Stability conditions on derived categories

Calabi-Yau三褶是一种特殊的六维弯曲空间，具有大量的几何结构。它们对研究代数和微分几何的数学家以及研究弦理论的物理学家都很有兴趣。基础物理学的最大问题是找到一种理论，它能成功地把爱因斯坦的广义相对论和量子论结合起来。广义相对论是关于非常大的物体，如星系的物理学，量子理论是关于非常小的物体，如原子的物理学。弦理论是解决这个问题的主要候选理论。它预言时空的维度不是4（3个空间加1个时间），而是10。额外的6个维度卷成一个卡拉比-丘3折叠，半径非常小。所以根据弦理论，卡拉比-丘三折描述了空间本身的真空。利用物理推理，弦理论家对卡拉比-丘三折做出了非凡的数学预测，被称为镜像对称，这些预测在许多情况下得到了验证，并在数学家中引起了极大的兴奋。镜像对称说两个完全不同的Calabi-Yau 3-fold X， X*可以有相同的量子理论（到目前为止，这还不是一个定义明确的想法），当这种情况发生时，我们可以在X和X*的几何方面之间建立对应关系，从而影响它们的量子理论。通常，这些对应关系涉及的对象看起来完全不同——一个非数学的类比是，肯尼亚的长颈鹿和赞比亚的香蕉之间存在一对一的对应关系。同调镜像对称猜想是部分镜像对称的数学精确版本。它说，与X和X*相关的两个数学结构T和T*，称为三角分类，应该是相同的。本课题研究了三角形范畴T上稳定性条件（Z，P）的存在性，这是对T结构的一种增强。我们的主要目标是在镜像对称图的一侧构造稳定性条件（Z，P）的例子。然后，我们可以使用稳定性条件来定义T中半稳定对象的不变量——数量——如果不使用稳定性条件来减少对象的数量，就无法定义T中对象的合理数量，因为数量将是无限的。了解这样的稳定性条件将使我们能够陈述一个更强大的版本的同调镜像对称猜想：三角形范畴T， T* (X, X*)应该具有稳定性条件（Z，P）和（Z*,P*），并且从T到T*的镜像映射应该识别（Z，P）和（Z*,P*）。因此，我们可以在每一边计算的不变量——半稳定对象的数量——应该在镜像映射下确定。也就是说，新猜想预测X上的一些几何物体的数目，我们有希望计算出来，应该和X*上的一些完全不同的几何物体的数目相同，我们也有希望计算出来。这可以在示例中进行检查。

Calabi-Yau 3-folds are a special kind of 6-dimensional curved space, with a lot of geometrical structure. They are of great interest to mathematicians working in Algebraic and Differential Geometry, and to physicists working in String Theory. The greatest problem in fundamental physics is to find a single theory which successfully combines Einstein's General Relativity -- the physics of very large things, such as galaxies -- and Quantum Theory -- the physics of very small things, such as atoms. String Theory is the leading candidate for doing this. It predicts that the dimension of space-time is not 4 (3 space plus one time), but 10. The extra 6 dimensions are rolled up in a Calabi-Yau 3-fold, with very small radius. So according to String Theory, Calabi-Yau 3-folds describe the vacuum of space itself. Using physical reasoning, String Theorists made extraordinary mathematical predictions about Calabi-Yau 3-folds, known as Mirror Symmetry , which have been verified in many cases, and cause much excitement among mathematicians. Mirror Symmetry says that two quite different Calabi-Yau 3-folds X, X* can have identical Quantum Theories (so far, this is not a well-defined idea), and when this happens, we can set up a correspondence between aspects of the geometry of X and X* which affect their Quantum Theories. Often these correspondences relate objects which seem quite different -- a non-mathematical analogue would be to conjecture a one-to-one correspondence between giraffes in Kenya and bananas in Zambia. The Homological Mirror Symmetry Conjecture is a mathematically precise version of part of Mirror Symmetry. It says that two mathematical structures T and T* associated to X and X*, called triangulated categories , should be the same.This research project concerns the existence of stability conditions (Z,P) on a triangulated category T, an enhancement of the structure of T. Our major goal is to construct examples of stability conditions (Z,P) on one side of the mirror symmetry picture. We can then use stability conditions to define interesting invariants -- numbers -- of semistable objects in T. Without using a stability condition to reduce the number of objects, there is no way to define sensible numbers of objects in T, as the number would be infinite. Knowing about such stability conditions would enable us to state a more powerful version of the Homological Mirror Symmetry Conjecture: the triangulated categories T, T* of X, X* should have stability conditions (Z,P) and (Z*,P*), and the mirror map from T to T* should identify (Z,P) and (Z*,P*). It follows that the invariants -- numbers of semistable objects -- we can compute on each side should be identified under the mirror map. That is, the new conjecture predicts that some numbers of geometric objects on X, which we can hopefully compute, should be the same as some numbers of quite different geometric objects on X*, which again we can hopefully compute. This could be checked in examples.