Speaker: Prof. Ezra Miller (Duke University)
Time: 9:30 -- 11: 00, September 17, 2025
Venue: Room 612, A6, Institute of Mathematics-VAST
Abstract: Families of modules or vector spaces indexed by an integer n, for instance, arising from powers of ideals, often have numerical invariants that grow polynomially. These invariants might be lengths, Betti or Bass numbers, or regularity, or combinations of these with deeper homological constructions.
Sometimes the growth is only quasipolynomial: there are polynomials P_1,...,P_r such that the numerical invariant takes the value P(n) = P_i(n) whenever n is congruent to i (mod r). What drives this kind of growth and periodicity? Joint work with Hailong Dao, Jonathan Montaño, Christopher O'Neill, and Kevin Woods provides a general answer in the multigraded setting -- that is, for families of monomial ideals and other finely graded modules over affine semigroup rings. The theory is clarified by thinking so generally that it yields for free the case of multiple parameters, when the families are indexed not merely by a single integer n but by many integers; in that context, a numerical function is called quasipolynomial when its values agree with one of several polynomials, one for each coset of a given integer lattice. The proofs and constructions rest on foundations from applied topology, specifically tame modules in persistent homology, combined with Presburger arithmetic. No familiarity with either of these theories is assumed. |
