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<a href="posts/2018-04-10-update-open-research.html"><h2> Updates on open research </h2></a>
<p>Posted on 2018-04-29</p>
<p>It has been 9 months since I last wrote about open (maths) research. Since then two things happened which prompted me to write an update.</p>
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<a href="posts/2017-08-07-mathematical_bazaar.html"><h2> The Mathematical Bazaar </h2></a>
<p>Posted on 2017-08-07</p>
<p>In this essay I describe some problems in academia of mathematics and propose an open source model, which I call open research in mathematics.</p>
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<a href="posts/2017-04-25-open_research_toywiki.html"><h2> Open mathematical research and launching toywiki </h2></a>
<p>Posted on 2017-04-25</p>
<p>As an experimental project, I am launching toywiki.</p>
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<a href="posts/2016-10-13-q-robinson-schensted-knuth-polymer.html"><h2> A \(q\)-Robinson-Schensted-Knuth algorithm and a \(q\)-polymer </h2></a>
<p>Posted on 2016-10-13</p>
<p>(Latest update: 2017-01-12) In <a href="http://arxiv.org/abs/1504.00666">Matveev-Petrov 2016</a> 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.</p>
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<a href="posts/2015-07-15-double-macdonald-polynomials-macdonald-superpolynomials.html"><h2> AMS review of 'Double Macdonald polynomials as the stable limit of Macdonald superpolynomials' by Blondeau-Fournier, Lapointe and Mathieu </h2></a>
<p>Posted on 2015-07-15</p>
<p>A Macdonald superpolynomial (introduced in [O. Blondeau-Fournier et al., Lett. Math. Phys. <span class="bf">101</span> (2012), no. 1, 27–47; <a href="http://www.ams.org/mathscinet/search/publdoc.html?pg1=MR&s1=2935476&loc=fromrevtext">MR2935476</a>; J. Comb. <span class="bf">3</span> (2012), no. 3, 495–561; <a href="http://www.ams.org/mathscinet/search/publdoc.html?pg1=MR&s1=3029444&loc=fromrevtext">MR3029444</a>]) 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</p>
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