3-Manifolds and Applications to Geometry

建议：DMS-0083097PI：Walter D.Neuman尽管代数几何有悠久的历史，但许多困难问题即使在低维也仍然存在。低维拓扑学的方法在这里被证明是非常有用的，这个项目的一部分是研究者成功地将这种方法应用于研究复杂曲面的奇点和平面仿射曲线的拓扑学。一个需要解决的问题是给定拓扑下奇点的显式解析描述。一般问题是出了名的难，但研究者和J.Wahl最近提出了一系列猜想，将具有有理同调球环的Gorenstein奇点有理地实现为完全交的显商，并在解决这些猜想方面取得了重大进展。3-流形拓扑学也被用于研究二元多项式的全局拓扑和解析分类问题。这些问题具有内在的意义，并且有可能有助于解决著名的雅各比猜想。在一个相当不同的方向上，研究者将研究与数论和代数有关的三维流形的不变量，即所谓的特征标和特征值簇以及Bloch不变量。代数几何本质上是研究多项式的零点集，它一直是基础研究的一个重要领域，在控制理论和通信等各种应用中也很重要。该项目将把拓扑学方法应用到代数几何中，以获得如何实现和分类导致拓扑结构的多项式族的结果。三维拓扑学的方法对这个项目特别重要，而且还有几何和数论对拓扑学的反馈，以及与理论物理和宇宙学的一些相互作用。该项目还研究了这些联系给所有这些领域带来的新见解。

Proposal: DMS-0083097PI: Walter D. NeumannDespite the venerable history of the algebraic geometry, many difficultproblems remain even in low dimensions. Methods of low dimensionaltopology have proved very helpful here, and part of this project concernsa continuation of the investigator's successful application of suchmethods to the study of singularities of complex surfaces and the topologyof plane affine curves. One issue to be addressed is explicit analyticdescription of singularities with given topology. The general question isnotoriously difficult, but the investigator and J. Wahl have recently madea series of conjectures on realising rationally Gorenstein singularitieswith rational homology sphere links as explicit quotients of completeintersections and are making significant progress towards theirresolution. 3-manifold topology is also being applied to the study of theglobal topology and analytic classification issues for two-variablepolynomials. These issues are of intrinsic interest as well as havingpotential to contribute to the resolution of the famous JacobianConjecture. In a rather different direction, the investigator will studyinvariants of 3-dimensional manifolds that relate to number theory andalgebra, namely the so-called character and eigenvalue varieties and theBloch invariant. Connections are emerging between these invariants thatwill inform both topology and algebra/number theory.Algebraic Geometry, which is essentially the study of the zero sets offamilies of polynomials, has always been an important area offundamental research, and is also important in such diverseapplications as control theory and communications. The project willapply topological methods to algebraic geometry, to obtain results onhow to realise and classify families of polynomials that lead toparticular topologies. The methods of 3-dimensional topology are ofparticular importance to this project, and there is also feedback fromalgebra and number-theory to topology, as well as some interactionwith theoretical physics and cosmology. The project also studiesnew insights that these connections bring to all these fields.