Academy of Mathematics and Systems Science, CAS Colloquia & Seminars
Speaker:
李利平 教授, 湖南师范大学
Inviter:
Title:
FI-MODULES AND HOMOLOGY OF CONGRUENCE SUBGROUPS
Time & Venue:
2018.1.15 10:30-11:30 N902
Abstract:
Let GLn(R) be the general linear group over a unital ring R, and I be a proper two-sided ideal of R. The congruence subgroup Gamma_n is defined to be the kernel of the natural map GLn(R)\to GLn(R/I). It has been observed that for a fixed i > 0, the sequence of homology groups {H_i(Gamma_n;A)| n > 0}, where A is an abelian group, does not satisfy the homological stability, but has certain representation stability pattern. That is, although homology groups in this sequence vary as n increases, one can still deduce an explicit description of H_i(Gamma_n;A) for arbitrary n > 0 whenever the first omega(i) terms are available, where omega(i) is a positive integer depending on i. In a jointed paper, Gan and Li proved that !(i) can be taken to be a linear function with slope 4. In this talk I will explain the highly creative machinery underlying the above result: representation theory of the category FI, whose objects are finite sets and morphisms are injections. Rooted from the fundamental work of Thomas Church, Jordan Ellenberg, and Benson Farb, representation theory of FI has been widely applied to investigate homology/cohomology groups of algebraic/topological objects appearing in algebraic topology, geometric group theory, number theory, commutative algebra, algebraic geometry, and combinatorics.