Conference: AGNES Summer School in Algebraic Geometry

Agnes关于模空间的交集理论暑期班将于2023年7月11日至14日在布朗大学举行。代数曲线是一维点的集合，可以用多项式方程来描述。例如，平面x^2+y^2=1中的单位圆定义了一条代数曲线。在代数曲线理论中，一个普遍存在的问题是了解有多少代数曲线满足给定的条件。这一信息对应于所谓的“模空间的交理论”：曲线的模空间是点对应于代数曲线的空间，它的交理论描述了该空间中的各种轨迹如何满足(对应于代数曲线上的各种条件同时满足)。模空间的Chow环将交理论的这些信息封装到一个代数结构中。这个暑期班将通过四个迷你课程、下午的锻炼课程和研究小组，向研究生和博士后介绍计算Chow环的最新发展。学生们将参加练习，以巩固迷你课程中的材料。他们还将在研究组中应用这些技术来计算曲线的模空间的Chow环的新例子(包括积分和有理)。更具体地说，四门迷你课程将讨论以下主题：(1)等变交理论：这描述了交集理论在群作用存在时的行为；(2)高Chow群：这些不变量捕捉到Chow环的割序列的精确度的失败；(3)稳定的尖曲线的模空间的重言环：这描述了Chow环的一个行为良好的子环；(4)2点亏格1曲线模的拼接技术和积分Chow环：通常将模空间扩大到包含奇异性较差的曲线，可以将有关地层Chow环的信息拼接到整个模空间的Chow环上。Https://sites.google.com/site/agneshomepage/brown-2023-agnes-summer-schoolThis奖反映了国家科学基金会的法定使命，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 AGNES Summer School on Intersection Theory on Moduli Spaces will be held at Brown University July 11-14, 2023. Algebraic curves are one-dimensional sets of points that can be described by polynomial equations. For example, the unit circle in the plane x^2 + y^2 = 1 defines an algebraic curve. A ubiquitous problem in the theory of algebraic curves is to understand how many algebraic curves there are satisfying a given list of conditions. This information corresponds to the so-called "intersection theory of their moduli space": a moduli space of curves is a space whose points correspond to algebraic curves, and its intersection theory describes how various loci in this space can meet (which corresponds to various conditions on the algebraic curve being satisfied simultaneously). The Chow ring of the moduli space packages this information of intersection theory into an algebraic structure. This summer school will introduce graduate students and postdocs to recent developments in computing Chow rings through four mini-courses, afternoon exercise sessions, and research groups. Students will participate in exercise sessions to reinforce the material from the mini-courses. They will also work in research groups to apply these techniques to compute new examples of Chow rings (both integral and rational) of moduli spaces of curves.More specifically, the four mini-courses will be on the following topics: (1) Equivariant intersection theory: this describes how intersection theory behaves in the presence of a group action; (2) Higher Chow groups: these invariants capture the failure of exactness of excision sequences of Chow rings; (3) The tautological ring of the moduli space of stable pointed curves: this describes a well-behaved subring of the Chow ring; and (4) Patching techniques and the integral Chow ring of the moduli of 2-pointed genus 1 curves: often enlarging moduli spaces to include curves with worse singularities can enable patching togetherinformation about Chow rings of strata to the Chow ring of the entire moduli space. https://sites.google.com/site/agneshomepage/brown-2023-agnes-summer-schoolThis 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.