Homotopy theory and derived categories

我将研究两个主题：配置空间和t-结构。流形M可以被认为是物体可能位置的空间，或者是机械系统的可能状态。例如M=R3，欧几里得三维空间就是我们生活的三维空间。k点F（M，k）上的位形空间由流形中不允许碰撞的k个粒子组成。所以它们是点的序列，其中没有两个是相同的。例如，它可以是具有许多手臂的机器人的状态。同样，如果我们认为欧几里德三空间，那么F（R3，k）可以被认为是宇宙中有k个行星的可能状态。我对M和F（M，k）之间的关系特别感兴趣，我正在研究一种数学方法，从M的信息中获得F（M，k）的信息。 t结构是一种通过将数学对象分割成更小的块来研究它们的方法。不同的t结构以不同的方式切割事物。同样，如果你拿一块石头，从不同的角度切割，你会看到不同的东西，所以使用不同的t结构会让你看到不同的信息。我正在努力了解可能的t结构是什么。我还试图看看哪些t结构给出了关于对象及其关系的完整信息。

I will work on two topics: configuration spaces and t-structures. A manifold M can be thought of as a space of possible locations of an object or as the possible states of a mechanical system. For example M=R3, Euclidean three space is the three dimensional space we live in. The configuration space on k points F(M,k) consists of k particles in the manifold that are not allowed to collide. So they are sequences of points no two of which can be the same. For example it could be the states of a robot with many arms. Again if we think of Euclidean three space, then F(R3,k) could be thought of as the possible states of a universe with k planets in it. I am particularly interested in the relation between M and F(M,k), am working on a mathematical way of getting information about F(M,k) from information about M. t-structures are a way to study some mathematical objects by cutting them up into smaller pieces. Different t-structures cut things in different ways. In the same what that if you take a piece of rock and cut it at different angles you will see different things, so using a different t-structure will let you see different information. I am working on understanding what the possible t-structures are. I also try to see which of the t-structures give complete information about the objects and their relationships.