Algebra, Geometry, and Mathematical Logic

This research area concerns three of the fundamental fields in pure mathematics: algebra, geometry and logic. Roughly speaking, algebra is the study of symmetry via mathematical structures with algebraic operations; geometry is concerned with properties of space including the concepts of distance and shape; and mathematical logic investigates the formal properties of mathematical systems including syntax (formal languages and calculi) and semantics (structures and models). The research groups working in this area at the University of Verona organise regular seminars, research activities, and host academic visitors from all over the world. Our research in algebra focuses on the representation theory of algebras, in particular, finite-dimensional algebras and Leavitt path algebras, localization theory, approximation theory, homological algebra, triangulated categories, t-structures and hearts, and spaces of Bridgeland stability conditions. In geometry we specialise in geometric mechanics, symplectic geometry and topology, including the study of Kähler and hyperkähler manifolds (in particular polygon and hyperpolygon spaces), moduli problems such as moduli of parabolic and parabolic Higgs bundles, equivariant cohomology, geometric quantization, cobordism, Gromov width and related topics. The logic group works on constructive algebra and analysis, Bishop set theory and type theory, higher-type computability theory and category theory, proof theory, Hilbert's program in abstract mathematics, especially the computational content of classical proofs with transfinite methods, logical methods in computer science, languages and logics for quantum computing.