As an amoeba is the set of lengths of points in a variety, its coamoeba is the set of angles (arguments) of its points. The set of limiting arguments forms its phase limit set. This combinatorial backbone of the coamoeba reflects the structure of the corresponding tropical variety. We give a recursive description of the phase limit set of a linear space/hyperplane complement in terms of the flats of the hyperplane arrangement. We use this to study the phase limit set of a reduced discriminant, showing that it is a union of prisms over discriminants of lower dimension. We conjecture that in dimension at least three the discriminant is a subset of its phase limit set, which implies that that coamoeba discriminant has a polyhedral structure.

Weak diameter coloring is the key notion used in the recent result that determines the asymptotic dimension of minor-closed families of graphs. We consider the list-coloring analog in this talk. For every graph H, we determine the minimum integer k such that every graph that does not contain H as a minor can be colored so that every monochromatic component has bounded weak diameter as long as every vertex has at least k available colors. This result is a common generalization of previous results about weak diameter coloring of graphs with excluded minors, about weak diameter list-coloring of graphs with bounded Euler genus, and about clustered coloring of graphs with bounded maximum degree and with excluded minors. It is joint work with Joshua Crouch.

This talk concerns singularities of local rings and numerical invariants used to detect mild singularities. Some classical notions of mild singularities include normal, Cohen-Macaulay and Gorenstein. However, when the ring has prime characteristic, the Frobenius map provides powerful new ways to detect mild singularities via the notions of F-purity and F-regularity. For instance, Hochster and Huneke proved that strongly F-regular rings are automatically normal and Cohen-Macaulay, but not always Gorenstein. In this talk, we will discuss a criterion for a strongly F-regular standard graded ring to be Gorenstein. This criterion was conjectured by Hirose, Watanabe and Yoshida and relies on the relationship between the F-pure threshold and another classical invariant of graded rings called the a-invariant. In this talk, we will discuss a proof of this conjecture, extending previous partial results of Singh, Takagi and Varbaro, and De Stefani and Núñez-Betancourt. We will review the relevant preliminary notions before presenting the proof.

The theory of quantum groups is essential to understanding quantum symmetries in noncommutative geometry and mathematical physics. We discuss a deformation theory to study various classes of algebras by lifting algebraic properties to categorical contexts via their universal quantum groups, where we can then apply the theory of tensor categories.