Conference: Betti Numbers in Commutative Algebra and Equivariant Homotopy Theory

Syracuse University, Syracuse NY

Investigators

Abstract

This award provides travel funding for US-based participants in the week-long workshop “Betti numbers in commutative algebra and equivariant homotopy theory” to be held September 23–27, 2024, at Bielefeld University, in Bielefeld, Germany. The workshop centers on a series of long-standing conjectures that appear in parallel in two major fields of mathematics. The goal of the workshop is to bring together researchers from these two fields to discuss recent advances on these conjectures. Another goal is to train more researchers to work on these important problems and help them build connections between the two fields. The overarching goal of this award would be to increase US participation in this highly active area of research, and to foster collaborations between US mathematicians and those from other countries. The funding is aimed especially at postdoctoral fellows and graduate students, as well as participants who do not have independent funding, to attend this workshop, and it will also be used to encourage participation by individuals from underrepresented groups in mathematics. A recent workshop held in Banff, Canada in 2022 initiated this goal, and funding for this event would cement the connections already made and build new ones for younger participants. The bridges we are building will not only connect researchers located in different countries but also between those working in different areas of mathematics. Algebra and topology are thriving branches of mathematics that are well represented in most math departments. Commutative algebra, as the algebraic underpinnings of algebraic geometry, and algebraic topology, with its strong focus on homology and homotopy, have occasional significant overlap in both methods and aims. The goal is to create a strong working alliance between the groups working on these conjectures and related problems, and also to get younger researchers involved in these problems. In fact, total Betti numbers appear in related, decades-old rank conjectures in commutative algebra and equivariant topology. On the topological side, Halperin and Carlsson conjectured that the total Betti number of a compact space with a free torus action or p-torus action of rank r is bounded below by 2r, which has inspired much research on the topological side of spaces with a group action. On the algebraic side, Avramov conjectured a similar lower bound for the total Betti number of finite length modules over a local ring. Recent work of Walker and VandeBogert-Walker resolves this conjecture positively for rings of prime characteristic, whereas counterexamples to a stronger conjecture show the subtlety of the questions. The web site for the workshop is at https://www.math.uni-bielefeld.de/birep/meetings/betti2024/index.php and includes a full speaker list. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.