## PhD research topic - Claudia Pinzari

#### Research areas

Operator algebras, Noncommutative geometry

#### Keywords

Compact quantum groups, quantum groups at roots of unity, fusion and modular categories, conformal field theory

#### Research topics

Quantum groups have been independently discovered in the mid 80s by Drinfeld and Jimbo and by Woronowicz, motivated, respectively, by the theory of integrable systems and Lie theory; and Connes noncommutative geometry. They have been shown to have deep connections with topology of framed knots and links (Jones, Reshetikhin-Turev) and conformal field theory in the framework of vertex operator algebras (Kohno, Drinfeld, Kazhdan-Lusztig, Finkelberg). Furthermore, they are a highly active research area in noncommutative geometry in the sense of Connes. More precisely they are involved in the construction of spectral triples in the sense of Connes, principal bundles, and generally speaking in the framework of quantum symmetries and their actions on non commutative spaces. A further important field of research is the classification problem of compact quantum groups with given fusion rules (Neshveyev-Yamashita).

My recent research in the area consists on two subjects: the study of classical notions of connectedness, local connectedness, topological dimension, in the spirit of a theory of quantum compact Lie groups in global terms, and secondly the formulation of an analytic theory more general than that of compact quantum groups, but still in the framework of operator algebras, as well as construction of new examples, with an eye to applications to conformal field theory, involving also the framework of conformal nets of the Rome school (Doplicher; Haag; Longo; Roberts).

#### Bibliography

[1] S. Neshveyev; L. Tuset: Compact Quantum Groups and Their Representation Categories, AMS 2013

[2] V. Chari; A. Pressley: A guide to quantum groups, Cambridge UP, 1994

[3] S. Carpi; Y. Kawahigashi; R. Longo; M. Weiner, from vertex operator algebras to conformal nets and back, Mem. of the AMS 254 (2018), n. 1213

[4] L. Cirio; A. D'Andrea; C. Pinzari; S. Rossi: Connected components of compact quantum groups and finiteness conditions, J. Funct Anal. (2014) n. 267, p. 3154-3204

[5] S. Carpi; S. Ciamprone; C. Pinzari: Weak quasi Hopf algebras, tensor C*-categories and conformal field theory, in preparation.