Student Topology Seminar

University of Kentucky Student Topology Learning Seminar

We are meeting (roughly) biweekly on Wednesday 15:15-16:15 at POT 715 (the fish bowl) usually, except Mar 11 and Apr 15 will be at POT 945 due to the grading parties

Please find a rough description of the topics below, titles and abstracts as well as notes will be updated.

Jan 22 Ang Li Intro and Background
Not that deep into topology yet? Not a thing! We will begin with homotopy, suspension and loop, then cohomology theories, then spectra, and a little bit category theory as a language we will use. The Note for this is here
Feb 5 Joe Cummings and Michael Morrow Operads and Associativity: an introduction by non-experts
Have you ever taken two isomorphic vector spaces and thought, "If I only had an associative algebraic structure on one of them, it would be sooooo easy to transfer that structure to the other"? If so, maybe you tried to take it one step further, and tried it with chain complexes which are only homotopic. In the first part of our talk, we will introduce how one can transfer an associative algebraic structure up to homotopy. Then one is naturally lead towards to operads, so we introduce operads, with numerous examples (from very trivial to the mother of all operads). At the end of the talk, we will mention Stasheff's associahedra and how it relates to operads and associativity.
Feb 19 Justin Barhite Operads and Commutativity
We will discuss the notions of an operad and an algebra over an operad, focusing in particular on topological operads. We will look at many examples (with an abundance of pictures), including the commutative operad, the little k-disks operad, and the linear isometries operad.
Mar 11 Kaelin Cook Powell Cyclic operads and more
This week we will focus on surveying several other types of operads coming from certain algebraic and geometric constructions, as well as an associated algebra over one of these operads. These constructions are meant to generalize the notion of an operad encoding operations. For example, a cyclic operad (probably the main focus) is something you obtain if you don't want to distinguish between inputs and outputs.
Apr 1 Carissa Slone Loop spaces, Bar construction
Ref: Dylan Wilson’s Talbot presemi note, available at here
J.F.Adams Infinite loop spaces, available at the library: here
Apr 15 All members Summarize and Diminish Confusions
Leave empty for members' interests, and unsolved history problems, confusion, etc.