Stabilization and destabilization of moduli spaces

在我的工作中，我将代数拓扑的方法应用于微分拓扑、低维拓扑和数论中的问题。我通过研究模空间的同伦类型来做到这一点。模空间包含给定类型的数学对象及其族的信息。通过将拓扑不变量应用于模空间，您可以了解很多有关所讨论对象的信息。许多模空间依赖于一个参数，当参数趋于无穷时，它们变得更容易理解，这是一个令人惊讶的现象。我研究的主题是你可以从这个极限值中恢复关于单个模空间的信息，反之亦然；“不稳定”和“稳定”。我的第一个目标是找到高维流形的模空间的有理模型，或者等价地说是它们的微分同构群。为此，我将以与Oscar Randal-Williams合作开发的方法为基础来理解磁盘的微分同态。圆盘的微分同态是给定流形的微分同态与自嵌入之间的普遍校正项。因此，我们可以通过添加手柄来“稳定”磁盘，研究由此产生的高维表面类似物的微分同构和自嵌入，然后通过比较这些来“破坏稳定”。一旦我们理解了校正项，我们就可以使用自嵌入空间的有理模型来给出微分同态的类似模型。通过一个显著的代数巧合，这对曲面的托雷利群有很深的应用。我将与我的合作者、博士后和学生一起制定这个计划，并探索它在流形和其他领域（如辛几何和黎曼几何）研究中的应用。我的第二个目标是了解环的一般线性群的同调与其代数k理论之间的关系。对于场，我想把代数k理论上的几个过滤联系起来。一方面，“动机过滤”包含了深层次的数论信息。另一方面，“秩过滤”与一般线性群的同调稳定性有关。在与Soren Galatius和Oscar Randal-Williams的联合工作中，我证明了Soule和Beilinson为动机过滤所推测的秩过滤具有消失线。因此，澄清这两种过滤之间的关系非常重要，为此我提出了两条调查路线。我将与我的合作者和博士后一起研究这个问题。对于整数环，我感兴趣的是在表示中具有系数的算术群的（co）同调，例如Steinberg模块。它们被推测在一定范围内消失，如果是这样，这就给出了同调稳定性和Vandiver猜想的新结果。我想把这些问题与Voronoi复合体和完美形式联系起来。扩展到其他环对学生来说是很好的项目。

In my work I apply the methods of algebraic topology to problems in differential topology, low-dimensional topology, and number theory. I do so by studying the homotopy type of moduli spaces. Moduli spaces contain information about mathematical objects of a given type, and families thereof. By applying topological invariants to moduli spaces you learn a lot about the objects in question. Many moduli spaces depend on a parameter, and it is a surprising phenomenon that they become easier to understand as we let the parameter go to infinity. The theme of my research is that you can recover information about the individual moduli spaces from this limiting value, and vice versa; "destabilization" and "stabilization". My first goal is to find rational models of moduli spaces of high-dimensional manifolds, or equivalently their diffeomorphism groups. To do so I will build on methods developed to understand diffeomorphisms of disks in joint work with Oscar Randal-Williams. Diffeomorphisms of disks are the universal correction term between diffeomorphisms and self-embeddings of a given manifold. We may thus "stabilize" the disk by adding handles, study the diffeomorphisms and self-embeddings of the resulting high-dimensional analogues of surfaces, and then "destabilize" by comparing these. Once we understand the correction term, we use rational models for spaces of self-embeddings to give similar models for diffeomorphisms. Through a remarkable algebraic coincidence, this has deep applications to Torelli groups of surfaces. Together with my collaborators, postdoctoral fellows, and students, I will work out this program and explore its applications to the study of manifolds and other fields such as symplectic and Riemannian geometry. My second goal is to understand the relationship between the homology of general linear groups of a ring and its algebraic K-theory. For fields, I want to relate several filtrations on algebraic K-theory. On the one hand, the "motivic filtration" contains deep number-theoretic information. On the other hand, the "rank filtration" is related to homological stability for general linear groups. In joint work with Soren Galatius and Oscar Randal-Williams I proved that the rank filtration has a vanishing line conjectured by Soule and Beilinson for the motivic filtration. Clarifying the relationship between these two filtrations is thus of great importance, and I propose two lines of investigation to do so. I will work on this with my collaborators and postdoctoral fellows. For rings of integers, I am interested in the (co)homology of arithmetic groups with coefficients in a representation, such as the Steinberg module. These are conjectured to vanish in a range, and if so, this gives new results on homological stability and Vandiver's conjecture. I want to relate these questions to Voronoi complexes and perfect forms. Extensions to other rings make good projects for students.