Derived Categories and Other Invariants of Algebraic Varieties

该项目专注于代数几何中的各种问题，松散地围绕上同调和范畴不变量的研究展开。代数几何是关于几何空间的研究，几何空间可以局部地建模为多个变量的多项式系统的解的集合，通常是在一个域上，但也在算术上感兴趣的环上，如整数。这种空间在数学中是无处不在的，出现在许多其他领域，如表示论、组合学和数学物理。它们还自然而然地出现在其他学科的各种应用中，如计算机科学和工程。在研究复杂对象时，数学中的一个共同主题，如代数簇，是考虑更容易处理但足够丰富的各种特征或不变量，以捕捉有意义的信息。这个项目将增进我们对几个这样的不变量的理解，并研究在代数和算术几何中的应用。大部分工作将与合作者和学生一起进行。(1)域上光滑射影代数簇上凝聚层的派生范畴，以及这些范畴之间等价的附加上同调特征。我们的目标是理解相干层的派生范畴以及附加结构在多大程度上决定了代数簇的同构类，或者可能是二元等价类。(2)对数凝聚层与对数方案的Hochschild同调与拓扑Hochschild同调。其目的是了解与原木方案的态射相关联的相干片层的适当DG类别。对这一范畴的研究对于研究相干滑轮的退化现象也具有更广泛的意义。(3)代数簇的重构定理。特别是，PI将调查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本群研究中的应用。除了这四个项目外，PI还将与研究生一起调查其他几个项目。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.