PAPERS

PREPRINTS

D. Dal Martello
Okamoto's symmetry on the representation space of the sixth Painlevé equation
arXiv (2024) 10.48550/arXiv.2411.17397

The sixth Painlevé equation (PVI) admits dual isomonodromy representations of type 2-dimensional Fuchsian and 3-dimensional Birkhoff. Taking the multiplicative middle convolution of a higher Teichmüller coordinatization for the Fuchsian monodromy group, we give Okamoto's symmetry w_2 of PVI a monodromic realization in the language of cluster \mathcal{X}-mutations. The explicit mutation formula is encoded in dual geometric terms of colored equilateral triangulations and star-shaped fat graphs. Moreover, this realization has a known additive analogue through the middle convolution for Fuchsian systems, and dual formulations for both the Birkhoff representation and its Stokes data exist. We give this quadruple of w_2-related maps a unified diagrammatic description in purely convolutional terms.

PUBLICATIONS

D. Dal Martello and M. Mazzocco
Generalised double affine Hecke algebras, their representations, and higher Teichmüller theory
Advances in Mathematics (2024) 10.1016/j.aim.2024.109763

Generalized double affine Hecke algebras (GDAHA) are flat deformations of the group algebras of 2-dimensional crystallographic groups associated to star-shaped simply laced affine Dynkin diagrams. In this paper, we first construct a functor that sends representations of the \tilde{D}_4-type GDAHA to representations of the  \tilde{E}_6-type one for specialised parameters. Then, under no restrictions on the parameters, we construct embeddings of both GDAHAs of type \tilde{D}_4 and \tilde{E}_6 into matrix algebras over quantum cluster \mathcal{X}-varieties, thus linking to the theory of higher Teichmüller spaces. For \tilde{E}_6, the two explicit representations we provide over distinct quantum tori are shown to be related by quiver reductions and mutations.

THESES

D. Dal Martello
GDAHA representations with Teichmüller, Stokes, and Painlevé
UBIRA (2024) etheses.bham.ac.uk/id/eprint/15029

The Painlevé VI equation governs the isomonodromic deformation problem of both 2-dimensional Fuchsian and 3-dimensional irregular types of linear systems of ODEs. Through Harnad duality, this feature turns into a map between the two systems, which translates to monodromy as a middle convolution operation. This thesis studies the quantum algebraic manifestation of the systems’ monodromy data by introducing a noncommutative analogue of the middle convolution functor. The Fuchsian data are known to quantize as the $C^\veeC_1$ DAHA; we construct a quantization of the irregular ones that match the $\tilde{E}_6$-type GDAHA, provided a specialization of the algebra parameters. Both quantum data are then shown to exhibit an alternative realization in higher Teichmüller terms. In particular, this framework advances the GDAHA representation theory by providing the first explicit representation of the universal GDAHA of type $\tilde{E}_6$, which can be reduced to the quantum irregular monodromy data by a new quiver-theoretical operation.

D. Dal Martello
Bi-Hamiltonian structures and generalizations
figshare (2020) 10.6084/m9.figshare.22310275.v1

Inspired by a class of infinite-dimensional integrable systems studied by Dubrovin, this thesis investigates a generalization of bi-Hamiltonian theory beyond the strictly Poisson formalism. Starting from the observation that the mechanism producing constants of motion does not require both tensors to be Poisson, we study alternative coupling conditions on the pair of bivectors that equally induce the bi-Hamiltonian property, using the language and toolkit of Lie algebroids (and their generalizations).