Probabilistic and Topological methods in Real Algebraic Geometry and Computational Complexity

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Algebraic geometry is concerned with the study of the zeros of polynomials (called varieties). Besides being prominent in pure mathematics, it has applications in a plethora of areas such as physics, computational geometry, and machine learning. This proposal aims to study topological properties of certain kinds of varieties, with a view toward applications in incidence combinatorics and computational complexity theory.Our first direction is the topological study of random polynomials. Studying random polynomials represents a shift in algebraic geometry - instead of worst-case analysis, which, for example, asks for the largest possible number of points in an intersection of curves, randomness gives an average-case understanding, thus providing a more realistic view. Via Erdos' astonishing 'probabilistic method', one can obtain deterministic results by introducing randomness into a question that apriori had nothing to do with randomness. We focus on a probability distribution on the space of homogeneous polynomials, which is an actively studied probability distribution with strong algebro-geometric significance.Specifically, we are interested in bounds on the topological complexity, as measured by Betti numbers, of random varieties restricted to sets in o-minimal geometry. O-minimal geometry is a framework in which sets (called definable sets) more general than varieties are allowed. The field of o-minimal incidence geometry, which involves combinatorial questions about definable sets, is badly in need of a polynomial partitioning theorem - a tool which has been a panacea in traditional incidence geometry. However, it has proved difficult because the worst-case topological complexity of definable sets restricted to varieties can be "bad".We propose to study the average-case complexity of definable sets restricted to random varieties instead. By doing so, we hope to demonstrate that topologically "bad" polynomials are unusual (we have already proved this for certain definable sets). Thus, if the measure of polynomials suitable for partitioning is large enough, we will have proved the existence of a partitioning polynomial, with "good" topological complexity, via the probabilistic method. This will provide a tremendous boost to o-minimal incidence geometry.Our second direction is algebro-topological questions with applications in computational complexity theory. Computational complexity theory is a mathematical area that classifies problems according to the resources required to solve them. The most important open question in the field is the exceedingly difficult P vs NP problem. There has been a recent impetus towards the problem, under the Geometric Complexity Theory (GCT) program, which involves casting related problems in algebraic language. The hope is that advanced tools in the fertile areas of algebraic geometry and representation theory will allow us to make progress on the problem.The central objective of the GCT program is to separate certain spaces called orbit closures. While that is obviously a lofty aim, our goal is to compute the topology of these orbit closures. Computing the topology helps in obtaining a coarse understanding of the geometry of the objects involved. Also, it is well known that obtaining quantitative bounds on the topology of a space helps in understanding the fundamental limitations of computational procedures that operate on the space.We shall also study the notion of Ulrich complexity, which quantifies the 'complexity' of polynomials in a different way. While it is conveniently defined using commutative-algebraic notions, and is evidently easier to work with, little is understood about it in general. We shall investigate the average Ulrich complexity of Kostlan polynomials. It is known that obtaining lower bounds on the Ulrich complexity of polynomials could potentially have implications on an important conjecture in commutative algebra, as well as the P vs NP problem.