Conference: AGNES Summer School in Algebraic Geometry

AGNES暑期学校的模空间相交理论将于2023年7月11日至14日在布朗大学举行。代数曲线是可以由多项式方程描述的一维点集。例如，平面x^2 + y^2 = 1中的单位圆定义了一条代数曲线。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列表。这些信息对应于所谓的“它们的模空间的相交理论”：曲线的模空间是一个其点对应于代数曲线的空间，其相交理论描述了这个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包装成一个代数结构。这个暑期学校将通过四个迷你课程，下午练习课程和研究小组向研究生和博士后介绍计算周环的最新发展。学生将参加练习课程，以加强从迷你课程的材料。他们还将在研究小组中应用这些技术来计算周环的新例子更具体地说，四个小型课程将讨论以下主题：（1）等变相交理论：这描述了相交理论在群作用存在时的行为;（2）高阶Chow群：（3）稳定点曲线模空间的重言式环：它描述了Chow环的一个良子环;（4）2-点亏格1曲线模的修补技巧和积分Chow环：通常，将模空间扩大到包括具有更差奇异性的曲线可以将关于地层的Chow环的信息修补到地层的Chow环。整个模空间https://sites.google.com/site/agneshomepage/brown-2023-agnes-summer-schoolThis奖项反映了NSF的法定使命，并被认为值得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的支持。

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.