Title:

Identities arising from coproducts on multiple zeta values and multiple polylogarithms

In this thesis we explore identities which can be proven on multiple zeta values using the derivation operators $ D_r $ from Brown's motivic MZV framework. We then explore identities which occur on multiple polylogarithms by way of the symbol map $ \mathcal{S} $, and the multiple polylogarithm coproduct $ \Delta $. On multiple zeta values, we consider Borwein, Bradley, Broadhurst, and Lisoněk's cyclic insertion conjecture about inserting blocks of $ \{2\}^{a_i} $ between the arguments of $ \zeta(\{1,3\}^n) $. We generalise this conjecture to a much broader setting, and give a proof of a symmetrisation of this generalised cyclic insertion conjecture. This proof is by way of the blockdecomposition of iterated integrals introduced here, and Brown's motivic MZV framework. This symmetrisation allows us to prove (or to make progress towards) various conjectural identities, including the original cyclic insertion conjecture, and Hoffman's $ 2\zeta(3,3,\{2\}^n)  \zeta(3,\{2\}^n,1,2) $ identity. Moreover, we can then generate unlimited new conjectural identities, and give motivic proofs of their symmetrisations. We then consider the task of relating weight 5 multiple polylogarithms. Using the symbol map, we determine all of the symmetries and functional equations between depth 2 and between depth 3 iterated integrals with 'coupledcross ratio' arguments $ [\mathrm{cr}(a,b,c,d_1), \ldots, \mathrm{cr}(a,b,c,d_k)] $. We lift the identity for $ I_{4,1}(x,y) + I_{4,1}(\frac{1}{x}, \frac{1}{y}) $ to an identity holding exactly on the level of the symbol and prove a generalisation of this for $ I_{a,b}(x,y) $. Moreover, we further lift the subfamily $ I_{n,1} $ to a candidate numerically testable identity using slices of the coproduct. We review Dan's reduction method for reducing the iterated integral $ I_{1,1,\ldots,1} $ to a sum in $ \leq n2 $ variables. We provide proofs for Dan's claims, and run the method in the case $ I_{1,1,1,1} $ to correct Dan's original reduction of $ I_{1,1,1,1} $ to $ I_{3,1} $ and $ I_4 $. We can then compare this with another reduction to find $ I_{3,1} $ functional equations, and their nature. We then give a reduction of $ I_{1,1,1,1,1} $ to $ I_{3,1,1} $, $ I_{3,2} $ and $ I_{5} $, and indicate how one might be able to further reduce to $ I_{3,2} $ and $ I_5 $. Lastly, we use and generalise an idea suggested by Goncharov at weight 4 and weight 5. We find $ \mathrm{Li}_n $ terms when certain $ \mathrm{Li}_2 $, $ \mathrm{Li}_3 $ and $ \mathrm{Li}_4 $ functional equations are substituted into the arguments of symmetrisations of $ I_{m,1}(x,y) $. By expanding $ I_{m,1}(\text{$\mathrm{Li}_k$ equation}, \text{$\mathrm{Li}_\ell $ equation}) $ in two different ways we obtain functional equations for $ \mathrm{Li}_5 $ and $ \mathrm{Li}_6 $. We make some suggestions for how this might work at weight 7 and weight 8 giving a potential route to $ \mathrm{Li}_7 $ and $ \mathrm{Li}_8 $ functional equations.
