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diff --git a/posts/2013-06-01-q-robinson-schensted-paper.md b/posts/2013-06-01-q-robinson-schensted-paper.md new file mode 100644 index 0000000..657412d --- /dev/null +++ b/posts/2013-06-01-q-robinson-schensted-paper.md @@ -0,0 +1,31 @@ +--- +template: oldpost +title: A \(q\)-weighted Robinson-Schensted algorithm +date: 2013-06-01 +comments: true +tags: RS, \(q\)-Whittaker_functions, Macdonald_polynomials +archive: false +--- +In [this paper](https://projecteuclid.org/euclid.ejp/1465064320) with [Neil](http://www.bristol.ac.uk/maths/people/neil-m-oconnell/) we construct a \\(q\\)-version of the Robinson-Schensted +algorithm with column insertion. Like the [usual RS +correspondence](http://en.wikipedia.org/wiki/Robinson–Schensted_correspondence) +with column insertion, this algorithm could take words as input. Unlike +the usual RS algorithm, the output is a set of weighted pairs of +semistandard and standard Young tableaux \\((P,Q)\\) with the same +shape. The weights are rational functions of indeterminant \\(q\\). + +If \\(q\\in\[0,1\]\\), the algorithm can be considered as a randomised +RS algorithm, with 0 and 1 being two interesting cases. When +\\(q\\to0\\), it is reduced to the latter usual RS algorithm; while +when \\(q\\to1\\) with proper scaling it should scale to directed random +polymer model in [(O'Connell 2012)](http://arxiv.org/abs/0910.0069). +When the input word \\(w\\) is a random walk: + +\\begin{align\*}\\mathbb +P(w=v)=\\prod\_{i=1}^na\_{v\_i},\\qquad\\sum\_ja\_j=1\\end{align\*} + +the shape of output evolves as a Markov chain with kernel related to +\\(q\\)-Whittaker functions, which are Macdonald functions when +\\(t=0\\) with a factor. + + diff --git a/posts/2014-04-01-q-robinson-schensted-symmetry-paper.md b/posts/2014-04-01-q-robinson-schensted-symmetry-paper.md new file mode 100644 index 0000000..38874bb --- /dev/null +++ b/posts/2014-04-01-q-robinson-schensted-symmetry-paper.md @@ -0,0 +1,19 @@ +--- +template: oldpost +title: Symmetry property of \(q\)-weighted Robinson-Schensted algorithms and branching algorithms +date: 2014-04-01 +comments: true +archive: false +tags: RS, growth_diagrams +--- +In [this paper](http://link.springer.com/article/10.1007/s10801-014-0505-x) a symmetry property analogous to the well known symmetry +property of the normal Robinson-Schensted algorithm has been shown for +the \\(q\\)-weighted Robinson-Schensted algorithm. The proof uses a +generalisation of the growth diagram approach introduced by Fomin. This +approach, which uses "growth graphs", can also be applied to a wider +class of insertion algorithms which have a branching structure. + + + +Above is the growth graph of the \\(q\\)-weighted Robinson-Schensted +algorithm for the permutation \\({1 2 3 4\\choose1 4 2 3}\\). diff --git a/posts/2015-01-20-weighted-interpretation-super-catalan-numbers.md b/posts/2015-01-20-weighted-interpretation-super-catalan-numbers.md new file mode 100644 index 0000000..6d9e75e --- /dev/null +++ b/posts/2015-01-20-weighted-interpretation-super-catalan-numbers.md @@ -0,0 +1,40 @@ +--- +template: oldpost +title: AMS review of 'A weighted interpretation for the super Catalan numbers' by Allen and Gheorghiciuc +date: 2015-01-20 +comments: true +archive: false +--- +The super Catalan numbers are defined as +\$\$ T(m,n) = {(2 m)! (2 n)! \over 2 m! n! (m + n)!}. \$\$ + + This paper has two main results. First a combinatorial interpretation +of the super Catalan numbers is given: +\$\$ T(m,n) = P(m,n) - N(m,n) \$\$ +where \\(P(m,n)\\) +enumerates the number of 2-Motzkin paths whose \\(m\\) -th step begins at an even level (called \\(m\\)-positive paths) and \\(N(m,n)\\) +those with \\(m\\)-th step beginning at an odd level (\\(m\\)-negative paths). The proof uses a recursive argument on the number of +\\(m\\)-positive and -negative paths, based on a recursion of the super Catalan +numbers appearing in \[I. M. Gessel, J. Symbolic Comput. **14** (1992), no. 2-3, 179–194; +[MR1187230](http://www.ams.org/mathscinet/search/publdoc.html?r=1&pg1=MR&s1=1187230&loc=fromrevtext)\]: +\$\$ 4T(m,n) = T(m+1, n) + T(m, n+1). \$\$ +This result gives an expression for the super Catalan numbers in terms +of numbers counting the so-called ballot paths. The latter sometimes are +also referred to as the generalised Catalan numbers forming the entries +of the Catalan triangle. + + Based on the first result, the second result is a combinatorial +interpretation of the super Catalan numbers \\(T(2,n)\\) +in terms of counting certain Dyck paths. This is equivalent to a +theorem, which represents \\(T(2,n)\\) +as counting of certain pairs of Dyck paths, in \[I. M. Gessel and G. +Xin, J. Integer Seq. **8** (2005), no. 2, Article +05.2.3, 13 pp.; +[MR2134162](http://www.ams.org/mathscinet/search/publdoc.html?r=1&pg1=MR&s1=2134162&loc=fromrevtext)\], +and the equivalence is explained at the end of the paper by a bijection +between the Dyck paths and the pairs of Dyck paths. The proof of the +theorem itself is also done by constructing two bijections between Dyck +paths satisfying certain conditions. All the three bijections are +formulated by locating, removing and adding steps. + +Copyright notice: This review is published at http://www.ams.org/mathscinet-getitem?mr=3275875, its copyright owned by the AMS. diff --git a/posts/2015-04-01-unitary-double-products.md b/posts/2015-04-01-unitary-double-products.md new file mode 100644 index 0000000..a2f8ccb --- /dev/null +++ b/posts/2015-04-01-unitary-double-products.md @@ -0,0 +1,10 @@ +--- +title: Unitary causal quantum stochastic double products as universal interactions I +date: 2015-04-01 +template: oldpost +comments: true +archive: false +--- +In [this paper](http://www.actaphys.uj.edu.pl/findarticle?series=Reg&vol=46&page=1851) with [Robin](http://homepages.lboro.ac.uk/~marh3/) we show the explicit formulae for a family of unitary +triangular and rectangular double product integrals which can be +described as second quantisations. diff --git a/posts/2015-04-02-juggling-skill-tree.md b/posts/2015-04-02-juggling-skill-tree.md new file mode 100644 index 0000000..be39822 --- /dev/null +++ b/posts/2015-04-02-juggling-skill-tree.md @@ -0,0 +1,33 @@ +--- +template: oldpost +title: jst +date: 2015-04-02 +comments: true +archive: false +--- +jst = juggling skill tree + +If you have ever played a computer role playing game, you may have +noticed the protagonist sometimes has a skill "tree" (most of the time +it is actually a directed acyclic graph), where certain skills leads to +others. For example, +[here](http://hydra-media.cursecdn.com/diablo.gamepedia.com/3/37/Sorceress_Skill_Trees_%28Diablo_II%29.png?version=b74b3d4097ef7ad4e26ebee0dcf33d01) +is the skill tree of sorceress in [Diablo +II](https://en.wikipedia.org/wiki/Diablo_II). + +Now suppose our hero embarks on a quest for learning all the possible +juggling patterns. Everyone would agree she should start with cascade, +the simplest nontrivial 3-ball pattern, but what afterwards? A few other +accessible patterns for beginners are juggler's tennis, two in one and +even reverse cascade, but what to learn after that? The encyclopeadic +[Library of Juggling](http://libraryofjuggling.com/) serves as a good +guide, as it records more than 160 patterns, some of which very +aesthetically appealing. On this website almost all the patterns have a +"prerequisite" section, indicating what one should learn beforehand. I +have therefore written a script using [Python](http://python.org), +[BeautifulSoup](http://www.crummy.com/software/BeautifulSoup/) and +[pygraphviz](http://pygraphviz.github.io/) to generate a jst (graded by +difficulties, which is the leftmost column) from the Library of Juggling +(click the image for the full size): + +[{width="38em"}](../assets/resources/juggling.png) diff --git a/posts/2015-05-30-infinite-binary-words-containing-repetitions-odd-periods.md b/posts/2015-05-30-infinite-binary-words-containing-repetitions-odd-periods.md new file mode 100644 index 0000000..8f2ff8e --- /dev/null +++ b/posts/2015-05-30-infinite-binary-words-containing-repetitions-odd-periods.md @@ -0,0 +1,94 @@ +--- +template: oldpost +title: AMS review of 'Infinite binary words containing repetitions of odd period' by Badkobeh and Crochemore +date: 2015-05-30 +comments: true +archive: false +--- +This paper is about the existence of pattern-avoiding infinite binary +words, where the patterns are squares, cubes and \\(3^+\\)-powers. + There are mainly two kinds of results, positive (existence of an +infinite binary word avoiding a certain pattern) and negative +(non-existence of such a word). Each positive result is proved by the +construction of a word with finitely many squares and cubes which are +listed explicitly. First a synchronising (also known as comma-free) +uniform morphism \\(g\\: \\Sigma_3^\* \\to +\\Sigma_2^\*\\) + +is constructed. Then an argument is given to show that the length of +squares in the code \\(g(w)\\) +for a squarefree \\(w\\) is bounded, hence all the squares can be obtained by examining all \\(g(s)\\) for \\(s\\) +of bounded lengths. The argument resembles that of the proof of, e.g., +Theorem 1, Lemma 2, Theorem 3 and Lemma 4 in \[N. Rampersad, J. O. +Shallit and M. Wang, Theoret. Comput. Sci. **339** +(2005), no. 1, 19–34; +[MR2142071](http://www.ams.org/mathscinet/search/publdoc.html?r=1&pg1=MR&s1=2142071&loc=fromrevtext)\]. +The negative results are proved by traversing all possible finite words +satisfying the conditions. + + Let \\(L(n_2, n_3, S)\\) be the maximum length of a word with \\(n_2\\) distinct squares, \\(n_3\\) +distinct cubes and that the periods of the squares can take values only +in \\(S\\) +, where \\(n_2, n_3 \\in \\Bbb N \\cup +\\{\\infty, \\omega\\}\\) +and \\(S \\subset \\Bbb N_+\\) +. + \\(n_k = 0\\) +corresponds to \\(k\\)-free, \\(n_k = \\infty\\) +means no restriction on the number of distinct \\(k\\)-powers, and \\(n_k = \\omega\\) +means \\(k^+\\)-free. + + Below is the summary of the positive and negative results: + +(1) (Negative) \\(L(\\infty, \\omega, 2 \\Bbb N) + < \\infty\\) + : \\(\\nexists\\) + an infinite \\(3^+\\) + -free binary word avoiding all squares of odd periods. + (Proposition 1) + +(2) (Negative) \\(L(\\infty, 0, 2 \\Bbb N + 1) \\le + 23\\) + : \\(\\nexists\\) + an infinite 3-free binary word, avoiding squares of even periods. + The longest one has length \\(\\le 23\\) + (Proposition 2). + +(3) (Positive) \\(L(\\infty, \\omega, 2 \\Bbb N + + 1) = \\infty\\) + : \\(\\exists\\) + an infinite \\(3^+\\) + -free binary word avoiding squares of even periods (Theorem 1). + +(4) (Positive) \\(L(\\infty, \\omega, \\{1, 3\\}) = + \\infty\\) + : \\(\\exists\\) + an infinite \\(3^+\\) + -free binary word containing only squares of period 1 or 3 + (Theorem 2). + +(5) (Negative) \\(L(6, 1, 2 \\Bbb N + 1) = + 57\\) + : \\(\\nexists\\) + an infinite binary word avoiding squares of even period containing + \\(< 7\\) + squares and \\(< 2\\) + cubes. The longest one containing 6 squares and 1 cube has length 57 + (Proposition 6). + +(6) (Positive) \\(L(7, 1, 2 \\Bbb N + 1) = + \\infty\\) + : \\(\\exists\\) + an infinite \\(3^+\\) + -free binary word avoiding squares of even period with 1 cube and 7 + squares (Theorem 3). + +(7) (Positive) \\(L(4, 2, 2 \\Bbb N + 1) = + \\infty\\) + : \\(\\exists\\) + an infinite \\(3^+\\) + -free binary words avoiding squares of even period and containing 2 + cubes and 4 squares (Theorem 4). + + +Copyright notice: This review is published at http://www.ams.org/mathscinet-getitem?mr=3313467, its copyright owned by the AMS. diff --git a/posts/2015-07-01-causal-quantum-product-levy-area.md b/posts/2015-07-01-causal-quantum-product-levy-area.md new file mode 100644 index 0000000..cbf2fcf --- /dev/null +++ b/posts/2015-07-01-causal-quantum-product-levy-area.md @@ -0,0 +1,13 @@ +--- +template: oldpost +title: On a causal quantum double product integral related to Lévy stochastic area. +date: 2015-07-01 +archive: false +comments: true +--- +In [this paper](https://arxiv.org/abs/1506.04294) with [Robin](http://homepages.lboro.ac.uk/~marh3/) we study the family of causal double product integrals +\\[ +\\prod_{a < x < y < b}\\left(1 + i{\\lambda \\over 2}(dP_x dQ_y - dQ_x dP_y) + i {\\mu \\over 2}(dP_x dP_y + dQ_x dQ_y)\\right) +\\] + +where $P$ and $Q$ are the mutually noncommuting momentum and position Brownian motions of quantum stochastic calculus. The evaluation is motivated heuristically by approximating the continuous double product by a discrete product in which infinitesimals are replaced by finite increments. The latter is in turn approximated by the second quantisation of a discrete double product of rotation-like operators in different planes due to a result in [(Hudson-Pei2015)](http://www.actaphys.uj.edu.pl/findarticle?series=Reg&vol=46&page=1851). The main problem solved in this paper is the explicit evaluation of the continuum limit $W$ of the latter, and showing that $W$ is a unitary operator. The kernel of $W$ is written in terms of Bessel functions, and the evaluation is achieved by working on a lattice path model and enumerating linear extensions of related partial orderings, where the enumeration turns out to be heavily related to Dyck paths and generalisations of Catalan numbers. diff --git a/posts/2015-07-15-double-macdonald-polynomials-macdonald-superpolynomials.md b/posts/2015-07-15-double-macdonald-polynomials-macdonald-superpolynomials.md new file mode 100644 index 0000000..efdd416 --- /dev/null +++ b/posts/2015-07-15-double-macdonald-polynomials-macdonald-superpolynomials.md @@ -0,0 +1,73 @@ +--- +template: oldpost +title: AMS review of 'Double Macdonald polynomials as the stable limit of Macdonald superpolynomials' by Blondeau-Fournier, Lapointe and Mathieu +date: 2015-07-15 +comments: true +archive: false +--- +A Macdonald superpolynomial (introduced in \[O. Blondeau-Fournier et +al., Lett. Math. Phys. <span class="bf">101</span> (2012), no. 1, 27–47; +[MR2935476](http://www.ams.org/mathscinet/search/publdoc.html?pg1=MR&s1=2935476&loc=fromrevtext); +J. Comb. <span class="bf">3</span> (2012), no. 3, 495–561; +[MR3029444](http://www.ams.org/mathscinet/search/publdoc.html?pg1=MR&s1=3029444&loc=fromrevtext)\]) +in \\(N\\) +Grassmannian variables indexed by a superpartition \\(\\Lambda\\) +is said to be stable if \\({m (m + 1) \\over 2} +\\ge |\\Lambda|\\) +and \\(N \\ge |\\Lambda| - {m (m - 3) \\over +2}\\) +, where \\(m\\) +is the fermionic degree. A stable Macdonald superpolynomial +(corresponding to a bisymmetric polynomial) is also called a double +Macdonald polynomial (dMp). The main result of this paper is the +factorisation of a dMp into plethysms of two classical Macdonald +polynomials (Theorem 5). Based on this result, this paper + +(1) shows that the dMp has a unique decomposition into bisymmetric + monomials; + +(2) calculates the norm of the dMp; + +(3) calculates the kernel of the Cauchy-Littlewood-type identity of the + dMp; + +(4) shows the specialisation of the aforementioned factorisation to the + Jack, Hall-Littlewood and Schur cases. One of the three Schur + specialisations, denoted as \\(s_{\\lambda, \\mu}\\), also appears in (7) and (9) below; + +(5) defines the \\(\\omega\\) + -automorphism in this setting, which was used to prove an identity + involving products of four Littlewood-Richardson coefficients; + +(6) shows an explicit evaluation of the dMp motivated by the most + general evaluation of the usual Macdonald polynomials; + +(7) relates dMps with the representation theory of the hyperoctahedral + group \\(B_n\\) + via the double Kostka coefficients (which are defined as the entries + of the transition matrix from the bisymmetric Schur functions \\(s_{\\lambda, \\mu}\\) + to the modified dMps); + +(8) shows that the double Kostka coefficients have the positivity and + the symmetry property, and can be written as sums of products of the + usual Kostka coefficients; + +(9) defines an operator \\(\\nabla^B\\) + as an analogue of the nabla operator \\(\\nabla\\) + introduced in \[F. Bergeron and A. M. Garsia, in *Algebraic methods and \\(q\\)-special functions* (Montréal, QC, 1996), 1–52, CRM Proc. Lecture + Notes, 22, Amer. Math. Soc., Providence, RI, 1999; + [MR1726826](http://www.ams.org/mathscinet/search/publdoc.html?r=1&pg1=MR&s1=1726826&loc=fromrevtext)\]. + The action of \\(\\nabla^B\\) + on the bisymmetric Schur function \\(s_{\\lambda, \\mu}\\) + yields the dimension formula \\((h + 1)^r\\) + for the corresponding representation of \\(B_n\\) + , where \\(h\\) + and \\(r\\) + are the Coxeter number and the rank of \\(B_n\\) + , in the same way that the action of \\(\\nabla\\) + on the \\(n\\) + th elementary symmetric function leads to the same formula for the + group of type \\(A_n\\) + . + +Copyright notice: This review is published at http://www.ams.org/mathscinet-getitem?mr=3306078, its copyright owned by the AMS. diff --git a/posts/2016-10-13-q-robinson-schensted-knuth-polymer.md b/posts/2016-10-13-q-robinson-schensted-knuth-polymer.md new file mode 100644 index 0000000..4d31e37 --- /dev/null +++ b/posts/2016-10-13-q-robinson-schensted-knuth-polymer.md @@ -0,0 +1,23 @@ +--- +template: oldpost +title: A \(q\)-Robinson-Schensted-Knuth algorithm and a \(q\)-polymer +date: 2016-10-13 +comments: true +archive: false +--- +(Latest update: 2017-01-12) +In [Matveev-Petrov 2016](http://arxiv.org/abs/1504.00666) a \\(q\\)-deformed Robinson-Schensted-Knuth algorithm (\\(q\\)RSK) was introduced. In this article we give reformulations of this algorithm in terms of Noumi-Yamada description, growth diagrams and local moves. We show that the algorithm is symmetric, namely the output tableaux pair are swapped in a sense of distribution when the input matrix is transposed. We also formulate a \\(q\\)-polymer model based on the \\(q\\)RSK and prove the corresponding Burke property, which we use to show a strong law of large numbers for the partition function given stationary boundary conditions and \\(q\\)-geometric weights. We use the \\(q\\)-local moves to define a generalisation of the \\(q\\)RSK taking a Young diagram-shape of array as the input. We write down the joint distribution of partition functions in the space-like direction of the \\(q\\)-polymer in \\(q\\)-geometric environment, formulate a \\(q\\)-version of the multilayer polynuclear growth model (\\(q\\)PNG) and write down the joint distribution of the \\(q\\)-polymer partition functions at a fixed time. + +This article is available at [arXiv](https://arxiv.org/abs/1610.03692). +It seems to me that one difference between arXiv and Github is that on arXiv each preprint has a few versions only. +In Github many projects have a "dev" branch hosting continuous updates, whereas the master branch is where the stable releases live. + +[Here]({{ site.url }}/assets/resources/qrsklatest.pdf) is a "dev" version of the article, which I shall push to arXiv when it stablises. Below is the changelog. + +* 2017-01-12: Typos and grammar, arXiv v2. +* 2016-12-20: Added remarks on the geometric \\(q\\)-pushTASEP. Added remarks on the converse of the Burke property. Added natural language description of the \\(q\\)RSK. Fixed typos. +* 2016-11-13: Fixed some typos in the proof of Theorem 3. +* 2016-11-07: Fixed some typos. The \\(q\\)-Burke property is now stated in a more symmetric way, so is the law of large numbers Theorem 2. +* 2016-10-20: Fixed a few typos. Updated some references. Added a reference: [a set of notes titled "RSK via local transformations"](http://web.mit.edu/~shopkins/docs/rsk.pdf). +It is written by [Sam Hopkins](http://web.mit.edu/~shopkins/) in 2014 as an expository article based on MIT combinatorics preseminar presentations of Alex Postnikov. +It contains some idea (applying local moves to a general Young-diagram shaped array in the order that matches any growth sequence of the underlying Young diagram) which I thought I was the first one to write down. diff --git a/posts/2017-04-25-open_research_toywiki.md b/posts/2017-04-25-open_research_toywiki.md new file mode 100644 index 0000000..1b9b45a --- /dev/null +++ b/posts/2017-04-25-open_research_toywiki.md @@ -0,0 +1,19 @@ +--- +template: oldpost +title: Open mathematical research and launching toywiki +date: 2017-04-25 +comments: true +archive: false +--- +As an experimental project, I am launching toywiki. + +It hosts a collection of my research notes. + +It takes some ideas from the open source culture and apply them to mathematical research: +1. It uses a very permissive license (CC-BY-SA). For example anyone can fork the project and make their own version if they have a different vision and want to build upon the project. +2. All edits will done with maximum transparency, and discussions of any of notes should also be as public as possible (e.g. Github issues) +3. Anyone can suggest changes by opening issues and submitting pull requests + +Here are the links: [toywiki](http://toywiki.xyz) and [github repo](https://github.com/ycpei/toywiki). + +Feedbacks are welcome by email. diff --git a/posts/2017-08-07-mathematical_bazaar.md b/posts/2017-08-07-mathematical_bazaar.md new file mode 100644 index 0000000..00a8724 --- /dev/null +++ b/posts/2017-08-07-mathematical_bazaar.md @@ -0,0 +1,208 @@ +--- +template: oldpost +title: The Mathematical Bazaar +date: 2017-08-07 +comments: true +archive: false +--- + +In this essay I describe some problems in academia of mathematics and +propose an open source model, which I call open research in mathematics. + +This essay is a work in progress - comments and criticisms are welcome! +[^feedback] + +Before I start I should point out that + +1. Open research is *not* open access. In fact the latter is a + prerequisite to the former. +2. I am not proposing to replace the current academic model with the + open model - I know academia works well for many people and I am + happy for them, but I think an open research community is long + overdue since the wide adoption of the World Wide Web more than two + decades ago. In fact, I fail to see why an open model can not run in + tandem with the academia, just like open source and closed source + software development coexist today. + +problems of academia +-------------------- + +Open source projects are characterised by publicly available source +codes as well as open invitations for public collaborations, whereas closed +source projects do not make source codes accessible to the public. How +about mathematical academia then, is it open source or closed source? The +answer is neither. + +Compared to some other scientific disciplines, mathematics does not +require expensive equipments or resources to replicate results; compared +to programming in conventional software industry, mathematical findings +are not meant to be commercial, as credits and reputation rather than +money are the direct incentives (even though the former are commonly +used to trade for the latter). It is also a custom and common belief +that mathematical derivations and theorems shouldn\'t be patented. +Because of this, mathematical research is an open source activity in the +sense that proofs to new results are all available in papers, and thanks +to open access e.g. the arXiv preprint repository most of the new +mathematical knowledge is accessible for free. + +Then why, you may ask, do I claim that maths research is not open +sourced? Well, this is because 1. mathematical arguments are not easily +replicable and 2. mathematical research projects are mostly not open for +public participation. + +Compared to computer programs, mathematical arguments are not written in +an unambiguous language, and they are terse and not written in maximum +verbosity (this is especially true in research papers as journals +encourage limiting the length of submissions), so the understanding of a +proof depends on whether the reader is equipped with the right +background knowledge, and the completeness of a proof is highly +subjective. More generally speaking, computer programs are mostly +portable because all machines with the correct configurations can +understand and execute a piece of program, whereas humans are subject to +their environment, upbringings, resources etc. to have a brain ready to +comprehend a proof that interests them. (these barriers are softer than +the expensive equipments and resources in other scientific fields +mentioned before because it is all about having access to the right +information) + +On the other hand, as far as the pursuit of reputation and prestige +(which can be used to trade for the scarce resource of research +positions and grant money) goes, there is often little practical +motivation for career mathematicians to explain their results to the +public carefully. And so the weird reality of the mathematical academia +is that it is not an uncommon practice to keep trade secrets in order to +protect one\'s territory and maintain a monopoly. This is doable because +as long as a paper passes the opaque and sometimes political peer review +process and is accepted by a journal, it is considered work done, +accepted by the whole academic community and adds to the reputation of +the author(s). Just like in the software industry, trade secrets and +monopoly hinder the development of research as a whole, as well as +demoralise outsiders who are interested in participating in related +research. + +Apart from trade secrets and territoriality, another reason to the +nonexistence of open research community is an elitist tradition in the +mathematical academia, which goes as follows: + +- Whoever is not good at mathematics or does not possess a degree in + maths is not eligible to do research, or else they run high risks of + being labelled a crackpot. +- Mistakes made by established mathematicians are more tolerable than + those less established. +- Good mathematical writings should be deep, and expositions of + non-original results are viewed as inferior work and do not add to + (and in some cases may even damage) one\'s reputation. + +All these customs potentially discourage public participations in +mathematical research, and I do not see them easily go away unless an +open source community gains momentum. + +To solve the above problems, I propose a open source model of +mathematical research, which has high levels of openness and +transparency and also has some added benefits listed in the last section +of this essay. This model tries to achieve two major goals: + +- Open and public discussions and collaborations of mathematical + research projects online +- Open review to validate results, where author name, reviewer name, + comments and responses are all publicly available online. + +To this end, a Github model is fitting. Let me first describe how open +source collaboration works on Github. + +open source collaborations on Github +------------------------------------ + +On [Github](https://github.com), every project is publicly available in +a repository (we do not consider private repos). The owner can update +the project by \"committing\" changes, which include a message of what +has been changed, the author of the changes and a timestamp. Each +project has an issue tracker, which is basically a discussion forum +about the project, where anyone can open an issue (start a discussion), +and the owner of the project as well as the original poster of the issue +can close it if it is resolved, e.g. bug fixed, feature added, or out of +the scope of the project. Closing the issue is like ending the +discussion, except that the thread is still open to more posts for +anyone interested. People can react to each issue post, e.g. upvote, +downvote, celebration, and importantly, all the reactions are public +too, so you can find out who upvoted or downvoted your post. + +When one is interested in contributing code to a project, they fork it, +i.e. make a copy of the project, and make the changes they like in the +fork. Once they are happy with the changes, they submit a pull request +to the original project. The owner of the original project may accept or +reject the request, and they can comment on the code in the pull +request, asking for clarification, pointing out problematic part of the +code etc and the author of the pull request can respond to the comments. +Anyone, not just the owner can participate in this review process, +turning it into a public discussion. In fact, a pull request is a +special issue thread. Once the owner is happy with the pull request, +they accept it and the changes are merged into the original project. The +author of the changes will show up in the commit history of the original +project, so they get the credits. + +As an alternative to forking, if one is interested in a project but has +a different vision, or that the maintainer has stopped working on it, +they can clone it and make their own version. This is a more independent +kind of fork because there is no longer intention to contribute back to +the original project. + +Moreover, on Github there is no way to send private messages, which +forces people to interact publicly. If say you want someone to see and +reply to your comment in an issue post or pull request, you simply +mention them by `@someone`. + +open research in mathematics +---------------------------- + +All this points to a promising direction of open research. A maths +project may have a wiki / collection of notes, the paper being written, +computer programs implementing the results etc. The issue tracker can +serve as a discussion forum about the project as well as a platform for +open review (bugs are analogous to mistakes, enhancements are possible +ways of improving the main results etc.), and anyone can make their own +version of the project, and (optionally) contribute back by making pull +requests, which will also be openly reviewed. One may want to add an +extra \"review this project\" functionality, so that people can comment +on the original project like they do in a pull request. This may or may +not be necessary, as anyone can make comments or point out mistakes in +the issue tracker. + +One may doubt this model due to concerns of credits because work in +progress is available to anyone. Well, since all the contributions are +trackable in project commit history and public discussions in issues and +pull request reviews, there is in fact *less* room for cheating than the +current model in academia, where scooping can happen without any +witnesses. What we need is a platform with a good amount of trust like +arXiv, so that the open research community honours (and can not ignore) +the commit history, and the chance of mis-attribution can be reduced to +minimum. + +Compared to the academic model, open research also has the following +advantages: + +- Anyone in the world with Internet access will have a chance to + participate in research, whether they are affiliated to a + university, have the financial means to attend conferences, or are + colleagues of one of the handful experts in a specific field. +- The problem of replicating / understanding maths results will be + solved, as people help each other out. This will also remove the + burden of answering queries about one\'s research. For example, say + one has a project \"Understanding the fancy results in \[paper + name\]\", they write up some initial notes but get stuck + understanding certain arguments. In this case they can simply post + the questions on the issue tracker, and anyone who knows the answer, + or just has a speculation can participate in the discussion. In the + end the problem may be resolved without the authors of the paper + being bothered, who may be too busy to answer. +- Similarly, the burden of peer review can also be shifted from a few + appointed reviewers to the crowd. + +related readings +---------------- + +- [The Cathedral and the Bazaar by Eric Raymond](http://www.catb.org/esr/writings/cathedral-bazaar/) +- [Doing sience online by Michael Nielson](http://michaelnielsen.org/blog/doing-science-online/) +- [Is massively collaborative mathematics possible? by Timothy Gowers](https://gowers.wordpress.com/2009/01/27/is-massively-collaborative-mathematics-possible/) + +[^feedback]: Please send your comments to my email address - I am still looking for ways to add a comment functionality to this website. |