Talk Abstracts and Slides

June 2: Introduction to this Summer’s Topics, Dora Woodruff

In my talk, I’ll mostly be giving an introduction to this summer’s topics. We’ll start by going over some starting definitions and relevant theorems, such as the Freudenthal Suspension Theorem, to make sure we’re all on the same page and to motivate the idea of stable homotopy. I’ll then try to give some intuition as to what the stable homotopy category is and why it’s natural and helpful to use it. Finally, I’ll discuss the Nilpotence Theorem, including its various different formulations and an application. Hopefully, there will be a good amount of time for questions and discussion.

June 9: Spectra and Cohomology Theories, Keita Allen

Last week, we were introduced to the stable homotopy category as “a nice place to do stable homotopy theory.” This week, I’d like to explore this in more detail; we’ll go make a definition of the category, and highlight some of the properties which make it so nice.

June 16: Bundles, Bordism, and Thom Spectra, Jonathan Buchanan

This week, we will introduce fiber bundles, principal \(G\)-bundles, classifying spaces, and vector bundles, with an interlude on representable functors. Then, we will discuss bordism and the Thom spectra, as well as computational results.

June 23: Bousfield Localization, Merrick Cai

This week, I’ll review and elaborate on some past topics such as Spanier-Whitehead duality, ring spectra, and module spectra, as well as universal coefficient theorem. Then I’ll discuss Bousfield localization, specifically applied to (co)homology theories (i.e. spectra), and basic results in that direction.

June 30: Steenrod Algebra and the Adams Spectral Sequence, Kenta Suzuki

This week, I’ll discuss the Steenrod Algebra and how it relates to stable cohomology operations. Cohomology groups can be given the structure of a module over the Steenrod algebra, which can be helpful in understanding cohomology better. As an example, I will discuss the Adams spectral sequence, which provides “approximations” to stable homotopy groups of spheres.

July 7: Formal Group Laws, Charley Hutchinson

This week, I’ll introduce formal group laws from the perspective of an analogue of Chern classes for certain generalized cohomology theories. I will discuss the relation between formal group laws and the complex cobordism spectrum \(MU\), and then will introduced the chromatic filtration of the \(p\)-local stable homotopy category.

July 14: The Ravenel Conjectures and the Chromatic Decomposition, Rushil Mallarapu

We’ll start putting together the machinery we’ve set up and discuss the Ravenel conjectures, which provide a global picture of the stable homotopy category. I’ll introduce Bousfield equivalence, talk about the Brown-Peterson spectrum and its relatives, set up the chromatic filtration, and walk through the conjectures and the story they tell. Along the way, I’ll include some vignettes relating this picture to the algebraic phenomena in the Adams-Novikov spectral sequence that inspired these conjectures.

August 11: The Proof of the Nilpotence Theorem, Rushil Mallarapu

We’re finally ready to prove the nilpotence theorem that we introduced all the way back in June! After discussing the various equivalent forms of the theorem and some key reductions down to a slightly simpler setup, we’ll walk through the core steps of the proof, roughly following Devinatz-Hopkins-Smith. Through this, I’ll discuss the various technical elements – from vanishing lines in the AdSSeq over various Hopf algebroids to the myriad constructions DHS does with Thom spectra – and try to illuminate the key ideas and insights behind these steps. Finally, I’ll discuss some implications of the nilpotence theorem and its role in chromatic homotopy theory.