## Periodic Reporting for period 2 - COCAN (Complexity and Condition in Algebra and Numerics)

Reporting period: 2020-07-01 to 2021-12-31

This project connects three areas that are usually considered quite distant from each other: computational complexity, algebraic geometry, and numerics. Computational complexity is a foundational part of computer science, provides formal models for investigating algorithms and tries to understand the inherent limits of efficient computation in a broad sense. Algebraic geometry is a highly developed branch of pure mathematics that studies the set of solutions of polynomial equations. Numerical mathematics is concerned with the development and analysis of efficient algorithms for the solution of mathematical problems such as those arising in natural sciences and engineering.

In the last decade, it became clear that the fundamental questions of computational complexity (P vs NP) are should be studied in algebraic settings, linking them to classic problems in algebraic geometry. Recent progress on this very challenging questions led to surprising progress in computational invariant theory, which also related to quantum information theory. The project's goals is to explore these connections. An essential new ingredient to this is to tackle the arising algebraic computational problems by means of approximate numeric computations, taking into account the concept of numerical condition.

A related goal of the proposal is to develop a theory of efficient and numerically stable algorithms in algebraic geometry that reflects the properties of structured systems of polynomial equations, possibly with singularities. While there are various heuristics, a satisfactory theory so far only exists for unstructured systems over the complex numbers (recent solution of Smale's 17th problem), which seriously limits its range of applications. In this framework, the quality of numerical algorithms is gauged by a probabilistic analysis that shows small average (or smoothed) running time. One of the main challenges here consists of a probabilistic study of random structured polynomial systems. We will also develop and analyze numerical algorithms for finding or describing the shape of the set of real solutions, e.g. in terms of their homology.

In the last decade, it became clear that the fundamental questions of computational complexity (P vs NP) are should be studied in algebraic settings, linking them to classic problems in algebraic geometry. Recent progress on this very challenging questions led to surprising progress in computational invariant theory, which also related to quantum information theory. The project's goals is to explore these connections. An essential new ingredient to this is to tackle the arising algebraic computational problems by means of approximate numeric computations, taking into account the concept of numerical condition.

A related goal of the proposal is to develop a theory of efficient and numerically stable algorithms in algebraic geometry that reflects the properties of structured systems of polynomial equations, possibly with singularities. While there are various heuristics, a satisfactory theory so far only exists for unstructured systems over the complex numbers (recent solution of Smale's 17th problem), which seriously limits its range of applications. In this framework, the quality of numerical algorithms is gauged by a probabilistic analysis that shows small average (or smoothed) running time. One of the main challenges here consists of a probabilistic study of random structured polynomial systems. We will also develop and analyze numerical algorithms for finding or describing the shape of the set of real solutions, e.g. in terms of their homology.

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. Our work points to intriguing open problems and suggests further research directions.

Surprisingly, during these investigations, a connection to to statistics popped up and is investigated as part of the project. This is concerned with maximum likelihood estimations in algebraic statistical models.

In another direction, we obtained a general result indicating that typically, solutions of structured polynomial systems, can be computed in polynomial time. This was achieved by the new method of rigid homotopy continuation. The model of structured systems we use, algebraic branching programs, is inspired by algebraic complexity theory. Moreover, the software „Homotopy Continuation.jl'' was further developed and represents the state of the art in numerical algebraic geometry.

In a third direction, 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.

In the area of random real algebraic geometry, we completed two papers on the number of reals zeros of random polynomials. The first one connects to complexity theory by showing that the claimed estimate of the real tau conjecture is typically true for random polynomials. The second contribution, for the first time, proves good upper bounds on the expected number of real zeros of random fewnomial system We work towards improving the approach and bounds here. In addition, we are developing an approach towards a probabilistic intersection theory in which zonoids, which are certain convex bodies, play a crucial role.

Surprisingly, during these investigations, a connection to to statistics popped up and is investigated as part of the project. This is concerned with maximum likelihood estimations in algebraic statistical models.

In another direction, we obtained a general result indicating that typically, solutions of structured polynomial systems, can be computed in polynomial time. This was achieved by the new method of rigid homotopy continuation. The model of structured systems we use, algebraic branching programs, is inspired by algebraic complexity theory. Moreover, the software „Homotopy Continuation.jl'' was further developed and represents the state of the art in numerical algebraic geometry.

In a third direction, 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.

In the area of random real algebraic geometry, we completed two papers on the number of reals zeros of random polynomials. The first one connects to complexity theory by showing that the claimed estimate of the real tau conjecture is typically true for random polynomials. The second contribution, for the first time, proves good upper bounds on the expected number of real zeros of random fewnomial system We work towards improving the approach and bounds here. In addition, we are developing an approach towards a probabilistic intersection theory in which zonoids, which are certain convex bodies, play a crucial role.

We already designed and analysed first and second order methods in the general framework of geodesically convex optimization on Riemannian manifolds that arise from the symmetries of non-commutative groups. However, when restricted to the commutative case, our algorithms' guarantees do not match those of cut methods in the spirit of the ellipsoid algorithm or of interior point methods. A project's major goal is to develop an analogue of interior point methods in the geodesic framework setting for non-commutative group actions. This would show that the general null cone membership problem has polynomial time algorithms and would be breakthrough. It also remains to investigate the potential consequences of this for the more general moment polytope membership problem. We plan to concentrate efforts on these problem.

A next step is to investigate the related orbit closure intersection problem. We obtained a satisfactory general result for the actions of commutative groups (tori), but the noncommutative case is wide open. Apparently, it is very challenging to bring in here numerical algorithms.

Another challenge is to investigate whether our homotopy methods can be modified to show that solutions of sparse polynomial systems, can be computed in polynomial time. Achieving such result would represent a breakthough. We believe that the idea of following along group orbits will be helpful for this.

We also seek to improve our results on the number of reals zeros of random polynomials. This is related to our work on probabilistic intersection theory that builds a new bridge from real algebraic geometry to convex geometry via zonoids.

A next step is to investigate the related orbit closure intersection problem. We obtained a satisfactory general result for the actions of commutative groups (tori), but the noncommutative case is wide open. Apparently, it is very challenging to bring in here numerical algorithms.

Another challenge is to investigate whether our homotopy methods can be modified to show that solutions of sparse polynomial systems, can be computed in polynomial time. Achieving such result would represent a breakthough. We believe that the idea of following along group orbits will be helpful for this.

We also seek to improve our results on the number of reals zeros of random polynomials. This is related to our work on probabilistic intersection theory that builds a new bridge from real algebraic geometry to convex geometry via zonoids.