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# Chaozhong Wu

INdAM-COFUND Incoming fellow between 2012-01-10 and 2014-01-09

### Project information

#### Title

Tau Functions of Integrable Systems and their Applications

#### Partner organisation

Scuola Internazionale Superiore di Studi Avanzati

via Bonomea, 265

34136 Trieste, Italy

#### Abstract

Tau function is a significant concept in the theoryof integrable systems and their applications in relevant researchareas of mathematical physics such like topological field theory,Gromov-Witten invariants and singularity theory. In this proposal we study tau functions of certain integrable systems and theirapplications.

Our key objective is a uniform definition of the tafunctions of all Drinfeld-Sokolov hierarchies, which are Hamiltonian systems associated to affine Kac-Moody algebras and have several equivalent versions in different contexts. Up to now, tau functions of these hierarchies have been considered separately in some particular cases. We plan to define tau functions of Drinfeld-Sokolov hierarchies by using appropriate tau-symmetric Hamiltonian densities, which is independent of representations of the Kac-Moody algebras. Such a definition is crucial in both the theory of integrable systems and their applications.

Furthermore we will investigate Virasoro symmetries acting on the tau function of the Drinfeld-Sokolov hierarchies. Such actions could be linear or nonlinear, and the corresponding tau function may have rich geometric illustrations. Progress in this direction will resolve a conjecture raised by B. Dubrovin and Y. Zhang for topological hierarchies associated to semisimple Frobenius manifold for Coxeter groups of ADE type. Moreover, it probably gives hint to generate the previous integrable-system method for semisimple Gromov-Witten invariants.

### Fellow information

#### Previous positions, awards

*2011-2012*: Research project “On certain integrable

hierarchies related to Gromov-Witten invariants” supported by

specific funding under the “Young SISSA Scientists’ Research

Projects” scheme*11 October 2010 – 10 January 2012*: Post doctor, SISSA, Trieste, Italy*3 September 2008 – 24 February 2009*: Visitor, SISSA, Trieste, Italy*1 September 2005 – 20 July 2010*: PhD student, Department of Mathematical

Sciences, Tsinghua University, Beijing, P.R. China*1 September 2001 – 15 July 2005*: Bachelor degree, Department of Mathematical

Sciences, Tsinghua University

#### Publications, preprints, other works

- Chao-Zhong Wu,
*Tau functions and Virasoro symmetries for Drinfeld – Sokolov hierarchies*, preprint (2012); arXiv:1203.5750 - Chao-Zhong Wu,
*From additional symmetries to linearization of Virasoro symmetries*, preprint (2011); arXiv:1112.0246 - Chao-Zhong Wu,
*$R$-matrices and Hamiltonian structures for certain Lax equations*, preprint (2010); arXiv:1012.5245 - Chao-Zhong Wu, Dingdian Xu,
*A class of infinite-dimensional Frobenius manifolds and their submanifolds*, Intern. Math. Res. Notices (2011); DOI:10.1093/imrn/rnr192 - Si-Qi Liu, Chao-Zhong Wu and Youjin Zhang,
*On the Drinfeld – Sokolov hierarchies of $D$ type*, Int. Math. Res. Not. IMRN 2011, no. 8, 1952–1996 - Chao-Zhong Wu and Dingdian Xu,
*Bihamiltonian structure of the two-component Kadomtsev-Petviashvili hierarchy of type B*, J. Math. Phys. 51 (2010), 063504 - Chao-Zhong Wu,
*A remark on Kac-Wakimoto hierarchies of D-type*, J. Phys. A: Math. Theor. 43 (2010), 035201 - Si-Qi Liu, Chao-Zhong Wu and Youjin Zhang,
*On properties of Hamiltonian structures for a class of evolutionary PDEs*, Lett. Math. Phys. 84 (2008), 47-63 - Chao-Zhong Wu,
*On solutions of the two-component Camassa-Holm system*, J. Math. Phys. 47 (2006), 083513

#### Conferences, schools, other events

*Contemporary Ways Of Integrability, A Workshop for*

Young Researchers in Integrable Systems and their Ramifications

Lisbon, Complexo Interdisciplinar da Universidade

de Lisboa, 16-19 May 2012