Probabilistic and Topological methods in Real Algebraic Geometry and Computational Complexity

代数几何研究多项式的零点（称为变量）。除了在纯数学中突出之外，它在物理、计算几何和机器学习等众多领域都有应用。本提案旨在研究某些种类的拓扑性质，以期在关联组合学和计算复杂性理论中的应用。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一个转变——而不是最坏情况分析，例如，要求在曲线的交叉点上有尽可能多的点，随机给出了一个平均情况的理解，从而提供了一个更现实的观点。通过Erdos令人惊讶的“概率方法”，人们可以通过将随机性引入一个先天与随机性无关的问题来获得确定性的结果。本文重点研究齐次多项式空间上的概率分布，这是一个被积极研究的概率分布，具有很强的代数几何意义。具体地说，我们感兴趣的是拓扑复杂度的界，用贝蒂数来衡量，这些随机变量被限制在0最小几何的集合中。最小几何是一个框架，在这个框架中，允许比变种更一般的集合（称为可定义集合）。在涉及可定义集的组合问题的0 -极小关联几何领域中，迫切需要多项式分割定理这一传统关联几何中的灵丹妙药。然而，由于限制于变量的可定义集的最坏情况拓扑复杂性可能是“坏”的，因此证明了它的困难。我们建议研究限制于随机变量的可定义集的平均情况复杂度。通过这样做，我们希望证明拓扑“坏”多项式是不寻常的（我们已经证明了某些可定义集）。因此，如果适合分区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”拓扑复杂度的分区多项式的存在性。这将为零最小入射几何提供巨大的提升。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。该领域最重要的开放问题是极其困难的P对NP问题。最近，在几何复杂性理论（GCT）项目的推动下，这个问题得到了解决，该项目涉及用代数语言描述相关问题。我们希望在代数几何和表示理论等丰富领域的先进工具将使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空间，称为轨道封闭。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。计算拓扑结构有助于获得对所涉及对象的几何形状的粗略理解。此外，众所周知，获得空间拓扑上的定量界限有助于理解在空间上操作的计算过程的基本限制。我们还将研究乌尔里希复杂度的概念，它以一种不同的方式量化多项式的“复杂性”。虽然使用交换代数概念可以方便地定义它，并且显然更容易处理，但通常对它知之甚少。我们将研究柯斯特兰多项式的平均乌尔里希复杂度。众所周知，获得多项式的乌尔里希复杂度的下界可能会对交换代数中的一个重要猜想以及P对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.