Quantum Topology in Dimension Three

第三维度的量子拓扑

• 批准号: 0508635
• 负责人: Charles Frohman
• 金额: $ 10.8万
• 依托单位: University of Iowa
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-07-15 至 2010-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0508635&HistoricalAwards=false
• 关键词: Quantum; Topology; Dimension; Three

AbstractAward: DMS-0508635Principal Investigator: Charles D. FrohmanThe principal investigator will extend and interpret quantuminvariants of three-manifolds utilizing the geometry of charactervarieties. The ideas he is exploring will combine tools fromgauge theory, representation theory, homological algebra, andthree-manifold topology. This entails work on severalproblems. With Kania-Bartoszynksa he will define a quantuminvariant of three-manifolds which will be a real analyticfunction on the open interval (-1,1). The invariant will beobtained by heat kernel regularization of the divergent formulafor the Turaev-Viro invariant. The power series expansion at 0,will be in terms of weighted signed counts of surfaces carried bya spine of the manifold. The normalized limit as you approach 1,will yield the total Reidemeister torsion of the SU(2)-charactervariety of the fundamental group of the manifold. Using ideasfrom matrix models he will develop an analogous invariant basedon the SL(2,C)-character variety of the three-manifold. With hisstudents he will continue to study the connection between theA-polynomial and quantum invariants, and explore the knot andlink homology theories of Khovanov and Khovanov-Rozansky. WithOliver Dasbach and Marta Asaeda he is looking at homologytheories underlying the Alexander polynomial. Finally, given athree-manifold and a Heegaard splitting there is an algebra whichis the Kauffman bracket skein module of the Heegaard surface anda bimodule over that algebra built from the skein modules of thetwo handle-bodies. With Mike McLendon, he will study whether thishomology is a three-manifold invariant, and if it is, what itsrelation to Khovanov homology is.The rational understanding of the path integrals of RichardFeynman stand as one of the major unresolved problems ofmathematics. Using his integrals Feynman was able to makecomputations in quantum electrodynamics that far exceededprevious work. The tools he developed allowed the construction ofmodern integrated circuits. The rules that physicists use forcomputing path integrals have never been made completelyrigorous. The major thrust of Frohman's work in recent years hasbeen about these integrals in a simplified setting where thingscan actually be computed. Specifically, the Yang-Mills measure inthe Kauffman bracket skein module assigns to a gauge field on thespace of flat connections on a surface a number. The formula forthe measure coincides with an asymptotic expansion that appearsthroughout the physical literature. However, in this situation itis actually a convergent series. Frohman is using this formula,and the estimates he used to prove it converged, to pursue theanalytic study of three-manifold invariants that were before onlycomputable using algebraic and combinatorial methods. The goal ofthe project is to reveal the geometric and toplogical nature ofquantum invariants of three-manifolds to the end of increasingthe understanding of three-manifolds, representation theory andquantum gravity.

摘要奖：DMS-0508635主要研究者：Charles D. Frohman主要研究者将扩展和解释量子不变量的三维流形利用几何的charactervarieties。他正在探索的思想将结合联合收割机工具，从规范理论，表示理论，同调代数，和三流形拓扑。这需要对几个问题的工作。与Kania-Bartoszynksa，他将定义一个量子不变量的三个流形，这将是一个真实的解析函数的开放区间（-1，1）。通过对Turaev-Viro不变量的发散公式进行热核正则化，得到该不变量。在0处的幂级数展开，将根据流形的脊所携带的表面的加权有符号计数。当你接近1时，归一化极限将产生流形基本群的SU（2）-特征簇的总雷德迈斯特挠率。利用矩阵模型的思想，他将发展一个类似的不变量的基础上SL（2，C）-字符簇的三个流形。 与他的学生，他将继续研究A-多项式和量子不变量之间的联系，并探索Khovanov和Khovanov-Rozansky的结和链接同源理论。他与奥利弗·达斯巴赫和玛尔塔·朝田一起研究亚历山大多项式的基础理论。最后，给定一个三流形和一个Heegaard分裂，存在一个代数，它是Heegaard曲面的Kauffman括号skein模和由两个曲面的skein模构成的代数上的双模。与迈克·麦克伦登，他将研究是否thismomology是一个三流形不变量，如果它是，它的关系Khovanov同调是什么。理查德费曼的路径积分的理性理解立场作为一个主要的数学未解决的问题。利用他的积分费曼能够使计算量子电动力学，远远超过以前的工作。他开发的工具使得现代集成电路的构建成为可能。物理学家用来计算路径积分的规则从来没有完全严格过。 主要推力弗罗曼的工作，近年来一直是关于这些积分在一个简化的设置thingcan实际上是计算。具体地说，杨米尔斯措施在考夫曼括号绞链模块分配给规范场的空间上的平面连接的表面上的一个数字。该测量公式与物理文献中出现的渐近展开式一致。然而，在这种情况下，它实际上是一个收敛级数。 弗罗曼是使用这个公式，估计他用来证明它收敛，追求thefrim流形不变量的分析研究，以前只能计算使用代数和组合方法。该项目的目标是揭示三维流形的量子不变量的几何和拓扑性质，以增加对三维流形，表示论和量子引力的理解。