Conference: AGNES Summer School in Algebraic Geometry

AGNES暑期学校将于2023年7月11日至14日在布朗大学举行。代数曲线是可以用多项式方程描述的一维点集。例如，平面x^2 + y^2 = 1上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，有多少代数曲线满足给定的一系列条件。这个信息对应了所谓的“它们的模空间的相交理论”：曲线的模空间是一个点对应于代数曲线的空间，它的相交理论描述了这个空间中各种轨迹如何满足（对应于代数曲线上同时满足的各种条件）。模空间的周氏环将这些交点理论的信息包装成一个代数结构。这个暑期学校将通过四门迷你课程、下午练习和研究小组向研究生和博士后介绍计算机周环的最新发展。学生将参加练习课程来强化迷你课程的内容。他们还将在研究小组中应用这些技术来计算曲线模空间的Chow环（积分和有理）的新例子。更具体地说，四门迷你课程将涉及以下主题：(1)等变交集理论：描述交集理论在群体行为中如何表现；(2)高Chow群：这些不变量捕获了Chow环切除序列的精确性失效；(3)稳定点曲线模空间的同义环：这描述了Chow环的一个良好的子环；(4)补片技术与2点1属曲线模的整体Chow环：通常将模空间扩大到包括奇点较差的曲线，可以将地层的Chow环信息一起补片到整个模空间的Chow环上。https://sites.google.com/site/agneshomepage/brown-2023-agnes-summer-schoolThis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 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.