We want to study the class of all finite structures together with the preorder given by primitive positive constructions as it plays an important role in the theory of constraint satisfaction problems. It is also equivalent to the homomorphism poset of the polymorphism minion of the structure, which is the right generalization of endomorphism monoid/ring in this context. It turns out that the first two layers of this order consist of a single equivalence class each, while the third layer consists of infinitely many points, represented by the transitive tournament on three vertices and structures in one-to-one correspondence with all finite simple groups.
Finite simple groups in the primitive positive constructability poset
Oct. 08, 2024 2:30pm (Math 3…
Lie Theory
John McHugh (University of Denver)
X
The "extended tensor product" is a relatively new operation in the representation theory of finite groups which generalizes both the familiar tensor product of representations over the ground field as well as the tensor product of bimodules. It was introduced in 2008 by Serge Bouc and studied extensively by Robert Boltje and Philipp Perepelitsky in the early part of this decade. I will attempt to motivate and explain the definition of the extended tensor product. If I am successful I will go on to discuss new results in the theory of modular representations of finite groups that are proved using the extended tensor product construction. The material will be presented in the most elementary ways possible: a potential audience member should only know what a finite group is and what an action of a finite group on a finite set looks like.
The Extended Tensor Product Sponsored by the Meyer Fund
Oct. 08, 2024 3:30pm (MATH 3…
Topology
Courtney Hauf
X
Born of Grothendieck’s brain, K-theory started as a way to discuss isomorphism classes of objects that live above some object of interest (in his case, vector bundles over algebraic manifolds). Today, K-theory can be an ambiguous term due to the various attempts to construct (higher) K-theory. In this talk we will look at the history of K-theory, discussing the motivations that led to its many variations and ways that K-theory has been used in various areas of mathematics (and physics!).
[Will not require background.]
Algebraic K-Theory I: Where did we come from? Where did we go?
Finite simple groups in the primitive positive constructability poset