Complete reducibility in algebraic groups

本研究研究“代数群”，即生活在代数和几何交叉点上的数学对象。它们同时是“群体”和“品种”。群是对称概念的数学抽象，而群是一组方程解的数学集合。让我们从分组开始。考虑一个正方形和它的对称性，我们可以对它进行几何运算，使它看起来像我们开始时的正方形。我们可以旋转0 90 180或270度。我们也可以把它反射成4条直线（2条连接角到角，2条连接边到边）。这8种对称是它们的全部。数学上的抽象是为了理解发生了什么我们要把所有的对称性放在一起考虑，而不是单独考虑。群论是关于研究这些对称集合的关键思想是当我们应用一种对称然后应用另一种对称时，我们会得到8种原始对称中的一种。这就是复合的概念。它具有特殊的性质。例如，在另一个对称前后旋转0度，使该对称保持不变。我们从小就有这样的想法：整数组成一个群，组成就是数字的加法。这里，单位元是0。如果我们给任何数加0，它还是一样的！方程解的研究是一个广阔的研究领域，自数学诞生以来一直是一个焦点。它是代数几何和数论等现代领域的基础。费马大定理是研究方程组解集的一个非常著名的例子，它需要在代数几何和数论方面非常深入的技术。我们很多人在学校都遇到过毕达哥拉斯定理和方程a^2 + b^2 = c^2。有许多整数解，称为毕达哥拉斯三元组，如（3,4,5）和（5,12,13）。与群论一样，数学家们发现研究这些问题的一个好方法是同时研究所有的解决方案。一个变量是一组特别好的被称为多项式的方程的解。因此，代数群的研究是高度跨学科的。事实上，它们的引入来自广义李群，这是一项分析发明，用于研究微分方程的连续解。这项研究将集中在故事的群论方面。代数群被划分为族，一个基本的开放性问题是如何理解“简单”族。这些简单群是所有代数群的基石，“简单”这个词掩盖了它们的真实本质。它们非常复杂，有着非常丰富和深刻的结构。它们本身就值得研究，更不用说它们在其他数学领域的应用了。我们将研究这些简单代数群的结构，其中仍有许多悬而未决的问题。我们的主要重点是研究与现代数学的另一个关键领域，表示理论的密切关系。为了研究一个对象，比如一个代数群，我们考虑它如何作用于更直接的东西，在这个例子中是线性空间v，线性空间很好，我们生活的世界就是一个线性空间，作为数学家我们理解它们。为了研究群如何作用于这样的线性空间，我们把它看作线性空间的对称全群GL(V)的子群。本研究的关键在于GL(V)本身是一个代数群。因此代数群的结构和表示理论是密切相关的。这是j.p。Serre提出了完全可约性的概念。我们将解决关于代数群的子群结构的开放问题，特别是某些类的子群，通过这个链接，与不完全可约的表示相关，这意味着它们有一个更复杂的结构，从更简单的表示建立起来。

This research studies `algebraic groups', mathematical objects which live at the intersection of algebra and geometry. They are simultaneously `groups' and `varieties'. A group is the mathematical abstraction of the idea of symmetry and a variety is the mathematical collection of solutions to a set of equations. Let us start with groups. Consider a square and its symmetries, geometric operations we can do to it that leave it looking like the square we started with. We can rotate it by 0, 90, 180 or 270 degrees. We can also reflect it in 4 straight lines (2 joining corners to corners and 2 joining sides to sides). And these 8 symmetries are all of them. The mathematical abstraction is to see that to understand what is going on we want to consider the symmetries all together, not just on their own. Group theory is about studying these sets of symmetries and the key idea is that when we apply one symmetry and then apply another one, we get back one of our 8 original symmetries. This is the notion of composition. It enjoys special properties. For example, rotating by 0 degrees before or after another symmetry leaves that symmetry unchanged. We have all interacted with this idea from an early age: the integers form a group where composition is just addition of numbers. There, the identity is 0. If we add 0 to any number, it stays the same!The study of solutions to equations is a vast area of study and has been a focal point since the inception of mathematics. It is the foundation of modern areas like algebraic geometry and number theory. Fermat's Last Theorem is a very famous example of studying sets of solutions of equations and required incredibly deep techniques in algebraic geometry and number theory. Many of us encounter Pythagoras' Theorem and the equation a^2 + b^2 = c^2 at school. There are many integer solutions to this, known as Pythagorean triples, like (3,4,5) and (5,12,13). As with group theory, mathematicians have found that a good way to study these problems is to look at all solutions at once. A variety is a set of solutions to especially nice equations called polynomials.The study of algebraic groups is therefore highly intradisciplinary. Indeed, their introduction came from generalising Lie groups, which were an analytic invention to study continuous solutions of differential equations. This research will concentrate on the group theoretic side of the story. Algebraic groups have been classified into families and a fundamental open problem is to understand the `simple' ones. These simple groups are the building blocks of all algebraic groups and the word simple obscures the true nature of them. They are incredibly complicated and have a very rich and deep structure. They deserve studying in their own right, let alone due to the applications to other fields of mathematics.We will study the structure of these simple algebraic groups, where many open problems remain. Our main focus is studying the close relationship with another key area of modern mathematics, representation theory. To study an object, like an algebraic group, we consider how it acts on something more straightforward, in this case a linear space V. Linear spaces are nice, the world we live in is a linear space and as mathematicians we understand them. To study how a group acts on such a linear space, we consider it as a subgroup of the full group of symmetries of the linear space, called GL(V). The crucial part for this research is that GL(V) is itself an algebraic group. And so the structure of algebraic groups and representation theory are intimately related. This was made precise by J.-P. Serre when he introduced the concept of complete reducibility. We will tackle open problems about the subgroup structure of algebraic groups, especially certain classes of subgroups related, through this link, to representations that are not completely reducible, meaning they have a more complicated structure built up from the simpler representations.