive child. I hope that my findings could be useful to specialists in mathematical. But I refrain from making any recommendations on teaching of mathematics. My project is motivated by the desire to get a better understanding of the specific nature of mathematical practice in the mainstream “research'” mathematics.

This is an archetypal personal story that I am looking for; it has been told to me by a woman with a PhD in Mathematics who works in education:

As a girl aged 6, EHK could easily solve “put a number in the box” problems of the type 7 + [ ] = 12, by counting how many 1’s she had to add to 7 in order to get 12 but struggled with [ ] + 6 = 11, because she did not know where to start. Worse, she felt that she could not communicate her difficulty to adults. She still remembers her frustration.

At a mainstream pedagogical level, such cases are well studied (Hiebert 1982). What makes my approach different is that I identify mathematical structures hiding behind innocuously looking elementary arithmetic. Dozens of stories that I have collected so far prove that children may feel the presence of hidden structures and can be deeply frustrated when adults around them do not share their vision. Many my correspondents retain a strong bond with their “inner child”— just listen to Michael Gromov, one of the greatest mathematicians of our times:

"My personal evaluation of myself is that as a child till 8-9, I was intellectually better off than at 14. At 14-15 I became interested in math. It took me about 20 years to regain my 7 year old child perceptiveness."

My project is a first attempt to understand this ``child perceptiveness’’.

References

J. Hiebert (1982). The position of the unknown set and children's solutions of verbal arithmetic problems, J. Res. Math. Ed. 13, no. 5, 341-349.
V. A. Krutetskii (1976). The Psychology of Mathematical Abilities in Schoolchildren. The University of Chicago Press. ISBN 0-226-45485-1.

This expository text contains an elementary treatment of finite groups generated by reflections. There are many good books on this subject. It has to be said that the theory is a very much alive and developing.

The main reason why we decided to write another text is not mathematical but pedagogical: we wished to emphasize the intuitive elementary geometric aspects of the theory of reflection groups. In the theory of reflection groups, the underlying ideas of many proofs can be presented by simple drawings much better than by a dry verbal exposition. Probably for the reason of their extreme simplicity these elementary arguments are mentioned in most books only briefly and tangentially.

This is what Yuri Manin said about the book:

This is an unusual and unusually fascinating book.

Readers who never thought about mathematics after their school years will be amazed to discover how many habits of mind, ideas, and even material objects that are inherently mathematical serve as building blocks of our civilization and everyday life.

A professional mathematician, reluctantly breaking the daily routine, or pondering on some resisting problem, will open this book and enjoy a sudden return to his or her young days when mathematics was fresh, exciting, and holding all promises.

And do not take the word "microscope" in the title too literally: in fact, the author looks around, in time and space, focusing in turn on a tremendous variety of motives, from mathematical "memes" (genes of culture) to an unusual life of a Hollywood star.

Introduction. Zilber’s original trichotomy conjecture proposed an explicit classification of all one-dimensional objects arising in model theory. At one point, classifying the simple groups of finite Morley rank was viewed as a subproblem whose affirmative answer would justify this conjecture. Zilber’s conjecture was eventually refuted by Hrushovski [9], and the classification of simple groups of finite Morley rank remains open today. However, these conjectures hold in two significant cases. First, Hrushovski and Zilber prove the full trichotomy conjecture holds under very strong geometric assumptions [10], and this suffices for various diophantine applications. Second, the Even & Mixed Type Theorem [1] shows that simple groups of finite Morley rank containing an infinite elementary abelian 2-subgroup are Chevellay groups over an algebraically closed field of characteristic two. In this paper, we clarify some middle ground between these two results by eliminating involutions from si...

In this paper, I discuss emotions related to a person's control (or lack of control) of his/her mathematics: sense of danger; sense of security; confidence, feeling of strength; feeling of power; these eventually lead to the ultimate emotion of mathematics: realisation that you know and understand something that no-one else in the world knows or understands – and that you can prove that. These higher level emotions are not frequently discussed in the context of mathematics education. Remarkably, they are known not only to professional research mathematicians, but also experienced by many children in their first encounters with mathematics. And I dare to suggest that there is another overarching emotion well known to many professional mathematicians: the feeling of a deep connection with the “inner child.” I will focus on a child's perception of mathematics but will start my narrative from a prominent episode in the history of “adult” mathematics.

We introduce the concept of orientation for Lagrangian matroids represented in the flag variety of maximal isotropic subspaces of dimension N in the real vector space of dimension 2N+1. The paper continues the study started in math.CO/0209100.

The author of this paper is a university teacher and a parent o f bilingual children. Using his and his children’s early reading experiences in Russian and English, he attempts to explain the advantages of using synthetic phonics as a method of teac hing to read. The paper is motivated, to some degree, by the recent White Paper on Educa tion [] where the Government has expressed strong support to synthetic phonics— the position that the author of this text shares and supports in his personal capacity as a pa rent. Disclaimer. Needless to say, all opinions expressed here are of the autho r and no-one else. 1. English orthography as a metaphor for everything that can go wrong in education There once was a man who for hiccough Tried all of the cures he could piccough, And the best without doubt, As at last he found oubt, Is warm water and salt in a ticcough. ‡ I can only look in horror at the suffering of children taught t o read English with its non-phonetic and non-transparent orthog...

The classical platonist/formalist dilemma in philosophy of mathematics can be expressed in lay terms as a deceptively naive question: is new mathematics discovered or invented? Using an example from my own mathematical life, I argue that there is also a third way: new mathematics can also be inherited -- and in the process briefly discuss a remarkable paper by W. Burnside of 1900.

Abstract. The present paper proposes a new and systematic approach to the so-called black box group methods in computational group theory. Instead of a single black box, we consider categories of black boxes and their morphisms. This makes new classes of black box problems accessible. For example, we can enrich black box groups by actions of outer automorphisms. As an example of application of this technique, we construct Frobenius maps on black box groups of untwisted Lie type in odd characteristic (Sec-tion 6) and inverse-transpose automorphisms on black box groups encrypting (P)SLn(Fq). One of the advantages of our approach is that it allows us to work in black box groups over finite fields of big characteristic. Another advantage is explanatory power of our methods; as an example, we explain Kantor’s and Kassabov’s construction of an involution in black box groups encrypting SL2(2n). Due to the nature of our work we also have to discuss a few methodological

Tuna Altınel Department of Mathematics, Rutgers University New Brunswick, New Jersey 08903, USA e-mail: altinel@math.rutgers.edu ... Alexandre Borovik Department of Mathematics, University of Manchester Institute of Science and Technology Manchester ...

Cauchy's contribution to the foundations of analysis is often viewed through the lens of developments that occurred some decades later, namely the formalisation of analysis on the basis of the epsilon-delta doctrine in the context of an Archimedean continuum. What does one see if one refrains from viewing Cauchy as if he had read Weierstrass already? One sees, with Felix Klein, a parallel thread for the development of analysis, in the context of an infinitesimal-enriched continuum. One sees, with Emile Borel, the seeds of the theory of rates of growth of functions as developed by Paul du Bois-Reymond. One sees, with E. G. Bjorling, an infinitesimal definition of the criterion of uniform convergence. Cauchy's foundational stance is hereby reconsidered.

In this paper we present a definition of oriented Lagrangian symplectic matroids and their representations. Classical concepts of orientation and this extension may both be thought of as stratifications of thin Schubert cells into unions of connected components. The definitions are made first in terms of a combinatorial axiomatisation, and then again in terms of elementary geometric properties of the Coxeter matroid polytope. We also generalise the concept of rank and signature of a quadratic form to symplectic Lagrangian matroids in a surprisingly natural 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⩾ n. We conclude that m=n, and the action is equivalent to the natural action of 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].

A construction of the real number system based on almost homomorphisms of the integers Z was proposed by Schanuel, Arthan, and others. We combine such a construction with the ultrapower or limit ultrapower construction, to construct the hyperreals out of integers. In fact, any hyperreal field, whose universe is a set, can be obtained by such a one-step construction directly out of integers. Even the maximal (i.e., On-saturated) hyperreal number system described by Kanovei and Reeken (2004) and independently by Ehrlich (2012) can be obtained in this fashion, albeit not in NBG. In NBG, it can be obtained via a one-step construction by means of a definable ultrapower (modulo a suitable definable class ultrafilter).

We examine Paul Halmos' comments on category theory, Dedekind cuts, devil worship, logic, and Robinson's infinitesimals. Halmos' scepticism about category theory derives from his philosophical position of naive set-theoretic realism. In the words of an MAA biography, Halmos thought that mathematics is "certainty" and "architecture" yet 20th century logic teaches us is that mathematics is full of uncertainty or more precisely incompleteness. If the term architecture meant to imply that mathematics is one great solid castle, then modern logic tends to teach us the opposite lession, namely that the castle is floating in midair. Halmos' realism tends to color his judgment of purely scientific aspects of logic and the way it is practiced and applied. He often expressed distaste for nonstandard models, and made a sustained effort to eliminate first-order logic, the logicians' concept of interpretation, and the syntactic vs semantic distinction. He 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.

Given a global exponent E for a black box group Y encrypting SL_2(F), where F is an unknown finite field of unknown odd characteristic, we construct, in probabilistic time polynomial in log E, the isomorphisms Y SL_2(K), where K is a black box field encrypting F. Our algorithm makes no reference to any additional oracles. We also give similar algorithms for black box groups encrypting PGL_2(F), PSL_2(F).

The present paper proposes a new and systematic approach to the so-called black box group methods in computational group theory. Instead of a single black box, we consider categories of black boxes and their morphisms. This makes new classes of black box problems accessible. For example, we can enrich black box groups by actions of outer automorphisms. As an example of application of this technique, we construct Frobenius maps on black box groups of untwisted Lie type in odd characteristic (Section 6) and inverse-transpose automorphisms on black box groups encrypting (P)SL_n(F_q). One of the advantages of our approach is that it allows us to work in black box groups over finite fields of big characteristic. Another advantage is the explanatory power of our methods; as an example, we explain Kantor's and Kassabov's construction of an involution in black box groups encrypting SL_2(2^n). Due to the nature of our work we also have to discuss a few methodological issues of the bl...

Our meetingWhere will the next generation of UK mathematicians come from? will concentrate on the education policy issues arising from our desire to nurture future mathematical talent. However, a brief look at the programme of the meeting shows that no discussion ofwhat mathematical abilities and talent are is scheduled. I hope that we have a shared understanding sufficient for a meaningful conversation. Nevertheless I believe that some coffee break chats about the nature of mathematical abilities and their early manifestations in children might be useful. To facilitate an informal discussion of a highly elusive topic, I have decided to offer my notes on mathematical thinking for the attention of the participants of the meeting. At this point, a disclaimer is necessary. I emphasise that I am not a psychologist nor a specialist in educational theory. My notes are highly personal and very subjective. They do not represent results of any systematic study. The notes are mostly based on ...

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 present a polynomial time Monte-Carlo algorithm for finite simple black box classical groups of odd characteristic which constructs all root SL(2,q)-subgroups associated with the nodes of the extended Dynkin diagram of the corresponding algebraic group.

We show that a minimal counter example to the Cherlin-Zilber Algebraicity Conjecture for simple groups of finite Morley rank has normal 2-rank at most two, which is a tameness free version of Borovik's original trichotomy theorem. This result serves as a bridge by showing that there are no groups found strictly between the generic and quasithin cases, i.e. between groups of Lie rank at least three, and groups of Lie rank one and two. Again this result depends upon previous work for the uniqueness case analysis.

We propose a simple one sided Monte-Carlo algorithm to distinguish, to any given degree of certainty, the symplectic group Cn(q) = PSp2n(q) from the orthogonal group Bn(q) = Ω2n+1(q) where q > 3 is odd and n and q are given. The algorithm does not use an order oracle and works in polynomial, of n log q, time. This paper corrects an error in the previously published version of the algorithm [1]. 2000 Mathematics Subject Classification: 20P05.

Coxeter matroids are combinatorial objects associated with finite Coxeter groups; they can be viewed as subsets M of the factor set W/P of a Coxeter group W by a parabolic subgroup P which satisfy a certain maximality property with respect to a family of shifted Bruhat orders on W/P . The classical matroids of matroid theory are exactly the Coxeter matroids for the symmetric group Symn (which is a Coxeter group of type An−1) and a maximal parabolic subgroup, while the maximality property turns out to be Gale’s classical characterisation of matroids [13]. The theory of Coxeter matroids sheds new light on the classical matroid theory and brings into the consideration a wider class of combinatorial objects [7]. Of this, we can specifically mention Lagrangian matroids, which are Coxeter matroids for the hyperoctahedral group BCn and a particular maximal parabolic subgroup. Lagrangian matroids are cryptomorphically equivalent to symmetric matroids or 2-matroids of Bouchet’s papers [9] an...

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 ${\rm PGL}_2(\mathbb{F})$, ${\rm PSL}_2(\mathbb{F})$.

I suggest a simple thought experiment. Science fiction books occasionally mention an imaginary device: a replicator. It consists of two boxes; you put an object in a box, close the lid, and instantly get its undistinguishable fully functional copy in the second box. In particular, a replicator can replicate smaller replicators. Now imagine the economy based on replicators. It needs two groups of producers: a very small group of engineers who build and maintain the biggest replicator and a very diverse, but still small, group of artisans, designers, and scientists who produce a single original prototype of each object. This hypothetical economy also needs service sector, mostly waste disposal. Next, try, if you can, imagine a sustainable, stable, equal, and democratic model of education that supports this lopsided economy. But this apocalyptic future is already upon us – in the information sector of economy, where computers act as replicators of information. Mathematics, due to its s...

In the present paper, we study homomorphic encryption as an area where principal ideas of black box algebra are particularly transparent. 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.

We introduce the concept of orientation for Lagrangian matroids represented in the flag variety of maximal isotropic subspaces of dimension N in the real vector space of dimension 2N+1. The paper continues the study started in math.CO/0209100.

We discuss the complexity of conjugacy problem in Miller’s groups. We stratify the groups in question and show that for “almost all”, in some explicit sense, elements, the conjugacy search problem is decidable in cubic time. It is worth noting that a Miller’a group may have undecidable conjugacy search problem; our results show that “hard” instances of the problem comprise a negligibly small part of the group.

* Quality of a course is, first of all, the quality of its content.

* Quality of course content is the quality of didactic transformation of the content.

* Quality of didactic transformation is the quality and depth of
mathematical work involved.