Algebraic bordism spectra: Computations, filtrations, applications

这些项目属于Motivic或A1-同伦理论领域，这是一个介于代数几何和同伦理论之间的数学领域。它是一门相对较新的、正在深入发展的学科，其基础大约建立在二十年前。尽管它年轻，但同伦理论已经在代数几何和经典同伦理论方面取得了惊人的结果和创新，包括Voevodsky对Milnor和Bloch-Kato猜想的证明，Isaksen对球面稳定同伦群结构的新见解，以及Hill、Hopkins和Ravenel在他们关于Kervaire不变量的工作中成功使用的切片过滤技术。Morel对模同伦球面的第零稳定同伦群的计算表明，域上二次型的Grothendieck-Witt环构成了模同伦理论中一个重要的不变量。在同伦理论中，为了更好地理解上同调理论，人们发展了各种各样的方法。基于复数边界谱MU的结构和相关上同调理论，给出了色同伦理论的一个具体例子。MU的代数模拟是由Voevodsky的代数边界谱MGL给出的。这个问题已经研究了一段时间了。特别地，Levine和Morel得到了相应上同调理论(部分)的几何表示，并且不同的作者将代数上同调应用于Motivic稳定上同伦范畴的研究和代数簇的研究(通过特征类和定向上同调理论)。与经典的同伦理论图景相反，基于MGL的方法在理据设置中遗漏了一些重要的信息，即二次取向。在本项目的过程中，我们计划研究与MGL不同的代数边界谱，如定向代数边界谱MSL、辛代数边界谱MSP、框架代数边界谱MSP，并研究这些谱与MGL之间的相互作用。假设的技术包括基于Witt理论和Hermite K-理论的连通和有效覆盖的谱序列以及显式几何构造。我们还计划应用与代数边界谱相关的上同调理论(代数Brown-Peterson上同调、代数Morava K-理论等)。涉及代数簇的研究，特别涉及齐次代数簇和上同调不变量的研究。

The projects belong to the field of motivic or A1-homotopy theory, an area of mathematics at the interface of algebraic geometry and homotopy theory. It is a relatively new and intensively developing discipline with the foundations established about twenty years ago. Despite its youth motivic homotopy theory has already led to striking results and innovations both in algebraic geometry and classical homotopy theory, including Voevodsky's proof of the Milnor and Bloch-Kato conjectures, new insights into the structure of stable homotopy groups of spheres by Isaksen, and techniques of slice filtrations that were successfully used by Hill, Hopkins and Ravenel in their work on the Kervaire invariant. Morel’s computation of the zeroth stable homotopy groups of motivic spheres implies that the Grothendieck-Witt ring of quadratic forms over a field constitutes an invariant of central importance in motivic homotopy theory.In homotopy theory an overwhelmingly rich variety of methods have been developed to gain a better understanding of cohomology theories. A particular example of such a technique is given by chromatic homotopy theory, which is based on the structure of the complex bordism spectrum MU and related cohomology theories. The algebraic analogue of MU is given by Voevodsky’s algebraic bordism spectrum MGL. This object has already been studied for some time. In particular, Levine and Morel obtained a geometric presentation for (a part of) the corresponding cohomology theory, and various authors applied algebraic cobordism to the study of the motivic stable homotopy category and to the study of algebraic varieties (via characteristic classes and oriented cohomology theories). Contrary to the classical homotopy theory picture, it turned out that in the motivic setting the approaches based on MGL miss some important piece of information, namely the quadratic orientation. This suggests one to look at other algebraic bordism spectra that take into account quadratic orientations.In the course of the current project we plan to study algebraic bordism spectra different from MGL, such as oriented algebraic bordism MSL, symplectic algebraic bordism MSp, framed algebraic bordism, and to investigate the interactions between these spectra and MGL. The supposed techniques involve spectral sequences based on connected and effective covers of Witt theory and hermitian K-theory as well as explicit geometric constructions. We also plan to apply cohomology theories related to algebraic bordism spectra (algebraic Brown-Peterson cohomology, algebraic Morava K-theories, etc.) to the study of algebraic varieties, in particular, to the study of homogeneous algebraic varieties and cohomological invariants.