Algebraic and Tropical Moduli Spaces and Brill-Noether Theory

代数几何学者研究由多项式方程（局部）定义的空间。研究这些空间的一种强有力的方法是退化法，它最近取得了令人兴奋的进展。应用这种方法可以将复杂的几何对象简化为简单的几何对象，用组合的几何方面代替某些几何方面。这样的退化对于几何问题的研究是有用的：非常粗略的哲学是，人们研究碎片的组合（即离散数据），以便推断出更复杂的原始空间的事情。该奖项支持的研究将集中于利用现代退化技术，特别是热带几何领域的退化技术，从代数几何中研究经典空间。热带几何的主要目标是将代数变异问题转化为多面体复合体问题。一种称为热带化的过程将多面体复合体附着在代数变体上。多面体复合体，一个组合对象，编码了一些原始代数变化的几何。第一个主要项目是利用热带几何和组合拓扑的新技术计算曲线模空间的顶权有理上同调，并在其中找到显式的新上同调类。这个项目的一个补充方面是发展热带模空间的堆栈理论基础。第二个课题研究曲线的Brill-Noether变型的几何，即曲线上线性级数的模空间。该方法建立在极限线性级数的模理论的最新进展上，并将产生对Brill-Noether变体的精细理解。该项目还将进一步揭示Brill-Noether理论与图和Young表的组合学之间的联系。然后，这些联系在第三个主要研究方向上被放大，这包括几个组合研究，这些研究将进一步阐明图和代数曲线之间的密切联系。

Algebraic geometers study spaces that are (locally) defined by polynomial equations. One powerful method of studying these spaces, which has seen exciting recent developments, is the method of degenerations. Applying this method allows to reduce complicated geometric objects to configurations of simple ones replacing some geometric aspects by combinatorial ones. Such degenerations can be useful for study of geometric problems: the very rough philosophy is that one studies the combinatorics (i.e., the discrete data) of the pieces in order to deduce things about the more complicated original space. The research supported by this award will center on using modern degeneration techniques, especially those from the field of tropical geometry, to study classical spaces from algebraic geometry. The main goal of tropical geometry is transforming questions about algebraic varieties into questions about polyhedral complexes. A process called tropicalization attaches a polyhedral complex to an algebraic variety. The polyhedral complex, a combinatorial object, encodes some of the geometry of the original algebraic variety. The first main project involves using new techniques from tropical geometry and combinatorial topology to compute top-weight rational cohomology of the moduli space of curves, finding explicit new cohomology classes therein. A complementary aspect of this project is to develop stack-theoretic foundations for tropical moduli spaces. The second project studies the geometry of Brill-Noether varieties of curves, i.e. moduli spaces of linear series on curves. The approach builds on recent advances in the moduli theory of limit linear series, and will yield a refined understanding of Brill-Noether varieties. This project will also uncover further connections between Brill-Noether theory and the combinatorics of graphs and Young tableaux. These connections are then amplified in the third main direction of research, which consists of several combinatorial investigations that will shed additional light on the close connection between graphs and algebraic curves.