CAREER: the Brauer group in algebraic and formal geometry

PI将致力于一系列与Brauer群在代数和形式几何中的作用有关的研究项目，包括三个关键部分：Brauer类的模理论和周期指数问题之间的相互作用，上同调和Azumaya Brauer群之间的比较，以及形式Brauer群在提升傅立叶-Mukai等价从正特征中的作用。首先，正如PI在他早期的研究中所表明的那样，将Brauer群的元素视为代数堆栈的同构类导致了经典代数中问题的拓扑和几何刚性。利用这些结构，人们可以将这些经典问题与有关局部到全局原理的现代问题联系起来，并与某些与向量丛的模空间密切相关的模空间的几何联系起来。PI和他的研究生将继续调查这些联系，并推动进入新的领域。其次，PI列出了一系列用于测试上同调代数空间和Azumaya Brauer代数空间群之间的差异的建议目标。要很好地理解这些目标，需要计算他们的代数K-理论，并将其与奇点分解的K-理论联系起来，PI希望与他的研究生一起研究这些例子。最后，PI将研究正特征中的傅里叶-Mukai等价，最初的重点是K3曲面。一个重要的工具将是Mukai Hodge结构的晶态和p元形式，他推测形式Brauer群将在正特征中发挥作用，类似于Mukai经典理论中的先验晶格所起的作用。作为一个特例，Pi和他的研究生们将研究K3曲面上的p阶向量丛的模空间的正特征HKR同构及其与新的特征p变形的联系。最初致力于研究多项式方程的解的代数几何在上个世纪极大地扩展了它的职权范围，涵盖了代数和几何之间的广泛相互作用以及在政府和工业中的大量应用。这个项目的研究部分将促进我们对具有密码学和理论意义的代数几何对象的理解。这笔补助金将用于资助私人侦探的工作和对其研究生的培训。除了进行研究外，PI还将在整个太平洋西北部地区推广高中数学推广工作，重点是城市中心和边远农村地区服务不足的人口。他将通过社交网络和搜索引擎进行讲座并组织有针对性的在线广告，努力发现和培养不同寻常的数学人才，包括在非传统的地方和通过非传统的手段。这笔赠款还将支持一批代数几何新博士的会议，重点是扩大这一群体的兴趣和数学联系。在一个资源有限和博士培训高度集中的时代，对于该领域的未来来说，我们继续鼓励代数几何学家社区的广泛视野并促进其成员之间的相互理解是至关重要的。

The PI will work on a range of research projects related to the role of the Brauer group in algebraic and formal geometry, with three key components: the interaction between moduli theory and the period-index problem for Brauer classes, the comparison between the cohomological and Azumaya Brauer groups, and the role of the formal Brauer group in lifting Fourier-Mukai equivalences from positive characteristic. First, as the PI has shown in his earlier research, viewing elements of the Brauer group as isomorphism classes of algebraic stacks gives rise to topological and geometric rigidifications of questions in classical algebra. Using these structures, one can relate these classical questions to modern questions about local-to-global principles and to the geometry of certain moduli spaces closely related to moduli spaces of vector bundles. The PI and his graduate students will continue to investigate these connections and push into new territory. Second, the PI has a list of proposed targets for testing the difference between the cohomological and Azumaya Brauer groups of algebraic spaces. A good understanding of these targets will involve computing their algebraic K-theory and relating it to the K-theory of resolutions of singularities, and the PI expects to work on these examples jointly with his graduate students. Finally, the PI will study Fourier-Mukai equivalences in positive characteristic, with an initial focus on K3 surfaces. One important tool will be crystalline and p-adic forms of the Mukai Hodge structure, and he conjectures that the formal Brauer group will play a role in positive characteristic analogous to that played by the transcendental lattice in Mukai's classical theory. As a special case, the PI and his graduate students will investigate the HKR isomorphism in positive characteristic and its connection with novel characteristic p deformations of moduli spaces of vector bundles of rank p on K3 surfaces coming from the tangent space to the formal Brauer group.Originally devoted to the study of solutions of polynomial equations, algebraic geometry has vastly expanded its mandate in the last century to encompass a large range of interactions between algebra and geometry and a raft of applications in government and industry. The research component of this project will advance our understanding of an algebro-geometric object with both cryptographic and theoretical significance. The grant will fund work by the PI and the training of his graduate students. In addition to performing research, the PI will extend high school mathematical outreach efforts throughout the Pacific Northwest, with a focus on underserved populations in urban centers and outlying rural areas. He will give lectures and organize targeted online advertising through social networks and search engines in an effort to find and foster unusual mathematical talent, including in untraditional places and through untraditional means. This grant will also support a conference for a cohort of new PhDs in algebraic geometry, with a focus on broadening the interests and mathematical connections within this cohort. In an era of constrained resources and highly focused PhD training, it is essential for the future of the field that we continue to encourage a broad view of the community of algebraic geometers and promote mutual understanding among its members.