Index theory assigns stable invariants to differential operators, linking analysis and geometry. Can it also reveal number-theoretic invariants associated with arithmetic groups? In joint work with Wenqing Wu, we study this question through Hilbert modular groups. The spaces associated with these groups have noncompact ends called cusps, which complicate index calculations. We compactify these ends and add a potential that grows toward the cusps, producing a higher index that remembers the group action. We show that, for the groups considered here, this index is determined by the cuspidal part of the operator and can be expressed in terms of Hilbert cusp forms. The talk will focus on the ideas connecting cusp geometry, index theory, and arithmetic.

王航，华东师范大学算子代数研究中心教授，国家级青年人才。 2006年毕业于复旦大学，2011年博士毕业于美国范德堡大学，2011-2013在清华大学丘成桐数学科学中心做博士后。2013-2017在澳大利亚阿德莱德大学工作，曾入选海外高层次青年人才计划。研究方向是非交换几何与算子代数，有多篇代表性论文发表在一流数学期刊上，如：Journal of Differential Geometry, Advances in Mathematics, Proceedings of the London Mathematical Society等