Singularities and symplectic topology

奇点无处不在：从你把光线照进咖啡杯时形成的尖刻的焦散曲线，到我们期望在弦和膜理论中以额外维度形成的微观黑洞。奇点的数学研究涉及代数几何。这是几何学的一个分支，它涉及到为感兴趣的几何空间写方程。例如，尖点曲线在平面上由方程y^2=x^3描述。对于代数几何来说，当你试图对你可以通过方程写下的所有可能的空间进行分类时，奇点就会自然而然地出现(进行这种分类的努力被称为“最小模型程序”)。我提出了一种研究奇点的新方法。如果使方程变形(例如，对于某个常量t，研究y^2=x^3+t)，有时可以平滑奇点，但平滑后的空间可以具有非常重要的拓扑(如果您将咖啡杯倾斜到光线中，就可以看到这种情况)。当t趋于零时，这种新的拓扑图被压回到奇点，称为零圈。让我们假设你想要证明两个不同的奇点不能同时形成。我将试图通过展示消失的周期不能彼此分开(“移位”)来做到这一点。如果消失圈是不可移动的，那么可以形成一个或另一个奇点，但这两个奇点不能同时形成。要使这种论证严谨，困难在于消失圈本身可能是奇异的！为了证明这样的结果，我将需要开发现有的技术来证明不可移位(“弗洛尔理论”)，以证明消失的循环具有奇点的情况。然后，这些技术应该应用于数学中发挥奇异圈作用的其他部分，例如，在研究被称为镜像对称性(由弦理论家在20世纪90年代初预测，至今仍未完全理解)的神秘几何对偶时出现的“奇异Syz纤维”，或几何分析中拉格朗日平均曲率流的奇异极限。

Singularities are everywhere we look: from the cuspy caustic curve that forms when you shine light into your coffee cup to the microscopic black holes that we expect to form in extra-dimensions in the theory of strings and branes.The mathematical study of singularities involves algebraic geometry. This is a branch of geometry which involves writing equations for the geometrical spaces of interest. For example, the cusp curve is described by the equation y^2=x^3 in the plane. For algebraic geometers, singularities appear naturally when you try to classify all the possible spaces you can write down via equations (the efforts to carry out this classification go by the name of "the minimal model program").I am proposing a new way to study singularities. If you deform the equation (for example you study y^2=x^3+t for some constant t) you can sometimes smooth out the singularity, but the smoothed space can have highly nontrivial topology (you can see this happen if you tilt your coffee cup into the light). This new piece of topology, which is crushed back down to the singular point as t goes to zero, is called a vanishing cycle. Let's suppose you want to show that two different singularities cannot form at the same time. I will try to do this by showing that the vanishing cycles cannot be moved apart from one another ("displaced"). If the vanishing cycles are nondisplaceable then one or other singularity can form, but both singularities cannot form at the same time.The difficulty in making this kind of argument rigorous is that the vanishing cycles can themselves be singular! To prove such results, I will need to develop existing techniques for proving nondisplaceability ("Floer theory") to the situation where the vanishing cycles have singularities. These techniques should then have applications in other parts of mathematics where singular cycles play a role, for example the "singular SYZ fibres" which arise in studying the mysterious geometric duality called Mirror Symmetry (predicted by string theorists in the early 1990s and still not fully understood) or the singular limits of Lagrangian mean curvature flows in geometric analysis.