1. We initiated a systematic development of a theory of non-commutative optimization. It develops and analyzes algorithms for natural geodesically convex optimization problems on Riemannian manifolds that arise from the symmetries of non-commutative groups. These algorithms minimize the moment map (a non-commutative notion of the usual gradient) and test membership in null cones and moment polytopes (a vast class of polytopes, typically of exponential vertex and facet complexity, which arise from a priori non-convex, non-linear setting). This setting captures a diverse set of problems in different areas of computer science, mathematics, and physics.
We designed and analysed first and second order methods in this general framework. However, when restricted to the commutative case, these algorithms' guarantees do not match those of cut methods in the spirit of the ellipsoid algorithm or of interior point methods. For this reason, we developed an analogue of interior point methods in the geodesic framework of non-commutative group actions on Hadamard manifolds, generalizing the notion of self-concordance. However, the outcomes in the quantitative regime are less favorable than what we hoped for.
Surprisingly, during these investigations in computational invariant theory, a connection to to statistics popped up and was investigated as part of the project.
We also investigated the related orbit closure intersection problems and obtained a satisfactory general result for the actions of commutative groups (tori). The aspect of numerical robustness led to a surprising new connection between computational complexity and number theory (abc-conjecture). However, the complexity status of orbit problems in the general noncommutative case remains wide open.
2. Smale's 17th problem asked whether n polynomial equations in n variables can be solved over the complex numbers in average polynomial time. This problem had been solved affirmatively around ten years ago. However, in this model, the input size is generously measured by the number of coefficients. Relying on the method of rigid homotopies, Bürgisser, Cucker, and Lairez showed that an analogue of Smale's 17th problem holds in a setting of structured polynomial systems. Solutions can be computed in average polynomial time, when the input polynomials are given in the data structure of algebraic branching programs, which allows to encode polynomials of large degrees with few input parameters. The randomness comes from independent standard Gaussian coefficients. On the practical side, the freely available software „Homotopy Continuation.jl'' of numerical algebraic geometry by Breiding and Timme was further developed.
3. In the different direction of computing real solutions to polynomial equations, we established provably efficient and numerically stable algorithms for computing the topology of semialgebraic sets. These algorithms run in weak exponential time, while all previously known algorithms have doubly exponential complexity. The techniques and results are of relevance in high-dimensional data analysis.
4. A considerable amount of work went into an emerging theory that may be called probabilistic intersection theory. The general goal is to understand the real zero set of random polynomial systems. We proved two quantitative results on counting the real zeros of random polynomial systems: one of them connects to complexity theory: the claimed estimate of the real tau conjecture is shown to be true for random polynomials. A second result, for the first time, proves good upper bounds on the expected number of real zeros of random polynomial systems in terms of the number of monomials.
