Almost any reasonable class of finite relational structures has the Ramsey property or a precompact Ramsey expansion. In contrast to that, the list of classes of finite algebras with the precompact Ramsey expansion is surprisingly short. In this talk we show that any nontrivial variety (that is, equationally defined class of algebras) enjoys various dual Ramsey properties. We develop a completely new set of strategies that rely on the fact that left adjoints preserve the dual Ramsey property, and then treat classes of algebras as Eilenberg-Moore categories for a monad. We show that finite algebras in any nontrivial variety have finite dual small Ramsey degrees, and that every finite algebra has finite dual big Ramsey degree in the free algebra on countably many free generators. As usual, these come as consequences of ordered versions of the statements.
Dual Ramsey properties for classes of algebras
Feb. 08, 2022 3pm (MATH 350)
Topology
Daniel Lyness (CU Boulder)
X
This is the first talk in the semester learning seminar on modular tensor categories and applications to topological phases of matter.
In this talk, we will introduce the concept of topological field theories and give a few fundamental examples.
An introduction to TQFTs
Feb. 08, 2022 5pm (MATH 350)
Grad Student Seminar
Howie Jordan (CU Boulder)
X
Information. What is it? Where does it come from? How can I give you information about getting information? By talking at you while pizza powers your focus. (A proof of this will be constructed over the course of the talk).
Information theory starts with the idea that we gain information by observing the outcome of something we were unsure about. If you learn which horse wins the race, you've gained information about the race. If you learn a politician lied, you didn't really gain any information about the politician. In this talk, I'll introduce the basic concepts and tools of classical information theory. In particular, we'll consider the notion of entropy and its origins in analyzing communication.
Dual Ramsey properties for classes of algebras