Welcome to the Algebra and Geometry Seminar at Iowa State University. During Fall 2026, the seminar runs on Fridays at 3:20pm–4:10pm in Carver 401. Graduate students are especially encouraged to attend!
If you would like to be added to the mailing list, or wish to suggest a speaker, please reach out to of the organizers: Jonas Hartwig, Jason McCullough, or Tathagata Basak.
Topics include:
The first meeting is a meet and greet where we introduce ourselves and share our interests.
Continuous groups of symmetries (Lie groups) appear in many parts of mathematics and physics. Their structure can be understood using Lie algebras, which are vector spaces equipped with a bilinear operation called the bracket. When representing (semi-simple) Lie algebras by matrices, there is a universal formula called the extremal projector with remarkable properties. It has applications in quantum mechanics, but can be found already in a paper by Clebsch from 1862. I will describe a natural way to obtain this formula, using the quadratic Casimir. No prior experience with Lie algebras will be assumed.
In the second talk, I’ll tell you about Clebsch’s result on harmonic functions from 1862, and how it relates to the extremal projector for the Lie algebra sp(2). We then look at how the Casimir element can be used to construct extremal projectors explicitly as a formal series, for any simple Lie algebra.
Given a graph D such that the second largest eigenvalue of its adjacency matrix is a positive integer, one can define an free abelian group L together with a integer valued dot product of Lorentzian signature such that the symmetries of D act on L preserving the dot product. This kind of construction goes back to at least [Neumaier and Seidel, 1983] and is analogous to building a root lattice from a Dynkin diagram. Our aim is to illustrate some applications of this sort of construction.
In the first talk, we'll talk about the general construction, discuss some possible applications to graph theory and give some examples specially for strongly regular graphs and describe the associated hyperbolic geometry.
In the second talk, we'll discuss a “complexification” of our graph-to-lattice construction, that produces interesting “complex reflection groups”. One of these examples is related to the space of cubic surface. We'll We’ll try to describe the new information we obtain about the space of cubic surfaces from this point of view.
We'll try to keep the prerequisites to a minimum. In particular, no background knowledge of graph theory or hyperbolic geometry or cubic surfaces will be assumed.
Some of this is joint work with Daniel Allcock and Eduard Looijenga and some of this is joint work with Jon McCammond.
The algebraic geometry of varieties in projective space is a classical and well-studied subject. One can generalize this to the setting of toric varieties, which have the advantage of having combinatorial data (e.g. a fan) to aid in their study, while also being very general. I will review the classical picture and introduce the construction of toric varieties. I aim to make the talk as accessible as possible.
Building on last week’s talk, we start from the classical perspective, where a projective variety has an associated graded ideal in the corresponding polynomial ring. Its free resolution encodes many of the properties of the variety (e.g. dimension, Hilbert function, multiplicity, singularity type,…). Hilbert’s basis Theorem says that such free resolutions are finite and have length at most the dimension of the ambient projective space. There is a parallel theory of multigrade Cox rings of toric varieties in which, until recently, it was an open question as to whether there were correspondingly short (virtual) resolutions. I will present a step in the proof of their existence called cellular resolutions, which are defined combinatorially defined and resolve the normalization of the associated sub variety.
There has been much work done on representations of quivers, as well as on Hopf actions on a quiver's path algebra. I blend these two areas of study by classifying the representations of cyclic oriented affine quivers which are equivariant with respect to certain Hopf actions on their path algebras. In particular, this work provides a partial extension to Demonet's 2009 work on skew group algebras of path algebras. Additionally, in the finite case, I classify the representation type.
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