Functors in Homotopy Theory

我的研究项目是关于同伦理论中的函子。拓扑学本身关注的是物理(和理论)对象的形状和全局性质，我们称之为空间。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，它们可能模拟物理系统可能所处的所有位置。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，可以连续变形。函子是一种给某个对象赋值的方式，在我们的例子中通常是一个空间，另一个对象通常是更代数的对象，如一个数字或一组数字。函子应该与一些其他结构相容，例如上述变形。我们将研究同伦理论中出现的一些函子。首先，我们想知道一些工作人员的形象。所以，如果我们从任意空间开始，应用我们的函子之一，特别是上同调函子，我们可以得到哪些相应的代数对象？这个问题已经有很长的历史了，至少可以追溯到20世纪60年代的S。目标对象总是分次环，这个问题已经在一些特殊情况下得到了解决，如多项式环和研究有理数时。我们将研究模扭转问题。这在一定程度上简化了问题，但丢失的信息不像我们研究有理数时那么多。特别是，如果我们的环有两个元素a和b，我们仍然想知道它们的乘积是否是其他一些元素的倍数。在有理数上，这总是正确的，但它并不总是正确的模扭转。第二个主题是流形微积分。它着眼于与流形相关的函子。问题是这样的函子通常是复杂的，所以我们把它分解成它所谓的泰勒塔。这类似于取微积分中的一个函数，然后用它的泰勒多项式来代替它。一般的函数可能非常复杂，但多项式更容易理解。我们最感兴趣的是可能出现什么塔，换句话说，在这种情况下，什么是多项式。

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.