LOWDIMTOPO · Low-dimensional quantum topology

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Project objective

The research topic of the proposal is Quantum Topology. This mathematical theory (which involves topology, algebra, and mathematical physics) was born in the 1990's and was inspired by the works of Jones, Witten, Drinfeld, Atiyah, Reshetikhin, Turaev, Kontsevich, and their followers. Structures which originally do not have any relation with topology (such as quantum groups, braided categories) appear of crucial importance in the construction of new topological invariants of knots, of 3-dimensional manifolds, and of fibre bundles over 3-manifolds. These non-classical quantum" invariants belong to the general framework of topological/homotopical quantum field theories (T/HQFT). I propose a research project, which is directed according to four major objectives. These objectives are the natural extensions of as well my PhD thesis and actual research as the recent works of the scientific supervisors of the project who are Reshetikhin (at UC Berkeley) and Benedetti (at Pisa).More precisely, during the outgoing phase at the Department of Mathematics at Berkeley (United-States), I plan to acquire new competencies quantum group theory, representation theory, and 4-dimensional topology in order to:1) construct non-semi-simple TQFT via "Kirby elements";2) explore the implications of the "graded quantum groups" (that I have recently defined) into the topology of flat fibre bundles over 3-manifolds;3) construct the first non-trivial examples of 4-dimensional HQFT. During the return phase at the Department of Mathematics at Pisa, I plan to acquire new competencies in geometric and hyperbolic topology in order to:4) construct new quantum invariants of fibre bundles endowed with a (not necessarily) flat connection.The proposed mobility will allow me to pursue dynamic research by actively training and collaborating with world experts in famous mathematical centres. It appears to be a major step towards my professional independence as a researcher."

Original text from CORDIS.

Participants

• UNIVERSITÃ'  DI PISA · PISACoordinatorCity levelItaly
• DEPARTMENT OF MATHEMATICS, UNIVERSITY OF CALIFORNIA, BERKELEY · BERKELEYCity levelUnited States

Links

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