Professor Mazzocco's field of research is Integrable Systems, an area of research strength in the UK that falls within the EPSRC remit of Intradisciplinary Research in Mathematical Sciences and has strong links to Theoretical Physics. In particular her interests include special functions, combinatorial aspects of Teichmüller theory and quantisation, Cherednik algebras, cluster algebras and applications of random matrix models.
At the start of her research career, Professor Mazzocco focused on the theory of the Painlevé differential equations and their generalisations - non linear ODEs whose solutions are so famous that a chapter has been dedicated to them in the Digital Library of Mathematical Functions (replacement edition of the famous handbook of special functions by Abramowitz and Stegun), in which many of Professor Mazzocco’s contributions are quoted.
Jointly with Dubrovin, Mazzocco proposed a method to classify the algebraic solutions of a special case of the sixth Painlevé (PVI) equation, a problem that was open for a hundred years (Inv. Math. 141 and Math. Ann. 321). This method consists in describing the procedure of analytic continuation of the solutions to PVI by a certain action of the braid group. She then extended this method to the general PVI to classify all rational solutions (J. Phys. A. 34).
Another major open problem in the area of Painlevé equations is to find new Painlevé–type equations of higher order. Dr Mazzocco attacked this problem (Int. Math. Res. Not. 2002), and with Dubrovin discovered new higher order analogues as Hamiltonian reductions of high dimensional monodromy preserving deformations (Comm. Math. Phys. 271).
Due to the key observation that the same Poisson structure with the same action of the braid group appears both in the description of the analytic continuation of solutions of monodromy pre- serving deformations equations and in the action of the mapping class group on the Teichmüller space of a non–compact Riemann surface, Professor Mazzocco started to work in (quantum)Teichmüller theory from 2008. She successfully applied to EPSRC for an Advanced Research Fellowship on this theme and for two further research grants to attract to the UK her collaborator, Prof. Leonid Chekhov from Steklov Institute, with whom she published several results to this area. These include the quantisation of the monodromy manifold of PVI and of the braid group action on it (J. Phys. A, 43), the discovery that the affinisation of the algebra of geodesic functions on certain non-compact Riemann surfaces is in fact the semi-classical limit of a quantum group, the twisted q–Yangian for the orthogonal Lie algebra discovered by Molev, Ragoucy and Sorba (Adv. Math. 226 and Russ. Math. Surv., 64) and the introduction of a completely new quantum algebra structure with a complete characterisation of its central elements and of the action of the braid group on it, solving an open problem proposed by Molev et al. (Comm. Math. Phys. 332).
Recently, Professor Mazzocco started to be interested in the monodromy manifolds of the Painlevé equations from a more algebraic point of view. She discovered that the quantisation of such manifolds leads to special degenerations of the Askey-Wilson algebra which regulate the q-Askey polynomials (Nonlinearity 29) . This work has led the way to several important new discoveries:
- She proposed a representation-theoretic approach to the theory of the Painelvé equations by showing that the Cherednik algebra of type Cˇ1C1 appears naturally as quantisation of the group algebra of the monodromy group associated to the PVI equation and that the action of the braid group discussed above corresponds to the automorphims of the Cherednik algebra (to appear in Adv. Pure Math., 2018).
- With L. Chekhov she introduced the notion of bordered cusp in a Riemann surface and found that generalised cluster algebras appear naturally in the Teichmüller theory of non-compact Riemann surfaces with bordered cusps. These bordered cusps arise naturally from colliding two boundary components or two sides of the same boundary component in a non-compact Riemann surface (Nonlinearity 31, 2017).
- In collaboration with L. Chekhov and V. Rubtsov, she introduced the concept of decorated character variety and showed that on the level of monodromy manifolds the confluence of the Painlevé differential equations corresponds to colliding two boundary components or two sides of the same boundary component in a Riemann sphere so that their monodromy manifolds arise as Poisson sub-algebras of the cluster algebra structure naturally defined on the character variety (Int. Math. Res. Not. 2016).
Her current research draws ideas from representation theory, geometry and topology, differential and q-difference equations and Poisson algebra to establish a new link between cluster algebra theory
and the theory of Macdonald polynomials, a remarkable family of multi-variable q--orthogonal polynomials associated with affine root systems. The ultimate target of her current research is to construct a quantum cluster algebra for each affine root system such that its representation theory involves the corresponding Macdonald polynomials.