Stabilization and destabilization of moduli spaces

在我的工作中，我应用代数拓扑的方法来解决微分拓扑、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。 模空间包含关于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，您可以了解有关对象的许多信息。许多模空间依赖于一个参数，当我们让参数变得无穷大时，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各个模空间的信息，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，或者等价地找到它们的自同构群。为了做到这一点，我将建立在与奥斯卡·兰德尔-威廉姆斯共同工作的方法上，以理解圆盘的超同构。圆盘的自同态是流形的自同态与自嵌入之间的普遍修正项。因此，我们可以通过添加把手来“稳定”圆盘，研究由此产生的高维曲面类似物的自同构和自嵌入，然后通过比较这些来“不稳定”。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类似的自同态模型。通过一个显着的代数巧合，这有深刻的应用Torelli群的表面。与我的合作者，博士后研究员和学生一起，我将制定这个计划，并探索其在流形和其他领域（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。对于域，我想联系代数K-理论上的几个滤子。一方面，“动机过滤”蕴含着深刻的数论信息;另一方面，“秩滤除”与一般线性群的同调稳定性有关。在与索伦·加拉修斯和奥斯卡·兰德尔-威廉姆斯的合作中，我证明了等级过滤有一条消失线，这条消失线是由索尔和贝林森为动机过滤而建立的。因此，澄清这两种过滤之间的关系是非常重要的，我提出了两条调查路线。我将与我的合作者和博士后研究员一起工作。对于整数环，我感兴趣的是算术群的（上）同调与系数在一个表示，如斯坦伯格模。这些被证明是消失的范围内，如果是这样，这给出了新的结果同调稳定性和Vandiver的猜想。我想把这些问题与Voronoi复形和完全型联系起来。对其他环的扩展为学生提供了很好的项目。

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.