In this talk I want to discuss the interrelationship between various properties of group rings of polycyclic-by-finite groups. In the first part of the talk we determine precisely which of these group rings satisfy the Dixmier-Moeglin Equivalence. This equivalence describes when the primitive ideas of an algebra are determined by certain other desirable properties.

This leads us to consider the famous old question on the noetherianity of group rings. (We do not solve this problem but hopefully the discussion is of interest.)

This is all joint work with Jason Bell and Kenny Brown.

Abstract: Landau--Ginzburg (LG) models are the natural mirror candidates to Fano manifolds and have an enumerative theory analogous to quantum cohomology, known as FJRW theory. In this talk, I will begin by giving an overview of these ideas, keeping a running comparison to quantum cohomology in order to explain some of the similarities and subtleties. I will then briefly…

Title: On strongly robust toric ideals: when Markov and Graver bases coincide.
Abstract: In the first part of the talk, I will give an introduction to Markov bases of toric ideals, which are one of the first connections between statistics and commutative algebra. In the second part of the talk, I will discuss recent work on strongly robust toric ideals. A toric ideal is called strongly robust when the Graver basis is a minimal system of generators. In the talk, I will explain how to build a strongly robust simplicial complex which determines the strongly robust property of toric ideals. I will then discuss our results on the strongly robust property in the case of monomial curves, as well as codimension 2 toric ideals and configurations in general position. This is joint work with A. Thoma and M. Vladoiu.