Simone Virili, Length functions in Grothendieck categories

Abstract:

Given a Grothendieck category G (e.g., G=Mod(R) for some ring R), a length function L on G is a non-negative real invariant (that may attain infinity) of the objects of G, that is additive on short exact sequences and "continuous" on direct unions. Examples of length functions are the dimension of vector spaces, the composition length, the torsionfree rank on Ore domains, or the logarithm of the cardinality of Abelian groups. All these are examples of discrete length functions but we will show with some examples that non-discrete length functions exist in natural situations. We will consider the spectrum of length functions on G, that is, the family of all irreducible length functions, and we will show that Ryo Kanda's atom spectrum is contained in the spectrum of length functions (atoms correspond bijectively with irreducible discrete length functions). We will also give a complete description of all length functions in categories with Gabriel dimension and for modules over valuation domains. If time allows, we will also show that, for a ring R, there is a bijective correspondence between exact Sylvester matrix rank functions on R and a suitable subfamily of length functions of Mod(R).

Simone Virili, Algebraic entropy of amenable group actions

Abstract:

In this second talk we will give some applications of the theory of length functions to the representation theory of infinite groups. In particular, given a ring R, an amenable group G and a crossed product R*G, we will show how to extend a given length function L on R-Mod, to a suitable length function of the category of representations R*G-Mod. This construction can be performed using a dynamical invariant associated with the original function L, called algebraic L-entropy. We apply our results to two classical problems on group rings: the Stable Finiteness and the Zero-Divisors Conjectures.