Ashani Dasgupta

Author: Ashani Dasgupta

  • Game of Mario in Heisenberg group

    Yesterday we were discussing the Heisenberg group in our weekly Geometric group theory workshop. The game of Mario came up! If you have not played Mario, then here is a the only thing you need to know. Mario can jump and hit floating bricks. These bricks may pop open to produce gold coins.

    The method of simplifying products of elements in Heisenberg group is similar. First a few definitions.

    Let \( \Gamma \) be a group generated by \( \{a, b, c\} \). Moreover set of relators contain \( ac = ca, bc = cb, ba = cab \). We say \(\Gamma\) is the Heisenberg group.

    One way to think about this is: \( c\) commutes with everything, \(a\) and \( b\) do not commute but they are \( c\) away from actually commuting. Hence we do not have \( ba = ab \) but we have \( ba = c \cdot ab \).

    We claim that every element of Heisenberg group looks like \( a^m b^n c^p \) where \( (m, n, p) \in \mathbb{Z}^3 \). We play a game of Mario with the elements of Heisenberg group to illustrate this.

    Choose two elements \( a^m b^n c^p \) and \( a^r b^s c^t \) in \( \Gamma \). Lets multiply them and see what happens.

    \[ a^m b^n c^p \cdot a^r b^s c^t \]

    Since \( c \) commutes with everything, hence we can pull the \( c \)’s to the right and get the first simplification.

    \[ a^m \cdot b^n \cdot a^r \cdot b^s \cdot c^t \cdot c^p \]

    Now we focus on the \( b^n \cdot a^r \) in the middle of the expression. Think of each \( b\) as Mario and each of the \(a\)’s in \(a^r \) as a gold coin bearing brick. We will push the \( b\) to the right replacing \( b \cdot a \) by \(c \cdot a \cdot b\). The gold coin is \(c\). That is if we push the \(b\) to the right through one \(a\) then one \(c \) is produced and \(ba\) becomes \(ab\).

    Now we can push that \( b \) through all the \( r \) copies of \(a\) so that it moves to the right and sits with \( b^s \). In this process \( r \) gold coins (\(c\)’s) will be produced. Hence we have \( c^r \) produced.

    We will need to do this \(n \) times (for each of the Mario’s or \(b\)’s in \(b^n \)). Hence \( n \times r \) gold coins or \(c\)’s are produced in the process. After this is done we have the following expression.

    \[ a^m \cdot a^r \cdot c^{nr} \cdot b^n \cdot b^s \cdot c^t \cdot c^p \]

    Finally we push all the \( c\)’s to the right and get the final expression.

    \[ a^{m+r} b^{n+s} c^{t + p + nr} \]

  • Not far not near groups

    We use \(NFNN\) as an abbreviation for ‘not far not near’.

    Annulus \(A(w, r, R)\) in a metric space \((X, d)\), for \(w \in X, 0 < r < R < \infty \) is defined as the set \[A =\{ x \in X | r \leq d(w, x) \leq R \} \].

    Definition

    Suppose \(P\) is a finitely generated, one-ended group. Let \(\Gamma(P, S)\) be its Cayley graph with respect to some finite, symmetric generating set \(S\). We say \(P\) is \(M-NFNN\) if there exists an integer \(M\) such that given any two points \(x,y \in \Gamma(P)\) with \(2M \leq r_x \leq r_y\), where \(r_x = d(e,x)\) and \(r_y = d(e,y)\), there exists an arc in \(A(e, \frac{r_x}{3}, 2 r_y) \subset \Gamma(P)\) that connects \(x\) and \(y\).

    Question

    Characterise NFNN groups.

    Remark

    Finitely presented groups are \(M-NFNN\) for \(M\) equal to the largest length of a relator.

  • About the ‘Creative Math’ Project

    About the ‘Creative Math’ Project

    I was speaking to Mr. Paul at a petrol pump in Kolkata. He lives in Birati and has an eight year old daughter. Mr. Paul noticed that the little girl has a keen interest in mathematics. People in Kolkata are chatty. Perhaps my attire and demeaner gives away my profession. He enquired briefly about the nature of my work (‘ki koren’) and then earnestly requested my counsel: ‘how should I help my daughter with mathematics?

    I tried to explain a few of my ideas. ‘She should do interesting problems, parents should informally chat about such ideas at dinner table, she should not enter into formal training’ were my usual suggestions. Mr. Paul told me about the kid’s involvement with abacus. ‘She is in 4th year. I am worried if it is creating excessive pressure on her brain.’

    Many kids do abacus at an early age. This ancient Chinese calculation tool has really made some inroads in Indian households. Personally I do not believe that calculation based exercises should have such a center-stage in children’s lives. It creates a wrong impression about the more important aspects of mathematics; deduction, imagination, construction and so on.

    ‘Why don’t you use’… I paused a little for the appropriate phrase…’creative problems?’

    I had the math circle experience of eastern Europe in my mind. Additionally I was thinking about the hands-on, constructivist approach suggested by Cedric Villani, the holistic approach to education suggested by Tagore and so on. I needed something catchy, possibly in at most two words. as an alternative to ‘abacus’ and routine mathematics.

    Apparently Mr. Paul found the phrase actionable. ‘Where can I find this thing; the creative problems.’

    The Thousand Flowers program and Math Olympiad program at Cheenta certainly uses what I have in mind. It has been implemented systematically over the last few years. We certainly noticed that the effect of this program on children is quite profound. It transforms the way they approach mathematics.

    This program at Cheenta draws from many resources. However there is no one-stop shop for all of these ideas and tools. Over the last few years I have contemplated several times about creating a book that will incorporate the ideas and tools that we use in this program.

    The ‘Creative Math’ project is an attempt to write such a book. Ideally there should be 12 books corresponding to 12 years of schooling for a child. It should contain theory, problems and projects that students, teachers and parents can use. The thrust of these books is to introduce non-routine problem solving and an attitude for research in children. Moreover, it should encourage activities that promotes a culture that celebrates intellectual and spiritual happiness over material well-being. After all, as Gabor Szego pointed out, a deep involvement in mathematical enquiry over a sustained period of time is possible only when there is a cultural preference of such pursuits.