Proving the Landau-Ginzburg/Conformal Field Theory correspondence. This project aims to provide the first precise mathematical statement and geometric proof of the Landau-Ginzburg/Conformal Field Theory (LG/CFT) correspondence for simple singularities, a physically motivated principle that relates hypersurface singularities in algebraic geometry to representations of vertex algebras in conformal field theory. The formalism developed here is expected to clarify the nature of the correspondence an ....Proving the Landau-Ginzburg/Conformal Field Theory correspondence. This project aims to provide the first precise mathematical statement and geometric proof of the Landau-Ginzburg/Conformal Field Theory (LG/CFT) correspondence for simple singularities, a physically motivated principle that relates hypersurface singularities in algebraic geometry to representations of vertex algebras in conformal field theory. The formalism developed here is expected to clarify the nature of the correspondence and lead directly to generalisations beyond simple singularities, as well as provide a dictionary to translate methods of CFT into singularity theory and vice versa. These results will further cement Australia's reputation as an international leader in pure mathematics and mathematical physics research.Read moreRead less
Classical and quantum invariants of low-dimensional manifolds. This project aims to advance our understanding of knots and 3-dimensional spaces, which arise naturally in fields as diverse as physics, computer graphics, chemistry and biology. Recent ideas from quantum field theory link physics to topology in dimensions 3 and 4, leading to powerful invariants of knots and 3-dimensional manifolds that include the Jones polynomial and the 3D-index. This project aims to resolve key questions relating ....Classical and quantum invariants of low-dimensional manifolds. This project aims to advance our understanding of knots and 3-dimensional spaces, which arise naturally in fields as diverse as physics, computer graphics, chemistry and biology. Recent ideas from quantum field theory link physics to topology in dimensions 3 and 4, leading to powerful invariants of knots and 3-dimensional manifolds that include the Jones polynomial and the 3D-index. This project aims to resolve key questions relating these quantum invariants to classical topology and geometry. The project will have a major impact in low-dimensional topology, and lead to deep and unexpected connections between mathematics and mathematical physics.Read moreRead less
Frobenius manifolds from a geometrical and categorical viewpoint. This project aims to provide connections between Frobenius manifolds obtained from algebraic curves in diverse ways. The different constructions, using complex geometry on the one hand and category theory on the other, provide, respectively, a quantitative and qualitative view on the same Frobenius manifold. Together, these distinct points of view allow for the calculation of previously inaccessible physical quantities, and point ....Frobenius manifolds from a geometrical and categorical viewpoint. This project aims to provide connections between Frobenius manifolds obtained from algebraic curves in diverse ways. The different constructions, using complex geometry on the one hand and category theory on the other, provide, respectively, a quantitative and qualitative view on the same Frobenius manifold. Together, these distinct points of view allow for the calculation of previously inaccessible physical quantities, and point to deep new relations between algebraic, complex and differential geometry. These relations are expected to guide new fundamental research on the border of mathematics and physics.Read moreRead less