Derived Categories and Other Invariants of Algebraic Varieties

该项目集中于代数几何中的各种问题，松散地围绕着不变量的研究，包括上同调和范畴。代数几何关注的是对几何空间的研究，这些几何空间局部可以被建模为几个变量多项式系统的解的集合，通常是在一个域上，但也在算术上有趣的环上，如整数上。这样的空间在数学中无处不在，出现在许多其他领域，如表示理论、组合学和数学物理。它们也自然地出现在其他学科的各种应用中，如计算机科学和工程。在研究复杂对象（如代数变量）时，数学中的一个共同主题是考虑各种更容易处理但足够丰富的特征或不变量，以捕获有意义的信息。该项目将促进我们对几个不变量的理解，并研究在代数和算术几何中的应用。更具体地说，该项目涉及以下主题。大部分工作将与合作者和学生一起进行。(1)在光滑射影代数变异场上导出相干束的范畴，并理解这些范畴间等价的附加上同调特征。目标是理解相干束的派生范畴，连同附加结构，在多大程度上决定了代数变体的同构类，或者可能是双等价类。(2)对数格式的对数相干束、Hochschild和拓扑Hochschild同调。目标是理解与对数格式的态射相关的相干束的合适dg类别。对这一范畴的研究也对相干束的退化现象的研究具有更广泛的意义。(3)代数变量的重构定理。特别是，PI将研究以前使用模型理论研究的重建结果的新方法和推广。(4)对数格式的同伦群及其在晶体基群研究中的应用。除了这四个项目外，PI还将与研究生一起研究其他几个项目。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The project is focused on a variety of questions in algebraic geometry, loosely centered around the study of invariants, both cohomological and categorical. Algebraic geometry is concerned with the study of the geometric spaces which locally can be modeled as the set of solutions of systems of polynomials in several variables, typically over a field but also over arithmetically interesting rings such as the integers. Such spaces are ubiquitous in mathematics, arising in many other areas such as representation theory, combinatorics and mathematical physics. They also arise naturally in a variety of applications in other disciplines such as computer science and engineering. A common theme in mathematics when studying complex objects, such as algebraic varieties, is to consider various features or invariants which are more tractable yet rich enough to capture meaningful information. The project will advance our understanding of several such invariants and study applications in algebraic and arithmetic geometry.More specifically the project is concerned with the following topics. Most of the work will be carried out with collaborators and students. (1) Derived categories of coherent sheaves on smooth projective algebraic varieties over a field and understanding additional cohomological features of equivalences between such categories. The goal is understand to what extent the derived category of coherent sheaves, together with additional structure, determines the isomorphism class, or possibly the birational equivalence class, of an algebraic variety. (2) Log coherent sheaves and Hochschild and topological Hochschild homology for log schemes. The objective is to understand a suitable dg category of coherent sheaves associated to a morphism of log schemes. The investigation of such a category also has broader implications for the study of degeneration phenomena for coherent sheaves. (3) Reconstruction theorems for algebraic varieties. In particular, the PI will investigate new approaches to, and generalizations of, reconstruction results previously studied using model theory. (4) Homotopy groups of log schemes and applications to the study of crystalline fundamental groups. In addition to these four projects, the PI will investigate several other projects together with graduate students.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.