Probabilistic and Topological methods in Real Algebraic Geometry and Computational Complexity

代数几何研究多项式的零点(称为簇)。除了在纯数学中表现突出外，它在物理、计算几何和机器学习等众多领域都有应用。本文旨在研究某些类型簇的拓扑性质，以期在关联组合学和计算复杂性理论中得到应用。我们的第一个方向是随机多项式的拓扑研究。研究随机多项式代表了代数几何的转变--而不是最坏情况分析，例如，要求在曲线交点上尽可能多的点，随机性提供了平均情况的理解，从而提供了更现实的观点。通过鄂尔多斯惊人的“概率方法”，人们可以通过将随机性引入一个先验与随机性无关的问题来获得确定性的结果。我们主要研究齐次多项式空间上的概率分布，它是一种研究很活跃的具有很强代数几何意义的概率分布。具体地说，我们对限制在o-极小几何中的随机簇的拓扑复杂性的界感兴趣，用Betti数来度量。O-极小几何是一种框架，在该框架中允许集合(称为可定义集合)比变体更一般。O-最小关联几何领域涉及关于可定义集合的组合问题，迫切需要多项式分割定理--这是传统关联几何中的灵丹妙药。然而，这已经被证明是困难的，因为限制于变种的可定义集的最坏情况的拓扑复杂性可能是“坏的”，我们建议研究限制于随机变种的可定义集的平均情况复杂性。通过这样做，我们希望证明拓扑上的“坏”多项式是不寻常的(我们已经对某些可定义集证明了这一点)。因此，如果适合于划分的多项式的度量足够大，我们将通过概率方法证明具有“良好的”拓扑复杂性的划分多项式的存在。这将极大地促进o-极小关联几何的发展。我们的第二个方向是代数拓扑问题及其在计算复杂性理论中的应用。计算复杂性理论是一个数学领域，它根据解决问题所需的资源对问题进行分类。该领域最重要的悬而未决的问题是极其困难的P与NP问题。最近，在几何复杂性理论(GCT)计划下，这个问题得到了推动，该计划涉及用代数语言铸造相关问题。希望在代数几何和表示论的丰富领域中的先进工具将允许我们在这个问题上取得进展。GCT计划的中心目标是分离某些称为轨道闭合的空间。虽然这显然是一个崇高的目标，但我们的目标是计算这些轨道闭合的拓扑结构。计算拓扑有助于粗略地了解所涉及的对象的几何体。此外，众所周知，获得空间拓扑的量化界限有助于理解在空间上操作的计算过程的基本限制。我们还将研究Ulrich复杂性的概念，它以不同的方式量化多项式的“复杂性”。虽然它可以很方便地用交换代数概念来定义，而且显然更容易使用，但人们对它的总体了解很少。我们将研究Kostlan多项式的平均Ulrich复杂性。众所周知，得到多项式的Ulrich复杂性的下界可能会对交换代数中的一个重要猜想以及P与NP问题产生潜在的影响。

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.