The 4th floor Seminar room, Howard House, Bristol, BS8 1SD
Abstract: Diptych varieties were defined, and then proved to exist, by Brown and Reid. It turns out that they have a certain type of cluster algebra structure. I will introduce diptych varieties and then explain how one can prove the existence of the diptych variety in a different way to Brown and Reid, by using this cluster algebra structure.
Abstract: The study of singularities has a long history in commutative algebra and algebraic geometry. One approach is to provide properties which give classes of rings or varieties whose singularities are manageable, and the class of Gorenstein rings is such a class. In this talk, I will introduce the trace of the canonical module, which measures how close a ring is to being Gorenstein. I will discuss recent results on this trace for rings in which we can exploit some combinatorial data, including Hibi rings and more general toric rings. This is joint work with Jürgen Herzog and Fatemeh Mohammadi.
Abstract: Classical theory of Newton polyhedra calculates topological invariants of a zero set of a general system of Laurent polynomials in terms of combinatorics of their Newton polyhedra. More precisely, for a fixed polytopes P_1,...,P_k there exists an open dense subset U of the space of Laurent polynomials with Newton polyhedra P_1,...,P_k such that topological invariant of interest is the same for any system from U and can be computed combinatorially.
It could be that polyhedra P_1,...,P_k defines an overdetermined systems, i.e. the generic system of Laurent polynomials with Newton polytopes P_1,...,P_k does not have any solutions. In this case one can be interested in invariants of generic non-empty zero set. Since in this case solvable systems are not generic, all results of classical Newton polyhedra theory are not applicable to them. In my talk I will explain how to extend theory of Newton polyhedra to the case of overdetermined systems.
Abstract: A particularly interesting family of varieties are given by configurations of points on line arrangements. These varieties are examples of matroid varieties and we can use techniques from combinatorics and commutative algebra to answer questions which are typically very difficult. In this talk I will define point/line configurations, their associated matroid varieties and give some techniques which can be used to study them.
Registration not required. Please write to me if you like to join us for dinner.