Abstract: |
A triangulation can be seen as a discrete analog of a Riemannian metric (gravitational field) on a space. I will discuss several directions of research which have led to triangulations in an unexpected way. In ``classical" algebra, triangulations govern discriminants of polynomials. In 2 dimensions, triangulations are related to associativity conditions in abstract algebra. An important example is provided by Hall algebras, associative algebras appearing in theories of quantum groups and automorphic forms. For Hall algebras related to algebraic curves, commutation relations involve zeta functions. Further, associativity of Hall algebras is a manifestation of a finer structure (2-Segal space) involving triangulated polygons and spaces of polygonal membranes. Other examples of 2-Segal spaces can be constructed using spaces of holomorphic maps from hyperbolic polygons into complex manifolds. |