Title: Localizations of E-theory
 

Abstract: Chromatic homotopy theory uses the theory of formal groups from algebraic geometry to construct new topological invariants. The tightest link between the two worlds is Morava E-theory, a homotopical avatar of the space of deformations of a formal group of fixed height. We study what happens when E-theory undergoes chromatic localization, forcing the height of this formal group to decrease. We give modular descriptions of the resulting objects, and applications to the study of power operations in homotopy theory.

Title: Nilpotence Height in The Steenrod Algebra

Abstract:
The Steenrod algebra is a powerful tool in algebraic topology, generated by elements called squares, which define stable cohomology operations. Due to their topological origins, the algebraic properties of these squares carry topological implications. I will be giving a brief overview of the Steenrod Algebra and its properties, with an emphasis on the nilpotence of certain families of squares. I will end with some recent progress on the height of the family Sq(2^n-2). This is joint work with Bert Guillou.
Be there or…be Square. 

Title: An introduction to model categories, or, a `to do' list before getting to the homotopy category

Abstract: Want to understand a Homotopy Category? You need a model structure first! Don't know what that is? Not a problem! Come to my talk and we will start from the definitions, doing some "clean hand" category theory and hopefully get to some examples without making our hands dirty.

http://www.ms.uky.edu/~topology/

Title: An introduction to equivariant cohomology

Abstract: Cohomology is a very useful and powerful tool that is used in the study of spaces. In the equivariant world, when the spaces come equipped with group actions, there are several distinct constructions that go by the name of equivariant cohomology. I will discuss these theories and the relationships between them.

http://www.ms.uky.edu/~topology/

Title: (Equivariant) (Dendroidal) Segal Spaces and Categorical Homotopy Theory

Abstract: Segal spaces offer a convenient model for "categories up to homotopy". In this talk, we will first introduce Segal spaces and their connection to generic homotopy theories. Then, using the technology of G-trees, we will build modern generalizations which present equivariant multicategories with norm maps.

http://www.ms.uky.edu/~topology/

Title: A whirlwind tour of complex cobordism in chromatic homotopy theory

Abstract: Complex cobordism is a surprisingly powerful cohomology theory with deep connections to classical questions in stable homotopy theory and the theory of formal groups in algebraic geometry. In this talk we will attempt to introduce these connections and some of the most important theorems regarding complex cobordism without getting bogged down in the technicalities (ie. expect few proofs or even precise definitions).

http://www.ms.uky.edu/~topology/

During the semester, the UK Math Topology Seminar usually meets on Thursdays at 3:30 PM in POT 745. However, there are two exceptions for the SP2018: February 9 and April 10. On April 10, the seminar will meet between 3:00-6:00. Please visit UK Math Topology Seminar for updates and information about presenters. 

Title:  The Algebraic K-Theory of Varieties 

Abstract:  The Grothendieck ring of varieties is a fundamental object of study for algebraic geometers. As with all Grothendieck rings, one may hope that it arises as π₀ of a K-theory spectrum, K(Var_k). Using her formalism of assemblers, Zahkarevich showed that this is in fact the case. I'll present an alternate construction of the spectrum that allows us to quickly see the E_∞-structure on K(Var_k) and produce various character maps out of K(Var_k). I'll end with a conjecture about K(Var_k) and iterated K-theory.

Title: My preferred proof of the Lefschetz fixed point theorem 

Abstract: There are many different proofs of the Lefschetz fixed point theorem.  The most familiar approach uses simplicial approximation and is often a first example of the power of simplicial homology.  I'll talk about a very different proof that I find much more useful.  This proof requires more input, but it generalizes easily.