CAREER: Algebraic Geometry of Moduli Spaces

研究者和他的同事将讨论一些关于模空间的几何和算术的基本问题：g属曲线的模空间的规范模型是什么？是否可以构造在规范模型和稳定曲线的模空间之间进行插值的自然空间？这些可以被解释为模空间吗？我们能数出曲线模空间中有理点的个数吗？用来表示高度的自然函数（有效因子）是什么？平面曲线奇点的变形空间的自然定义层是什么？人们如何解释沿着这样的地层发生的爆炸？代数几何是研究多项式方程组解的学科。解集的几何性质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，有希望的假设可以用明确的例子来检验。几何的计算方法正日益改变着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该学科技术复杂性的年轻人。

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.