Topological invariants of 3-manifolds

• Grant number: 25/07476-8
• Support Opportunities: Scholarships abroad - Research Internship - Doctorate
• Start date: August 01, 2025
• End date: July 31, 2026
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: Luiz Roberto Hartmann Junior
• Grantee: Pedro Henrique Muller Bortolucci
• Supervisor: Louis Funar
• Host Institution: Centro de Ciências Exatas e de Tecnologia (CCET). Universidade Federal de São Carlos (UFSCAR). São Carlos , SP, Brazil
• Institution abroad: Institut Fourier, France
• Associated to the scholarship: 23/08170-4 - Topological invariants of 3-manifolds, BP.DR

Abstract

This project aims to study quantum invariants of 3-manifolds arising fromTopological Quantum Field Theories (TQFTs), following the foundational work ofReshetikhin and Turaev. These TQFTs are constructed from modular categories-algebraic structures that contain simple objects, a braiding, a twist, and a duality-providing the necessary data to define topological invariants of closed 3-manifolds,and consequently, a TQFT.Modular categories can arise from two main frameworks: one from the representa-tion theory of quantum groups, and another from skein theory. In particular, the TQFTassociated with the quantum group Uq(sl2) is known to produce the same invariantsas the one constructed via skein modules using the Kauffman skein relation and thecorresponding Jones-Wenzl idempotents.In this BEPE research project, we propose to investigate the generalization of skeinmodules in order to construct, in an analogous manner, a TQFT associated with theexceptional Lie algebra of type G2. This study will deepen our understanding ofquantum invariants and their potential connections to classical topological invariants. (AU)