My research interests reside in the overlap between Geometric Analysis and Topological Quantum Field Theory (TQFT). In particular, my research is oriented towards the study of logarithmic structures of invariants of manifolds and their representations (characterizations) as logarithmic TQFTs. In fact, following S. Scott’s definition, a logTQFT is a simplicial map from the nerve of the cobordism category to its representations into an additive category, and many topological invariants can be obtained as the composition of a logTQFT and a suitable choice of trace. This is the case of the signature and Euler characteristic of a manifold, possibly with boundary, and can be generalized to higher structures, such as Novikov’s signatures. This theory naturally fits within the intersections of many areas of mathematics, and thus involves techniques from index theory and elliptic boundary value problems, algebraic topology and differential geometry, category theory and functional analysis, and interacts with cutting-edge topics, such as noncommutative geometry.
I am also interested in the study of topological invariants of secondary type, such as Reidemeister torsion and analytic torsion, for manifolds with and without boundary, and for families. In particular, following my supervisor’s work, I focused on a non-equivalent counterpart of torsion, called residue torsion, which can be defined also for families of manifolds (via the Family Index Theorem) and characterized as the character of a logTQFT.