# Professor Marta Mazzocco PhD FIMA

## In 'People'

Back to 'School of Mathematics'School of Mathematics

Professor of Mathematics

## Contact details

- Address
- School of Mathematics

Watson building

University of Birmingham

Edgbaston

Birmingham

B15 2TT

Professor Mazzocco is a specialist in the area of integrable systems, namely mathematical problems often motivated by Mathematical Physics that present unexpected beauty - a serendipity of geometric/algebraic/analytical structures that make a very complicated problem solvable in some sense. Her research brings geometry, quantum algebra and analysis together to describe and tackle problems which have so far resisted all other methods.

Professor Mazzocco has an outstanding record in attracting external funding. In 2003, she obtained an EPSRC First Grant while still on a temporary job; according to the EPSRC administrators, this is very rare, if not the unique case they know of. In 2006, she was awarded an EPSRC Advanced Research Fellowship for 5 years, then in 2008, 2011, 2017 three further major EPSRC research grants.

Professor Mazzocco is a keen advocate of the importance of basic research and its role in society. She believes that in the current funding landscape that emphasises socio-economic impact, it is paramount to nurture interdisciplinary research as well as to raise the profile of the unique contribution that basic research (and in particular Mathematical Sciences) brings to the overall research base, the economy and society - this has driven her to become a member of the Council of the European Mathematical Society; of the Institute of Mathematics and Applications Research Committee; and of the London Mathematical Society Nominating Committee.

Professor Mazzocco is also a passionate in promoting Equality and Diversity initiatives both within her own University and internationally. In her previous position at Loughborough University she has successfully led my department to win an Athena SWAN silver award (one of only 5 mathematics departments in the UK to have achieved this), and has acted as director of Equality and Diversity for the School of Science.

## Qualifications

Professor of Mathematics:

- March 2004: French qualification to apply for full professorships in pure and applied mathematics.

- October 1998: Ph.D. in Mathematical Physics, SISSA, International School for Advanced Studies, Trieste, Italy.

- July 1994: Laurea in Mathematical Physics (equivalent to MPhil.), University of Padova, Italy.

## Biography

After graduating with the highest honours (cum laude) in Theoretical Physics at the University of Padova, in 1998 Professor Mazzocco took a PhD in Mathematical Physics at the International School of Advanced Studies (SISSA, Trieste) one of the two most famous PhD schools in Italy. She then moved to Berkeley where she was a post doc at MSRI and to Oxford where she was an EPSRC funded RA under the supervision of Professor Hitchin for three years. In 2002 she was appointed as temporary University Lecturer in DPMMS, Cambridge University, and three years after as lecturer in Applied Mathematics at the University of Manchester. In 2008 she accepted a readership at Loughborough University, where she was promoted to a personal chair in 2014. In February 2018 she was appointed as Professor of Mathematics at UoB.

## Research

Professor Mazzocco 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.

## Other activities

Member of the Council of the European Mathematical Society

Member of the Institute of Mathematics and its Applications research Committee

Member of the London Mathematical Society Nominating Committee

## Publications

M. Mazzocco,

Embedding of the rank 1 DAHA into Mat(2,Tq) and its automorphisms.

arXiv:1603.03770, to appear in Adv. Pure Maths. (2018)

L. Chekhov and M. Mazzocco,

Colliding holes in Riemann surfaces and quantum cluster algebras.

Nonlinearity, Volume 31, Number 1

L. Chekhov and M. Mazzocco,

On a Poisson homogeneous space of bilinear forms with a Poisson Lie action.

arXiv:1404.0988, to appear in Russ. Math. Surveys 2018.

Chekhov L., Mazzocco M. and Rubtsov V., Painlevé monodromy manifolds, decorated character varieties and cluster algebras Int. Math. Res. Not.(2016).

M. Mazzocco,

Confluences of the Painlevé equations, Cherednik algebras and q-Askey scheme.

Nonlinearity 29 2565-2608 (2016).

M. Mazzocco,

Non-Symmetric Basic Hypergeometric Polynomials and Representation Theory for Confluent Cherednik Algebras,

SIGMA10, 116, 10 pages, (2014),arXiv:1409.4287.

L. Chekhov and M. Mazzocco,

Quantum ordering for quantum geodesic functions of orbifold Riemann surfaces,inTopology, Geometry, Integrable Systems, and Mathematical Physics: Novikov's Seminar 2012-2014,

American Mathematical Society Translations--Series 2, 234 (2014)

arXiv:1309.3493.

L. Chekhov and M. Mazzocco,

Poisson algebras of block-upper-triangular bilinear forms and braid group action,

Comm. Math. Phys. 322, no.1:49-71 (2013)

M. Mazzocco and R. Vidunas,

Cubic and quartic transformations of the sixth Painleve equation in terms of the Riemann-Hilbert correspondence,

Studies in App. Math., 130, 1:17–48 (2013)

L. Chekhov and M. Mazzocco,

Block triangular bilinear forms and braid group action,

in Tropical Geometry and Integrable Systems,

Contemporary Mathematics 580 (2012)

L. Chekhov and M. Mazzocco, Teichmüller spaces as degenerated symplectic leaves of the Dubrovin-Ugaglia bracket, Phisica D, 241, Issues 23–24:2109–2121 (2012)

L. Chekhov and M. Mazzocco,

Isomonodromic deformations and twisted Yangians arising in Teichmüller theory,

Adv. Math. 226, no.6:4731-4775 (2011)

L. Chekhov and M. Mazzocco,

Shear coordinate description of the versal unfolding of the D4 singularity,

J. Phys. A, 43, no. 44:442002-442015 (2010)

L. Chekhov and M. Mazzocco,

Orbifold Riemann surfaces:Teichmüller spaces and algebras of geodesic functions,

Russian Math. Surveys, 64, no.6:1079-1130 (2009)

K. Kajiwara, M. Mazzocco and Y. Ohta,

A remark on the Hankel determinant formula for solutions of the Toda equation,

J. Phys. A. 40, no.42:12661-12675 (2007).

M. Mazzocco and Man Mo Yue,

The Hamiltonian structure of the second Painlevé hierarchy,

Nonlinearity, 20, no. 12:2845-2882 (2007)

B. Dubrovin and M.Mazzocco,

On the Reductions and Classical Solutions of the Schlesinger equations,

in Differential equations and quantum groups, Andrey A. Bolibruch memorial volume,

IRMA Lectures in Mathematics and Theoretical Physics 9, 157--187 (2006).

B. Dubrovin and M.Mazzocco,

Canonical structure of the Schlesinger systems,

Comm. Math. Phys. 271, no.2:289-373 (2007)

N. Joshi, K. Kajiwara and M. Mazzocco,

Generating Function Associated with the Determinant Formula for the Solutions of the Painlevé IV Equation,

Funkcial. Ekvac. 49, no.3:451-468 (2006).

N. Joshi, K. Kajiwara and M. Mazzocco,

Generating Function Associated with the Determinant Formula for the Solutions of the Painlevé II Equation,

in Analyse complexe, systèmes dynamiques, sommabilité des séries divergentes et théories galoisiennes (II); Volume en L'Honneur de Jean-Pierre Ramis,

Asterisque 297 (2004), 67--78.

M. Mazzocco,

Irregular isomonodromic deformations for Garnier systems and Okamoto's canonical transformations,

J. London Math. Soc. (2), 70 (2004), no. 2, 405--419.

N. Joshi and M.Mazzocco,

Existence and uniqueness of tri-tronquee solutions of the second Painlevé hierarchy,

Nonlinearity 16 (2003), no. 2, 427--439.

M. Mazzocco,

Painlevé sixth equation as isomonodromic deformations equation of an irregular system.

in The Kowalevski property (Leeds, 2000), 219--238,

CRM Proc. Lecture Notes, 32, Amer. Math. Soc., Providence, RI, 2002.

M. Mazzocco,

The geometry of the classical solutions of the Garnier systems,

Int. Math. Res. Not. 2002, no. 12, 613--646.

M. Mazzocco,

Picard and Chazy solutions to the Painlevé VI equation,

Math. Ann. 321 (2001), no. 1, 157--195.

M. Mazzocco,

Rational solutions of the Painlevé VI equation,

Kowalevski Workshop on Mathematical Methods of Regular Dynamics (Leeds, 2000).

J. Phys. A 34 (2001), no. 11, 2281--2294.

B. Dubrovin and M. Mazzocco,

Monodromy of certain Painlevé VI transcendents and reflection groups,

Invent. Math. 141 (2000), no. 1, 55--147.

M. Mazzocco,

KAM theorem for generic analytic perturbations of the Euler system,

Z. Angew. Math. Phys. 48 (1997), no. 2, 193--219.