Geometric analysis via manifolds with corners

从破波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，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（PDE）解的方式，例如热方程或波动方程。 在这个提议中，我们打算使用带角流形的概念来研究这类问题。 事实上，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。 当人们想要求解偏微分方程时，这些边界超曲面对于给出偏微分方程的渐近模型是非常有用的。 为了求解偏微分方程，一个自然的策略是首先在每个边界超曲面上求解模型问题，然后将它们拼接在一起以获得偏微分方程的良好近似解，最后使用泛函分析方法精确求解偏微分方程。 利用这一方法，本提案将特别侧重于开发分析工具，以研究磁单极子的空间。 这是一个微妙的空间，因为单极子有时可以被视为不同的粒子，但通常可以合并成一个不可分割的对象。 然而，有强有力的证据表明，存在一个带有角的流形来描述这个空间以及单极子在无穷远处散射的方式。 该提案的一个重要目标是为这样一个空间构造一个合适的算子演算，这将允许求解其上的自然几何PDE。对这些解的良好理解将允许验证来自理论物理和基于弦理论中S-对偶原理的预测。 这种算子演算对于研究其他位形空间也是有用的。 与此同时，它将提供新的例子卡-丘空间，这是空间与消失的里奇曲率，这意味着他们的解决方案的黎曼模拟爱因斯坦的方程在广义相对论。 在另一个方向上，仍然使用带有角点的流形，该提案将研究存在奇点的谱不变量。 就像蝙蝠或海豚的回声定位一样，几何算子的频谱和由其构建的频谱不变量在许多情况下允许“听到”空间的形状。 谱不变量的一个重要例子是解析挠率，它可以检测空间的拓扑信息。 该提案将研究特别是如何分析扭转的影响存在的边缘奇点。 受双曲几何和数论的启发，它还将研究具有尖点奇点的空间上的解析挠率，并试图确定它可以从空间中恢复哪些拓扑信息。 所开发的方法也将是有用的，以研究其他谱不变量，如eta不变量或正则化行列式的椭圆算子。

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.