Abstract

Real algebraic (and semi-algebraic) geometry studies subsets of R^n defined by a finite number of polynomial equalities and inequalities. Such sets occur ubiquitously in practice both inside and outside of mathematics. While being easy to define, semi-algebraic sets can be complicated topologically, which restricts the application of many algorithms. In recent years, there has been progress in proving much stronger results – both quantitative and algorithmic -- when the problem under consideration involves the invariance under some group action.

In this workshop, we plan to focus on two situations where this phenomenon happens.

The first one is the statistical study of the topology of random real algebraic varieties as well as semi-algebraic sets, where the polynomials defining these objects are picked from a distribution invariant under the action of a certain group (usually the orthogonal group) acting on the space of variables. The behavior of the set of zeros (or more generally level sets) of such random polynomial systems is an extremely active topic with deep connections with classical enumerative problems in algebraic geometry (over real as well as complex fields), theoretical computer science, statistical physics, and machine learning.

The second one is the deterministic behavior of the topology of semi-algebraic sets admitting a group action (say a finite reflection group or a Lie group). The cohomology modules of such symmetric semi-algebraic sets then acquire the structure of a finite-dimensional representation of the corresponding group. This added structure is extremely useful in studying the topology as well as developing algorithms for computing topological invariants of such sets. The study of symmetric semi-algebraic sets has promising connections with several other areas, including representational and homological stability, sums-of-squares proofs and lower-bound questions in theoretical computer science.