In this essay I describe some problems in academia of mathematics and propose an open source model, which I call open research in mathematics.

As an experimental project, I am launching toywiki.

(Latest update: 2017-01-12) In Matveev-Petrov 2016 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.

A Macdonald superpolynomial (introduced in [O. Blondeau-Fournier et al., Lett. Math. Phys. 101 (2012), no. 1, 27–47; MR2935476; J. Comb. 3 (2012), no. 3, 495–561; MR3029444]) 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

In this paper with Robin we study the family of causal double product integrals \begin{equation*} \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) \end{equation*}

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 \(g\: \Sigma_3^* \to \Sigma_2^*\)

jst = juggling skill tree

In this paper with Robin we show the explicit formulae for a family of unitary triangular and rectangular double product integrals which can be described as second quantisations.

The super Catalan numbers are defined as $$ T(m,n) = {(2 m)! (2 n)! \over 2 m! n! (m + n)!}. $$

In this paper 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.

In this paper with Neil we construct a \(q\)-version of the Robinson-Schensted algorithm with column insertion. Like the usual RS 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 \(q\).