Geometric analysis via manifolds with corners

从破浪到爱因斯坦相对论中的黑洞概念，奇点在科学和日常生活中无处不在。数学，更具体地说是几何和分析，提供了正确的语言来描述和研究它们，特别是它们影响偏微分方程组(PDE)的解的方式，例如热方程或波动方程。在这个提案中，我们打算使用带角的流形的概念来研究这类问题。实际上，带角的流形通常可以通过添加合适的边界超曲面来解决空间的奇点问题。当人们想要求解一个偏微分方程组时，这些边界超曲面对于给出偏微分方程组的渐近模型变得非常有用。对于偏微分方程组的求解，一个自然的策略是首先求解每个边界超曲面上的模型问题，然后将它们拼接在一起得到偏微分方程组的一个良好的近似解，最后用泛函分析方法精确地求解该偏微分方程组。利用这一方法，本提案将特别侧重于开发分析工具来研究磁单极子的空间。这是一个微妙的空间，因为单极有时可以被视为不同的粒子，但通常可以结合在一个不可分割的物体中。然而，有强有力的证据表明，有一个带角的流形来描述这个空间和单极子在无穷远的散射方式。该建议的一个重要目标是为这样的空间构造一个合适的算子演算，以允许求解其上的自然几何偏微分方程组。对这些解的良好理解将使我们能够验证来自理论物理的、基于弦理论中的S对偶原理的预测。这种算符演算对研究其他构型空间也是有用的。同时，它将提供Calabi-Yau空间的新例子，这些空间是Ricci曲率为零的空间，这意味着它们是广义相对论中爱因斯坦方程的黎曼模拟的解。在另一个方向上，仍然使用带角的流形，该提议将研究存在奇点时的谱不变量。就像蝙蝠或海豚的回声定位一样，几何运算符的光谱和由其构建的光谱不变量在许多情况下可以‘听到’空间的形状。谱不变性的一个重要例子是解析扭转，它可以检测到关于空间拓扑的信息。该提案将特别研究边奇点的存在对解析扭转的影响。在双曲几何和数论的启发下，它还将研究具有尖点奇点的空间上的解析扭转，并试图确定它可以从空间中恢复哪些拓扑信息。所发展的方法也将有助于研究其他谱不变量，如ETA不变量或椭圆算子的正则化行列式。

From breaking waves to the notion of black holes in Einstein's theory of relativity, singularities are omnipresent in science and every day life. Mathematics, more specifically geometry and analysis, provide the right language to describe and study them, especially the way they affect solutions to partial differential equations (PDE's), for instance the heat equation or the wave equation. In this proposal, we intend to use the notion of manifolds with corners to study this sort of questions. Indeed, manifolds with corners can often be used to resolve the singularities of a space by adding suitable boundary hypersurfaces. When one wants to solve a PDE, these boundary hypersurfaces become very useful to give asymptotic models of the PDE. To solve the PDE, a natural strategy is then to first solve the model problems at each boundary hypersurfaces, then patch them together to obtain a good approximate solution of the PDE, and finally solve exactly the PDE using functional analytical methods. Using this approach, this proposal will focus in particular on developing analytical tools to study the space of magnetic monopoles. This is a subtle space, since monopoles can sometime be seen as distinct particles, but typically can coalesce in one indivisible object. Nevertheless, there are strong evidences that there is a manifold with corners describing this space and the way monopoles scatter at infinity. One important goal of the proposal is to construct a suitable calculus of operators for such a space that will allow to solve natural geometric PDE's on it. A good understanding of these solutions will allow to verify predictions coming from theoretical physics and based on the principle of S-duality in string theory. This calculus of operators will also be useful to study other configuration spaces. At the same time, it will provide new examples of Calabi-Yau spaces, which are spaces with vanishing Ricci curvature, meaning that they are solutions of the Riemannian analog of Einstein's equation in general relativity. In another direction, still using manifolds with corners, the proposal will study spectral invariants in the presence of singularities. Like echolocation for bats or dolphins, the spectrum of a geometric operators and spectral invariants built out of it allow in many instances to `hear' the shape of space. One important example of spectral invariant is analytic torsion, which can detect information about the topology of the space. The proposal will study in particular how analytic torsion is affected by the presence of edge singularities. Motivated by hyperbolic geometry and number theory, it will also study analytic torsion on spaces with cusps singularities and try to determine what topological information it can recover from the space. The methods developed will also be useful to study other spectral invariants, like the eta invariant or the regularized determinant of elliptic operators.