From Calabi-Yau 3-folds to Gorenstein rings

Abstract: In physics, String Theory posits a symmetry between certain pairs of geometric objects: Calabi-Yau threefolds. In algebra, Gorenstein rings are objects possessing an intrinsic, internal symmetry. We’ll discuss connections between the two sets of objects, going from the basic algebraic construction, to computation, culminating in smooth Calabi-Yau threefolds with new Hodge numbers.

The talk will start from ground zero, and anything we need along the way will be defined (by example). Joint work with Mike Stillman (Cornell) and Beihui Yuan (Swansea).

A counter-example to the Schenck-Stiller "2r+1" conjecture

Abstract: To approximate a function over a region, it is useful to consider a subdivision of the region and then approximate the function by a piecewise polynomial. In this talk, I would like to talk about commutative algebra tools we use to study this subject, conjectures on splines and a counter-example to the Schenck-Stiller "2r+1" conjecture. This talk is based on joint work with Mike Stillman and Hal Schenck.

Symmetrically Colored Gaussian Graphical Models with Toric Vanishing Ideals

Abstract: Gaussian graphical models are multivariate Gaussian statistical models in which a graph encodes conditional independence relations among the random variables. Adding colors to this graph allows us to describe situations where some entries in the concentration matrices in the model are assumed to be equal. In this talk, we focus on RCOP models, in which this coloring is obtained from the orbits of a subgroup of the automorphism group of the underlying graph. We show that when the underlying block graph is a one-clique-sum of complete graphs, the Zariski closure of the set of concentration matrices of an RCOP model on this graph is a toric variety. We also give a Markov basis for the vanishing ideal of this variety in these cases.

Polynomials Over Hyperfields

Abstract: Kapranov’s theorem is a foundational result in tropical geometry. It states that the set of tropicalisations of points on a hypersurface coincides precisely with the tropical variety of the tropicalisation of the defining polynomial. My talk will focus on introducing the multivalued setting of hyperfields, along with the relevant polynomial definitions and discuss my results towards a generalisation of Kapranov’s theorem for class of hyperfield homomorphisms, whose target is the tropical hyperfield and satisfy a relative algebraic closure condition. To provide an example of such a hyperfield homomorphism, the map from the complex tropical hyperfield to the tropical hyperfield is investigated.