CAREER: Algebraic Geometry of Moduli Spaces

研究人员和他的同事们将讨论模空间的几何和算法的一些基本问题：亏格g的曲线的模空间的正则模型是什么？人们能构造出介于正则模型和稳定曲线的模空间之间的自然空间吗？这些空间本身能被解释为模空间吗？人们能计算出曲线的模空间关于不同高度有界的有理点的数目吗？人们可以用什么自然函数(有效因子)来指定高度？在平面曲线奇点的变形空间中，自然定义的层是什么？人们如何解释沿着这样的层的爆破？代数几何是研究多项式方程组的解的学科。解集的几何特征转化为方程的代数性质，反之亦然。这种方法的优点是，计算机可以非常有效地处理方程。因此，提出的假设可以在明确的例子上得到检验。越来越多的几何计算方法正在改变本科生和研究生的教育，无论是在纯数学领域还是在应用数学的相关领域。这种对具体例子和明确计算的强调为年轻人创造了新的研究机会，特别是那些刚刚开始学习这门学科的技术复杂性的人。

The investigator and his colleagueswill address a number offundamental problems aboutthe geometry and arithmetic of moduli spaces:What is the canonical model of the moduli space of curves of genus g?Can one construct natural spaces that interpolate between thecanonical model and the moduli space of stable curves?Can these be interpretted as moduli spaces in their own right?Can one count the number of rationalpoints of moduli spaces of curves, boundedwith respect to various heights? What are the naturalfunctions (effective divisors) one might use to specify theseheights?What are the naturally defined stratain the deformation space of a plane curve singularity?How can one interpret blow-ups along such strata?Algebraic geometry is the study of the solutionsto systems of polynomial equations. Geometricaspects of the solution sets translate into algebraicproperties of the equations, and vice versa. Thisapproach has the advantage that computers canmanipulate equations very efficiently. Promisinghypotheses can thus be checked on explicit examples.Increasingly, computational approaches to geometryare transforming the education of undergraduateand graduate students, both in pure mathematicsand in related fields where mathematics is applied.This emphasis on concrete examples and explicit computationcreates new research opportunities for young people,especially those who have only just started to learnthe technical intricacies of the subject.