"This proposal aims at strengthening the connections between more fundamentally oriented areas of mathematics like algebra, geometry, analysis, and topology, and the more applied oriented and more recently emerging disciplines of discrete mathematics, optimization, and algorithmics.The overall goal of the project is to obtain, with methods from fundamental mathematics, new effective tools to unravel the complexity of structures like graphs, networks, codes, knots, polynomials, and tensors, and to get a grip on such complex structures by new efficient characterizations, sharper bounds, and faster algorithms.In the last few years, there have been several new developments where methods from representation theory, invariant theory, algebraic geometry, measure theory, functional analysis, and topology found new applications in discrete mathematics and optimization, both theoretically and algorithmically. Among the typical application areas are networks, coding, routing, timetabling, statistical and quantum physics, and computer science.The project focuses in particular on:A. Understanding partition functions with invariant theory and algebraic geometryB. Graph limits, regularity, Hilbert spaces, and low rank approximation of polynomialsC. Reducing complexity in optimization by exploiting symmetry with representation theoryD. Reducing complexity in discrete optimization by homotopy and cohomologyThese research modules are interconnected by themes like symmetry, regularity, and complexity, and by common methods from algebra, analysis, geometry, and topology."