We'll discuss some mathematics that is motivated by a card trick. We'll then touch on some applications to cryptography. To avoid any disappointment, I should mention that this magic trick is not one you would pay to see.
A “form” is a category equipped with abstract subobject posets for its objects and “connections” between those posets induced by morphisms of the category. In this talk we present an application of this structure in developing a self-dual approach to homomorphism theorems for group-like universal algebras. We call a form satisfying the axioms necessary for obtaining the standard isomorphism theorems a “noetherian form”. Examples of noetherian forms abound (in some of the examples, though, the isomorphism theorems trivialize). For instance, any bounded lattice can be viewed as a noetherian form with a trivial base category. Furthermore, not only group-like universal algebras (e.g., modules, groups, rings with or without unit, loops, etc.), but even any variety of universal algebras can be viewed as a noetherian form, thanks to the fact that the category of sets has the structure of a noetherian form (a non-trivial fact, since the two natural forms, that of subsets and that of equivalence relations, are both non-neotherian). If time allows, we will also discuss some combinatorial problems suggested by the notion of a noetherian form.
Full heaps are an interesting collection of posets defined by Richard Green, associated to an extended Dynkin diagram and a choice of two extending nodes. They were introduced for Lie-theoretic applications. I will describe a different way to construct them (in the simply-laced cases) based on the representation theory of quivers. I will not assume that the audience is familiar with the theory of quiver representations, so I will try to give a bit of an introduction to them. This is based on joint work with Al Garver and Becky Patrias, and is motivated by our study of affine type generalizations of the RSK correspondence. I won't have time to say much about that, but I will at the very least attempt to explain how we use full heaps to define an affine version of reverse plane partitions. (The talk will be given on Zoom at https://cuboulder.zoom.us/j/94126918788.)
Given the ubiquity of vector bundles, it is perhaps surprising that there are so many open questions about them -- even on projective spaces. In this talk, I will outline some results about vector bundles on projective spaces, including my ongoing work on complex rank 3 topological vector bundles on CP^5. In particular, I will describe a classification of such bundles which involves a surprising connection to topological modular forms; a concrete, rank-preserving additive structure which allows for the construction of new rank 3 bundles on CP^5 from "simple" ones; and future directions related to this project.
Topic: CU Topology Seminar Time: This is a recurring meeting Meet anytime