Motives Associated to Graphs

这个项目关注的动机在算术代数几何的意义上，可以与图形.与这些动机相关的周期作为量子场论中费曼积分微扰展开的系数出现。这些时期经常与多种zeta值有关。研究人员将调查这种关系的各个方面，特别是重整化问题及其与限制混合Hodge结构的关系。这项工作将试图澄清图论，量子场论，数论和动机理论之间的一个有趣的关系。物理学家们已经证明，描述基本粒子行为的相关函数可以通过用图形索引的某些序列来计算。在许多特殊情况下，这些展开式中的系数是Riemann zeta或多个Riemann zeta值。这些系数的计算涉及到图形拉普拉斯，它首先出现在世纪的电路研究中。图拉普拉斯算子的零点定义了一个代数簇，研究的中心对象将是相应的“动机”，它用于统一不同的组合和算术结构。

This project concerns motives in the sense of arithmetic algebraicgeometry which can be associated to graphs. Periods associated tothese motives arise as coefficients in the perturbative expansion forthe Feynman integral in quantum field theory. These periods arefrequently related to multiple zeta values. The researcher willinvestigate aspects of this relationship; particularly therenormalization problem and its relations with limiting mixed Hodgestructures. This work will try to clarify an interesting relation between graph theory,quantum field theory, number theory, and the theory ofmotives. Physicists have shown that correlation functions describingthe behavior of elementary particles can be calculated by certainseries indexed by graphs. In many special cases, the coefficients inthese expansions are Riemann zeta or multiple Riemann zeta values. Thecalculation of these coefficients involves the graph laplacian, apolynomial which first arose in the 19th century study of electriccircuits. The zeroes of the graph Laplacian define an algebraicvariety, and the central object of study will be the corresponding"motive" which serves to unify the diverse combinatorial andarithmetic structure.