Mathematical Sciences: Moduli Spaces of Riemann Surfaces

罗伯特彭纳将研究装饰泰希穆勒理论， 几个方向，如下所示。 米勒- Riemann曲面模空间上的Morita-Mumford类 应该是可以计算的，使用交叉理论在一个适当的V- 模空间的流形紧化;特别是，所有的 实施这一计划的必要因素似乎已经到位 一次穿孔的Weil-Petersson体积的计算 表面。 也有一个“普遍”的模拟装饰 Teichmuller空间，以及基本上所有的经典结构 他希望对宇宙空间的研究 将有助于解释经典理论 他还希望， 计算在坐标系中模的复结构。 计算机工作将继续推导出组合结构 亏格3的一次穿孔曲面的模空间，以及 他希望明确地计算出这个空间的同调。 最后，一个程序已经发展到扩展他的一些工作， 封闭表面的设置。 Teichmuller理论是一个七十岁的混合代数 几何学、微分几何学和拓扑学。 因此，进步 会引起众多数学家的兴趣 部分客房还设有 与弦理论有关，所以一些理论物理学家 这项研究也是如此。

Robert Penner will study decorated Teichmuller theory in several directions, as follows. Integrals of monomials of Miller- Morita-Mumford classes over the moduli space of Riemann surfaces should be computable using intersection theory on an appropriate V- manifold compactification of moduli space; in particular, all the necessary ingredients seem to be in place for applying this scheme to the computation of Weil-Petersson volumes for once-punctured surfaces. There is also a "universal" analog of the decorated Teichmuller space, and essentially all of the classical structures extend to this setting; he hopes the study of this universal space will shed light on the classical theory. He furthermore hopes to compute the complex structure of moduli in his coordinates. Computer work will be pursued to derive the combinatorial structure of the moduli space of once-punctured surfaces of genus three, and he hopes explicitly to compute the homology of this spaces. Finally, a program has evolved for extending some of his work to the setting of closed surfaces. Teichmuller theory is a seventy year-old blend of algebraic geometry, differential geometry, and topology. As such, advances will interest a broad spectrum of mathematicians. Some also have a bearing on string theory, so some theoretical physicists have a stake in this research as well.