Research in universal algebra: constraint satisfaction and residual properties

普通代数研究普通数字的加法、减法、乘法和除法的规律。现代代数研究在各种情况下出现的“非标准”代数系统。一个简单的例子是布尔代数，它是由逻辑运算符AND、OR和NOT在两个真值0（“假”）和1（“真”）上操作时建模的定律系统。更复杂的代数系统，其中许多是奇怪的，其中一些在物理，化学，理论计算机科学和工程中很有用，可以被发明，研究和建模。通用代数是对非标准代数定律及其模型的模式和极限的一般研究。这项提案资助的研究有三个主要部分。首先，我和我的团队将研究并重新制定由西蒙弗雷泽大学的Andrei Bulatov和罗蒙诺索夫莫斯科国立大学的Dmitriy Zhuk在2017年独立宣布的解决方案，这是一个20年前的理论计算机科学猜想。这个猜想被称为约束满足问题二分猜想，它断言，对于某一类计算问题，该类中的每个问题要么在计算上很容易，要么很难（在精确的意义上）；换句话说，课堂上没有问题具有中等难度。通用代数的工具在证明这个猜想时特别有用。我这部分研究的目标是在通用代数中发展新的理论，以更好地解释布拉托夫和朱克的发现。在第二部分中，我和我的团队将研究两个40年来未解决的关于涉及非标准代数定律及其有限模型的模式的猜想。第一个猜想描述的情况（它被认为）应该意味着一个非标准代数系统的定律都可以从一些固定的、有限的基本定律集演绎出来。这个猜想在很多情况下都是正确的；我们的研究将旨在扩展已知猜想为真的领域。第二个猜想涉及非标准代数系统的某些“不可约”模型的分布。在相当弱的假设下，当一个代数系统没有无限大小的不可约模型时，有限大小系统的不可约模型似乎服从某些我们目前无法解释的规则模式。我和我的团队希望通过寻找理由来证明观察到的模式，从而揭开这些谜团。本计画的最后一部分涉及对通用代数中另外两个开放问题的探索性研究。这项研究将服务于全球的纯数学和理论计算机科学家社区，他们寻求理解由代数建模的抽象数学现象。它将培养学生从事尖端研究，并通过解决备受瞩目的问题为加拿大带来声望，从而为加拿大服务。

Ordinary algebra studies the laws of addition, subtraction, multiplication and division of ordinary numbers. Modern algebra studies "nonstandard" systems of algebra which arise in various contexts. A simple example is Boolean algebra, which is the system of laws modeled by the logical operators AND, OR, and NOT as they operate on the two truth values 0 ("false") and 1 ("true"). Much more complicated systems of algebra, many of them bizarre, some of them useful in physics, chemistry, theoretical computer science and engineering, can be invented, studied, and modeled. Universal algebra is the general study of patterns in, and the limits of, nonstandard laws of algebra and their models. The research to be funded by this proposal has three major parts. First, my team and I will study and reformulate the solutions, announced independently in 2017 by Andrei Bulatov at Simon Fraser University and Dmitriy Zhuk at Lomonosov Moscow State University, of a 20-year-old conjecture in theoretical computer science. This conjecture, called the Constraint Satisfaction Problem Dichotomy Conjecture, asserts that, for a certain class of computational problems, each problem in the class is either computationally easy, or is impossibly hard (in a precise sense); in other words, no problem in the class has intermediate difficulty. The tools of universal algebra were especially useful in proving this conjecture. The goal of this part of my research will be to develop new theories within universal algebra to better explain the discoveries of Bulatov and Zhuk. In the second part, my team and I will work on two 40-year-old unsolved conjectures concerning patterns involving nonstandard laws of algebra and their finite models. The first conjecture describes circumstances which (it is believed) should imply that the laws of a nonstandard system of algebra will all be deducible from some fixed, finite set of basic laws. This conjecture is known to be true in a very large number of cases; our research will aim to extend the domain in which the conjecture is known to be true. The second conjecture concerns the distribution of certain "irreducible" models of a nonstandard system of algebra. Under rather weak assumptions, when an algebraic system has no irreducible models of infinite size, the irreducible models of the system of finite size seem to obey certain patterns of regularity that we currently cannot explain. My team and I hope to shed light on these mysteries by finding reasons to justify the observed patterns. The final part of this project involves exploratory investigations of two additional open problems in universal algebra. This research will serve the world-wide community of pure mathematicians and theoretical computer scientists who seek to understand abstract mathematical phenomena modeled by algebra. It will serve Canada by training students in cutting-edge research, and in bringing prestige to Canada through the solution to high-profile problems.