Mini-workshop in homological algebra

Schedule (9 April):

• 12:00: Lunch
• 13:15-14:15: Talk by Lars Winther Christensen
• 14:15-15:00: Fika
• 15:00-16:00: Talk by Sergio Estrada
• 17ish: out somewhere

Title (Lars): Generic type 2 quotients of the polynomial algebra in 3 variables

Abstract: Let k be a field and I an ideal in the polynomial algebra Q= k[x, y, z] such that (x, y, z)^p ⊆ I ⊆ (x, y, z)^2 for some p≥2. We are interested in the quotient ring R= Q/I. Such quotients can be classified based on differential graded (dg) algebra structures on their minimal free resolutions over Q. If the quotient has type 1, then R is Gorenstein. If the quotient has type 2, then it belongs, generically, somewhere on a spectrum between Gorenstein and Golod. Here the spectrum is one of (dg) algebra structures, and the the exact “position” of a type 2 ring on this spectrum is determined by its socle polynomial.

Title (Sergio): Approximation Theory in Algebraic Contexts

Abstract: The study of covers and envelopes of modules began in the early 1980s, driven by the contributions of Enochs, Auslander, and Smalø. This theory is founded on the fundamental idea of approximating arbitrary objects in a nice way by more accessible ones belonging to well-understood classes. In this talk, we will outline the historical development of the theory of covers and envelopes, culminating in what became known as the Flat Cover Conjecture. We will discuss the key mathematical tools that led to the conjecture’s resolution (published in 2001) and explore the broader impact of these techniques, which continue to play a crucial role in ongoing research in homological algebra.