Singularities and symplectic topology

奇点无处不在：从你将光线照进咖啡杯时形成的尖锐焦散曲线，到我们在弦理论和膜理论中期望在额外维度中形成的微观黑洞，奇点的数学研究涉及代数几何。这是几何学的一个分支，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。对于代数几何学家来说，当你试图对所有可能的空间进行分类时，奇点自然会出现，你可以通过方程来写下（进行这种分类的努力被称为“最小模型程序”）。如果你对方程进行变形（例如，你研究了某个常数t的y^2=x^3+t），有时你可以平滑奇点，但平滑后的空间可能具有高度非平凡的拓扑（如果你把咖啡杯倾斜到光线中，你可以看到这种情况）。这个新的拓扑，当t为零时，被压缩回奇点，称为消失环。假设你想证明两个不同的奇点不能同时形成。我将通过展示消失的循环不能彼此分开（“移位”）来尝试做到这一点。如果消失环是不可置换的，那么可以形成一个或另一个奇点，但两个奇点不能同时形成。使这种论证严格的困难在于消失环本身可以是奇点！为了证明这样的结果，我将需要发展现有的技术来证明非线性（“弗洛尔理论”）的情况下，消失的周期有奇点。这些技术应该可以应用于奇异圈发挥作用的数学的其他部分，例如在研究称为镜像对称的神秘几何对偶（由弦理论家在20世纪90年代早期预测，但仍未完全理解）或几何分析中拉格朗日平均曲率流的奇异极限时出现的“奇异SYZ纤维”。

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.