Research interests, and work

I am interested in Multiple Zeta Values, and Multiple Polylogarithms, including the various more generalised versions of these objects. I have been using some quite recently developed algebraic tools, and lots of computer assistance, to find, prove or motivate identities. These tools include the symbol map of Goncharov on multiple polylogarithms, and Brown's coproduct and derivations on (motivic) multiple zeta values.

Read a full version of my research statement. Below you can find an small overview of my research.

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Multiple Polylogarithms

Polylogarithms, and their multi-variable versions, the multiple polylogs, are generalisations of the usual logarithm \( -\log(1-x) \), defined by the following power series \[ \Li_{s_1, s_2, \ldots, s_n}(x_1, x_2, \ldots, x_n) := \sum_{1 \leq n_1 < n_2 < \cdots < n_k} \frac{x_1^{n_1} x_2^{n_2} \cdots x_k^{n_k}}{n_1^{s_1} n_2^{s_2} \cdots n_k^{s_k}} \, . \] For \( x_1 = \cdots = x_k = 1 \), we recover the multiple zeta values defined above. Polylogarithms are interesting for their connections to \( K \)-theory and Zagier's conjecture, by way of their functional equations. MPLs are interesting for their applications to calculations in particle physics, but also in how they will assist in understanding aspects of the corresponding polylogarithm, at higher weights.

Weight 5 symmetries

Following Gangl's success in analysing the symmetries and relations of weight 4 iterated integrals (essenially weight 4 multiple polylogarithms), I am performing a similar analysis the weight 5 case (and higher).

Using the symbol map, some good choices for arguments and a lot of computer time doing multilinear algebra, one can systemaically find all funcional equations, and relations between various MPLs. Two examples of these type of relations include a 3-term relation for \( I_{4,1} \) \[ I_{4,1}(x,y) + I_{4,1}(\tfrac{1}{1-x}, \tfrac{1}{1-y}) + I_{4,1}(1 - \tfrac{1}{x}, 1 - \tfrac{1}{y}) \equiv_\delta 0 \, , \] and a a 4-term relation for \( I_{3,2} \) \[ I_{3,2}(1,2,3,4,5) + I_{3,2}(1,2,4,3,5) \equiv_\delta I_{3,2}(1,2,3,5,4) + I_{3,2}(1,2,5,3,4) \, . \] Here \( \equiv_\delta \) means to ignore product terms, and \( \Li_5 \)-like terms including Nielsen polylogarithms. Also \( I_{3,2}(1,2,3,4,5) \) is short-hand notation meaning \[ I_{3,2}(\cr(1,2,3,4), \cr(1,2,3,5)) \, , \] where \( \cr \) is the usual cross-ratio of 4 points.

Numerically testable idenities

By finding the product terms, and using successive slices of the MPL coproduct \( \Delta_{n-1,1}, \Delta_{n-2,1,1}, \ldots \), Duhr succeeds in lifting one of Gangl's relations, a relation between \( I_{3,1}(x,y) \) and \( I_{3,1}(\frac{1}{x}, \tfrac{1}{y}) \), from a symbol level idenity to a full numerically testable functional equation. This is apparently the first such numerically testable functional equation for genuine weight 4 MPLs in two variables. My analysis of the weight 5, and weight 6 cases has lead to a proven general expression for the product terms in every such relation between \( I_{a,b}(x,y) \) and \( I_{a,b}(\tfrac{1}{x}, \tfrac{1}{y}) \), which appears to be the first of its kind. And even a numerically testable functional equation in the case \( b = 1 \). It is worth noting how difficult it is to actually find functional equations for (multiple) polylogarithms because of the difficult in finding good choices of arguments; currently no non-trivial functional equations known for \( \Li_8 \), or higher.

With Duhr, Dulat and Gangl, we have given a method by which mod-products symbol level identities can be lifted automatically to numerically testable identities amongst clean single-valued MPL's

Reduction of \( I_{1,1,1,1,1} \)

By combining Nicusor Dan's method of reducing a general hyperlogarithm from \( n \) variables to \( n-2 \) variables, with several of the relations found via the above procedure, I can give explicit expressions for the general weight 5 iterated integral in terms of \( I_5 \), \( I_{3,2} \), and \( I_{3,1,1} \) only. This required verifying Nicusor Dan's reduction method, and implementing it in Mathematica.

Conjecurally indices \( 1 \) are not necessary and can be replaced, this reduces the task at weight 5 to expressing \( I_{3,1,1} \) purely in terms of \( I_5 \), \( I_{3,2} \). Recent work with Gangl and Radchenko on geometric functional equations for MPL's, generalising Goncharov and Rudenko's \( \mathbf{Q}_4 \) relation, has given the required reduction of \( I_{3,1,1} \) to \( I_{3,2} \) and \( I_{5} \). We also have a redution of \( I_{1,1,1,1,1,1} \) in weight 6, to \( I_{4,1,1}, I_{4,2} I_{5,1} \) and \( I_{6} \).

Functional equations of Nielsen polylogarithms

With Gangl and Radchenko, we show that the Nielsen polylogarithm \[ S_{3,2}(x) = \frac{1}{4} \int_{0}^{1}\log^{2}(t)\log^{2}(1-zt)\frac{dt}{t}\,. \] satisfies the dilogarith 5-term relation modulo explicit \( \Li_5 \) terms, and give some new evaluations. \begin{align*} S_{3,2}(\phi ^{-2})={}& \frac{1}{33} \Li_5\big({}-8 [\phi ^{-3}]{}+{}780 [\phi ^{-1}]{}+{}804 [-\phi ]{}+{}8 [-\phi ^3]\big) + \Li_4(\phi ^{-2})\log (\phi ) \\ & +\frac{1}{2}\zeta (5) +\frac{481}{11} \zeta (4) \log (\phi ) -\zeta(3)\Li_2(\phi ^{-2}) +\frac{50}{11} \zeta (2) \log ^3(\phi ) +\frac{14}{15}\log ^5(\phi ) \,. \end{align*}

We expect \[ S_{4,2}(x) = -\frac{1}{12} \int_{0}^{1}\log^{3}(t)\log^{2}(1-zt)\frac{dt}{t}\,. \] to satisfy the 22-term trilogarithm relation modulo \( \Li_6 \). As evidence for this, we use the motivic coproduct of iterated integrals to find candidate evaluations for some \( S_{4,2} \) Nielsen polylgoarithms. These evaluations follow from trilogarithm duplication so give evidence that \( S_{4,2} \) satisfies it modulo \( \Li_6 \). \begin{align*} S_{4,2}(-1) \overset{?}{=}{} & \frac{1}{13} \bigg(\frac{1}{3}\Li_6\Big(-\frac{1}{8}\Big)-162 \Li_6\Big(-\frac{1}{2}\Big)-126 \Li_6\Big(\frac{1}{2}\Big)\bigg) -\frac{1787 }{624}\zeta (6) +\frac{3}{8} \zeta (3)^2 \\& {}+\frac{31}{16} \zeta (5) \log(2) -\frac{15}{26} \zeta (4) \log ^2(2) +\frac{3}{104} \zeta (2) \log^4(2) -\frac{1}{208} \log ^6(2) \,, \\[1ex] S_{4,2}(\phi ^{-2}) \overset{?}{=}{} & \frac{1}{396} \Li_6\Big( 2 \big[ \phi ^{-6} \big] -128 \big[ \phi ^{-3}\big]+801 \big[\phi ^{-2}\big]-576 \big[\phi^{-1} \big] \Big) +\frac{35 }{99}\zeta (6) +\frac{2}{5} \zeta (3)^2 \\& + \Li_5 \left(\phi ^{-2}\right) \log (\phi) -\zeta (5)\log (\phi ) +\frac{2}{11} \zeta (4) \log ^2(\phi ) -\zeta (3) \Li_3\left(\phi ^{-2}\right) \\& +\frac{10}{33} \zeta (2) \log ^4(\phi ) -\frac{79}{990} \log ^6(\phi ) \end{align*}

Grassmannian polylogarithms and an explicit 4-ratio

Goncharov and Rudenko arXiv:1803.08585 [math.NT] prove Zagier's conjecture for \( \zeta_F(4) \) and show abstract the 4-ratio must exist. With Gangl and Radchenko we find an explicit expression for it in terms of Gangl's 122-term reduction of \( I_{3,1}(\mathrm{fiv term } \Li_2, z) = \sum \Li_4 \)'s.

(C, Gangl, Radchenko) Denote by \( \mathrm{GR}_4 \) the multi-valued weight 4 Grassmannian polylogarithm. Denote by \( V(z; x,y) \) the \( \Li_4 \) terms in (a symmetrisation of) Gangl's 122-term reduction. Modulo products, the combination \[ \frac{7}{144} \mathrm{GR}_4 + 2 \mathrm{Alt}_8 \widetilde{I_{3,1}}(\mathrm{cr}(34\mid2567), \mathrm{cr}(67\mid1345)) \] can be written as a sum of 7 terms \( V(z; x, y) \).

Denote the formal linar combination of the arguments of the \( \Li_4 \) terms appearing by \( \mathcal{Q}_(v_1,\ldots,v_8) \). Then the single-valued \( \Li_4 \) satisfies the functional equation \[ \sum_{j=0}^8 (-1)^j \mathcal{L}_4(\mathcal{Q}(v_0, \ldots, \widehat{v_j}, \ldots, v_8)) = 0 \,. \] The combination \( \mathcal{Q} \) is the 4-ratio. This functional equation is an analogue of the 5-term relation for \( \Li_2 \) and Goncahrov's 840-term relation for \( \Li_3 \).

We have also been able to give explicit expressions for the Grassmannian \( n \)-logarithm in terms of the classical iterate integrals, as a starting point to tackle Zagier's conjecture on \( \zeta_F(n) \) for higher \( n \).

Geometric functional equations after Goncharov-Rudenko

Goncharov and Rudenko arXiv:1803.08585 [math.NT] give the following fundamental relation \( \mathbf{Q}_4 \) between weight 4 polylogarithms, and use it to prove Zagier's conjecture for \( \zeta_F(4) \). \begin{align*} {\mathrm{Cyc}}_7 \left( I_{3,1}(- [1234, 4671] + [1234,4571] - [1234,4561]) + \Li_4([1246] + 6 [123456]) \right) = 0 \,, \end{align*} or pictorially \[ \Rule{0px}{60px}{60px} {\mathrm{Cyc}}_7 \Bigg( I_{3,1} \bigg( \smash{ - \mypic{96px}{96px}{i31x5.png} + \mypic{96px}{96px}{i31x6.png} - \mypic{96px}{96px}{i31x7.png} } \bigg) + \Li_4 \bigg( \smash{ \mypic{96px}{96px}{l4c4.png} + 6 \mypic{96px}{96px}{l4c6.png} } \bigg) \Bigg) = 0 \,. \] Here \[ [1234] \leftrightarrow \mathrm{cr}(x_1,x_3,x_2,x_4) = \frac{(x_1-x_2)(x_3-x_4)}{(x_2-x_3)(x_4-x_1)} \,, \] is a renormalisation of the cross-ratio and \[ [123456] \leftrightarrow [x_1,\ldots,x_6] = -\frac{(x_1-x_2) (x_3-x_4) (x_5-x_6)}{(x_2-x_3) (x_4-x_5) (x_6-x_1)} \]

With Gangl and Radchenko we are attempting to generalise these type of relations to higher weight. We have candidates for \( \mathbf{Q}_5 \) and \( \mathbf{Q}_6 \), which already lead to some highly non-trivial reductions of weight 5 and weight 6 polylogarithms. In particular \( \mathbf{Q}_5 \) implies every weight 5 MPL is expressed in terms of \( I_{4,1} \) and \( I_{5} \), and \( \mathbf{Q}_6 \) implies every weight 6 MPL can be expressed in terms of \( I_{4,1,1} \), \( I_{4,2} \), \( I_{5,1} \) and \( I_6 \).

Multiple Zeta Values

Multiple zeta values are a set of real numbers defined by infinite series of the form \[ \zeta(s_1, s_2, \ldots, s_k) := \sum_{1 \leq n_1 < n_2 < \cdots < n_k} \frac{1}{n_1^{s_1} n_2^{s_2} \cdots n_k^{s_k}} \, , \] where \( s_1, \ldots, s_k \) are positive integers, and \( s_k > 1 \) for convergence. Multiple zeta values satisfy a huge number of relations, but the global structure is not very well understood, and largely remains conjectural.

Original cyclic insertion conjecture

One well known identity proven by Broadhurst, and conjectured by Zagier, states \[ \zeta(\{1, 3\}^n) = \frac{1}{2n+1} \zeta(\{2\}^{2n}) = \frac{1}{2n+1} \frac{\pi^{4n}}{(4n+1)!} \, . \] This result is a special case of the Cyclic Insertion conjecture of Borwein, Bailey, Broadhurst, and Lisoněk, which more generally claims

(Cyclic insertion) Given \( a_0, \ldots, a_{2n} \), non-negative integers, the sum \[ \sum_{\text{cyclic shifts of \( a_i \)}} \zeta(\{2\}^{a_0}, 1, \{2\}^{a_1}, 3, \ldots, 1, \{2\}^{a_{2n-1}}, 3, \{2\}^{a_{2n}}) \overset{?}{=} \zeta(\{2\}^\wt) = \frac{\pi^\wt}{(\wt+1)!} \, . \] (Here \( \wt \) refers to the weight of the MZVs appearing on the left hand side.)

The Bowman-Bradley Theorem shows that inserting blocks of \( 2 \) for each (weak) composition \( a_0 + a_1 + \cdots + a_{2n} = N \) gives an explicit rational multipel of \( \pi^\wt \).

But with Brown's coproduct and derivations on motivic MZVs, I have shown that inserting blocks of \( 2 \) for all permutations of \( a_i \) is sufficient to get a rational multiple of \( \pi^\wt \), although the rational is not made explicit, as here

(Symmetric Insertion, C) Given \( a_0, \ldots, a_{2n} \), non-negative integers, the sum \[ \sum_{\text{permutations of \( a_i \)}} \zeta(\{2\}^{a_0}, 1, \{2\}^{a_1}, 3, \ldots, 1, \{2\}^{a_{2n-1}}, 3, \{2\}^{a_{2n}}) \in \pi^\wt \Q \, . \]
In particular, the MZV evaluation \[ \zeta(\{ \, \{2\}^m, 1, \{2\}^m, 3 \}^n, \{2\}^m) \in \pi^{4n + 2m(2n+1)} \Q \, , \] indeed holds, for any non-negative integers \( n, m \).

Hoffman's conjectural identity

Another family of conjectural identities is listed on Hoffman's info page (using the other argument convention). The identity claims \[ 2\zeta(3, 3, \{2\}^c) - \zeta(3, \{2\}^c, 1, 2) \overset{?}{=} -\frac{\pi^\wt}{(\wt+1)!} \, . \]

Again a motivic proof of this, up to a rational, is possible. The motivic proof even shows the way to a much more general identity with any even number of \( 3 \)s, of which the following is a simple instance

(C) Given \( a, b, c \), non-negative integers, we have \begin{align*} & 2\zeta(\{2\}^a, 3, \{2\}^b, 3, \{2\}^c) + 2\zeta(\{2\}^b, 3, \{2\}^a, 3, \{2\}^c) - \\ & - \zeta(\{2\}^a, 3, \{2\}^c, 1,2, \{2\}^b) - \zeta(\{2\}^b, 3, \{2\}^c, 3,\{2\}^a) \in \pi^\wt \Q \end{align*}
Moreover, numerical testing suggests that this identity breaks down further into irreducible identities of the following type \begin{align*} &\zeta(\{2\}^a, 3, \{2\}^b, 3, \{2\}^c) - \zeta(\{2\}^b, 3, \{2\}^c, 1, 2, \{2\}^a) \\ & + \zeta(\{2\}^c, 1, 2, \{2\}^a, 1, 2, \{2\}^b) \overset{?}{=} -\frac{\pi^\wt}{(\wt+1)!} \end{align*}

Generalised cyclic insertion conjecture

It now appears that both of these identities are just special cases of a more general version of cyclic insertion conjecture, which holds for even and odd weights. Calculations with Brown's derivations, and further numerical experiments have helped me to pin down a precise candidate identity which can be generated from any MZV who arguments consists of \( 1 \)s, \( 2 \)s, and \( 3 \)s with no consecutive pair \( 1, 1 \).

For example, I can show

(C) For any non-negaive integer \( n \), we have \begin{align*} & \zeta(\{2\}^n, 1, 3, 3, 1, 2) + \zeta(3, 1, 2, 1, \{2\}^n, 3) - \zeta(1, 2, 1, \{2\}^n, 3, 1, 2) \\ & + \zeta(1, 2, 1, 3, 3, \{2\}^n) - \zeta(3, \{2\}^n, 1, 3, 3) = \alpha \frac{\pi^\wt}{(\wt+1)!} \, , \end{align*} for some \( \alpha \in \Q \), and numerically I expect \( \alpha = 1 \).
I can also show symmetrised versions of a more general identity. I am currently preparing a draft version of this work, and trying to further generalise it.

Checkerboard Schur MZV's

With H. Bachmann we established some evaluations of checkerboard Schur MZV's.

(3x3 square) We have the following evaluation for the 3x3 square Schur MZV's \[ \zeta\left( \begin{array} {|r|r|}\hline 3 & 1 & 3 \\ \hline 1 & 3 & 1 \\ \hline 3 & 1 & 3 \\ \hline \end{array} \right) = \frac{1}{32} \det \begin{pmatrix} \zeta(3) & \frac{\pi^4}{180} & \zeta(7)\\ \frac{\pi^4}{72} & \zeta(5) & \frac{17\pi^8}{90720} \\ \zeta(7) & \frac{13\pi^8}{226800} & \zeta(11) \end{pmatrix}\,. \]

We were able to understand this evaluation as a generalised Jacobi-Trudi formula from an outside decomposition of the Young tableau, and generalise it to all skew diagonal Schur MZV's. From there we answered a number of questions posed by Bachmann and Yamasaki arXiv:1711.04746 [math.NT].