AF: Small: Geometric Complexity Theory Approach to the P vs NP problem

本项目开发了解决P vs. NP问题的几何方法，称为几何复杂性理论。P vs. NP问题是数学科学的基本猜想，它说定理证明不能自动化。几何复杂性理论（GCT）是通过代数几何和表示理论来解决这个问题和相关问题的方法。这种方法将这些问题在复数领域的变体减少到证明障碍物存在的问题，障碍物是代数几何和表示论中的对象，可以作为所考虑的下限问题的硬度证明。这些障碍的存在与代数几何和表示论中的某些正性假设密切相关。本项目的目标是深入研究复杂性理论中的困难性和数学中的积极性之间的联系，从而进一步扩展这一方法。由于科学和工程中易处理和难处理问题之间的界限取决于P vs. NP问题，因此本项目所进行的研究对复杂性理论以及其他几个科学和工程领域具有重要的智力意义。此外，由于GCT揭示了复杂性理论中的P vs. NP和相关问题以及纯数学中的基本正性问题之间的深刻联系，因此这项研究可能会导致复杂性理论与代数几何和表示论等几个数学分支之间的广泛合作。

This project develops a geometric approach to the P vs. NP problem, called geometric complexity theory.The P vs. NP problem is the foundational conjecture of mathematical sciences, which says that theorem proving cannot be automated. Geometric complexity theory (GCT) is an approach to this and related problems via algebraic geometry and representation theory. This approach reduces variants of these problems over the field of complex numbers to the problem of proving existence of obstructions, which are objects in algebraic geometry and representation theory that serve as proof certificates of hardness in the lower bound problems under consideration. The existence of these obstructions is deeply connected with certain positivity hypotheses in algebraic geometry and representation theory. The goal of this project is to study this connection between hardness in complexity theory and positivity in mathematics in depth and thereby extend the approach further.Since the boundary between the tractable and intractable problems in science and engineering depends on the P vs. NP problem, the study undertaken in this proposal is of central intellectual relevance to complexity theory as well as several other areas of science and engineering. Furthermore, since GCT reveals a deep connection between the P vs. NP and related problems in complexity theory and fundamental positivity problems in pure mathematics, this study could lead to extensive collaboration between complexity theory and several branches mathematics, such as algebraic geometry and representation theory.