Functors in Homotopy Theory

我的研究项目是关于同伦理论中的函子。拓扑学关注的是我们称之为空间的物理（和理论）对象的形状和整体属性。简单的例子是甜甜圈（或其表面），地球（或其表面），或者更一般的流形，它可以模拟物理系统可能处于的所有位置。代数拓扑学使用代数来理解这些对象。在同伦理论中，我们认为物体是由橡胶制成的，并且可以连续变形。函子是赋值给某个对象的一种方式，在我们的例子中通常是一个空间，另一个对象通常更具有代数性质，比如一个数字或一组数字。函子应该与其他一些结构兼容，比如上面提到的变形。我们将研究同伦理论中出现的几个函子。首先我们想知道一些函子的像。所以如果我们从一个任意空间开始应用一个函子，特别是上同函子，我们能得到哪些相应的代数对象？这个问题有很长的历史，至少可以追溯到20世纪60年代。目标对象通常是分级环，这一问题在一些特殊情况下，如多项式环和有理数上都得到了解决。我们将研究模扭转问题。这在某种程度上简化了问题，但不像处理有理数时那样丢失太多信息。特别是如果我们有环上的两个元素a和b，我们仍然想知道它们的乘积是否是其他元素的倍数。在有理数上这总是成立的，但模扭转并不总是成立。第二个主题是流形微积分。它关注与流形相关的函子。问题是这样的函子通常很复杂，所以我们把它分解成所谓的泰勒塔。这类似于在微积分中取一个函数并用它的泰勒多项式代替它。一般的函数可能很复杂，但多项式却容易理解得多。我们最感兴趣的是可能出现的塔，换句话说，在这种情况下多项式是什么。

My research program is concerned with functors in homotopy theory. Topology concerns itself with the shapes and global properties of physical (and theoretical) objects which we refer to as spaces. Simple examples are a donut (or its surface), the earth (or its surface), or more generally a manifold, which might model all the positions that a physical system can be in. Algebraic topology uses algebra to understand these objects. In homotopy theory, we consider the objects as if they are made out of rubber and can be continuously deformed. A functor is a way of assigning to some object, in our case often a space, another object which is often more algebraic such as a number or a set of numbers. The functor should be compatible with some other structures such as the deformations mentioned above. We will study some functors that arise in homotopy theory. First we wish to know the image of some functors. So if we start with an arbitrary space and apply one of our functors, in particular the cohomology functor, which corresponding algebraic objects can we get? This problem has a long history dating back at least to the 1960's. The target objects are always graded rings and this problem has been solved in some special cases such as polynomial rings and also when working over the rational numbers. We will study this problem modulo torsion. This simplifies the problem somewhat but not as much information is lost as when we work over the rational numbers. In particular if we have two elements a and b of our ring, we still want to know if their product is a multiple of some other elements. Over the rational numbers this is always true, but it is not always true modulo torsion. The second topic is known as Manifold Calculus. It looks at functors associated to manifolds. The problem is such a functor is often complicated, and so we resolve it into its so-called Taylor tower. This is analogous to taking a function in calculus and replacing it by its Taylor polynomials. A general function can be very complicated, but polynomials are much easier to understand. We are mostly interested in what possible towers can occur, in other words, what are the polynomials in this context.