## PhD research topic - Domenico Fiorenza

#### Research areas

Topological quantum field theories

#### Keywords

Topological quantum field theories, symmetric monoidal categories, factorization algebras, elliptic cohomology, topological modular forms

#### Research topics

Topological quantum field theories are linear representations of (extended) cobordism. On the one hand they are a powerful source of algebraic invariants of manifolds (classical examples are the Jones polynomial, Turaev-Viro, and Dijkgraaf-Witten invariants). On the other hand, the intuition coming from the geometric nature of cobordism provides a clear explanation for otherwise mysterious algebraic constructions. An intriguing example is provided by elliptic cohomology. This is known in the physics folklore to be related to a certain supersymmetric quantum field theory, yet a rigorous construction remains elusive and an appealing challenge to mathematicians.

#### Bibliography

[1] Edward Witten, Elliptic Genera And Quantum Field Theory , Commun.Math.Phys. 109 525-536 (1987)

[2] Matthew Ando, Michael Hopkins, Neil Strickland, Elliptic spectra, the Witten genus, and the theorem of the cube, Inventiones Mathematicae, 146:595–687, 2001

[3] Stephan Stolz, Peter Teichner, What is an elliptic object? in Topology, geometry and quantum field theory , London Math. Soc. LNS 308, Cambridge Univ. Press (2004), 247-343

[4] P. Hu, I. Kriz, Conformal field theory and elliptic cohomology, Advances in Mathematics, Volume 189, Issue 2, 20 December 2004, Pages 325–412

[2] Matthew Ando, Michael Hopkins, Neil Strickland, Elliptic spectra, the Witten genus, and the theorem of the cube, Inventiones Mathematicae, 146:595–687, 2001

[3] Stephan Stolz, Peter Teichner, What is an elliptic object? in Topology, geometry and quantum field theory , London Math. Soc. LNS 308, Cambridge Univ. Press (2004), 247-343

[4] P. Hu, I. Kriz, Conformal field theory and elliptic cohomology, Advances in Mathematics, Volume 189, Issue 2, 20 December 2004, Pages 325–412