I am a mathematical physicist specializing in Integrable Systems, namely mathematical problems often motivated by Theoretical Physics that present unexpected beauty - a serendipity of geometric/algebraic/analytical structures that lead solvability in some sense. My scientific interests include isomonodromic deformations and their quantization, (quantum) Teichmüller theory, double affine Hecke algebras and basic orthogonal polynomials.
Painlevé equations and their generalizations:
At the start of my research career, I focused on the theory of the Painlevé differential equations and their generalizations - 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 my contributions are quoted.
Jointly with Dubrovin, we 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. I 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. I 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).
Teichmüller theory:
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-preserving deformations equations and in the action of the mapping class group on the Teichmüller space of a non-compact Riemann surface, I started to work in (quantum) Teichmüller theory from 2008.
I successfully applied to EPSRC for an Advanced Research Fellowship on this theme and for two further research grants to attract to the UK my collaborator, Professor Leonid Chekhov at the time based at Steklov Institute, with whom I published several results to this area:
The quantization of the monodromy manifold of PVI and of the braid group action on it (J. Phys. A, 43), the discovery that the affinization 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).
We introduced the notion of bordered cusp in a Riemann surface and found that generalized 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). A video about this can be found here: https://www.birs.ca/events/2016/5-day-workshops/16w5027/videos/watch/201610041100-Mazzocco.html
Quantum algebra:
My work in this topic sprung from my discovery that the quantization of the monodromy manifolds of the Painlevé equations leads to special degenerations of the Askey-Wilson algebra which regulate the q-Askey polynomials (Nonlinearity 29, 2016). This lead me to propose a representation-theoretic approach to the theory of the Painlevé equations by showing that the Cherednik algebra of type Č1C1 appears naturally as quantization 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 automorphisms of the Cherednik algebra (Adv. Pure Math., 2018). A video about this topic can be found here: https://www.birs.ca/events/2018/5-day-workshops/18w5078/videos/watch/201809111330-Mazzocco.html
Following these results, I secured another Research Grant from EPSRC to explore the relationship between (quantum) cluster algebras, double affine Hecke algebras and orthogonal polynomials and, in collaboration with L. Chekhov and V. Rubtsov, we introduced the generalized Sklyanin-Painlevé algebra and characterized its PBW/PHS/Koszul properties. This algebra contains as limiting cases the generalized Sklyanin algebra, Etingof-Ginzburg and Etingof-Oblomkov-Rains quantum del Pezzo and the quantum monodromy manifolds of the Painlevé equations. According to the referees, these results provide a first step towards the more ambitious goal of describing all Gross-Hacking and Keel theta functions in the non-commutative world (Adv. Math. 376). A video about this topic can be found here: https://www.youtube.com/watch?v=lPKXyaA5TD4
Representation theory of (G)DAHA
Double Affine Hecke Algebras (DAHA) are algebraic structures that generalize Hecke algebras and affine Hecke algebras. They play a crucial role in representation theory, algebraic geometry, mathematical physics, and special functions, particularly in connection with Macdonald polynomials and integrable systems. My contribution is this are is
Work on the representation theory of the confluent Cherednik algebras with implications to the theory of non-symmetric basic orthogonal polynomials (see below).
Work in collaboration with my former PhD student Davide Dal Martello on the representation theory of the generalized DAHA, flat deformations of the group algebras of 2-dimensional crystallographic groups associated to star-shaped simply laced affine Dynkin diagrams. We have faithful represnetations of the E6 case in terms matrix algebras over quantum cluster X-varieties.
Basic orthogonal polynomials:
My research in this field concerns families of orthogonal polynomials belonging to the q-Askey scheme. These are either standard or Laurent polynomials in one variable, say z, that depend on some parameters. They appear as eigenfunctions of a second order q-difference operator L and of a second order difference operator with eigenvalue z+1/z (or z in the case of standard polynomials). The operator L and the operator of multiplication by z+1/z (or z) generate a non-commutative algebra, which in the most general case this is the Zhedanov algebra and in the other cases a confluent version of it.
My contributions to this area are:
Producing new non symmetric versions of the continuous dual q-Hahn, Al Salam–Chihara, continuous big q–Hermite and continuous q-Hermite polynomials.
In collaboration with Tom Koornwinder, we produced new dualities between different families of polynomials in the q-Askey scheme and their non-symmetric generalizations by exploiting the morphisms of the corresponding confluent Zhedanov algebras as well as of the confluent double affine Hecke algebra.
In collaboration with Tom Koornwinder, new contiguity relations between Askey Wilson polynomials, their non-symmetric extensions as well as their function counterparts, by exploiting the automorphisms of the Zhedanov algebra (and its confluences) a well as of the double affine Hecke algebra of type Č1C1.
Moduli spaces of meromorphic connections
My recent research focuses on meromorphic connections on Riemann surfaces, capturing a vastly wide range of phenomena including those underpinning contemporary questions in theoretical physics.
Starting from the classical problem of merging simple poles of a differential equation to create higher order ones, I have attacked the problem of characterizing the moduli spaces of meromorphic connections on a Riemann surface, and its naturally homeomorphic Betti moduli space of representations of its fundamental group.
In collaboration with L. Chekhov and V. Rubtsov, we introduced the concept of decorated character variety (now dubbed bordered cusped 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).
In collaboration with V. Rubtsov and our joint former PhD student Ilia Gaiur, we studied the isomonodromic deformations of systems of differential equations with poles of any order on the Riemann sphere as Hamiltonian flows on the product of co-adjoint orbits of the truncated current algebra, also called generalised Takiff algebra. Our motivation wss to produce confluent versions of the celebrated Knizhnik–Zamolodchikov equations and explain how their quasiclassical solution can be expressed via the isomonodromic tau-function. A video of my presentation of this work can be found here: https://www.youtube.com/watch?v=5O3dMAThExc