# Advanced Courses

## The \infty-categorical point of view for Thom spectra

*By Omar Antolín, Universidad Nacional Autónoma de México.Language: English.*

I will describe the \infty-categorical approach developed by Ando, Blumberg, Gepner, Hopkins, and Rezk to Thom spectra of stable spherical fibrations. I will also mention joint work with Tobias Barthel that uses this approach to study the multiplicative structure of such spectra. This multiplicative structure is characterized by a universal property, which is more useful than might seem given its simplicity. As an application we will discuss E_{n}-orientations and the Thom isomorphism.

I won’t assume familiarity with either \infty-categories or with E_n-ring spectra; on the contrary, I hope this mini course will serve as an introduction to these topics.

## Foundations of homological algebra applied to modules

*By Sean Sather-Wagstaff, Clemson University.Language: English.*

Each module M over a commutative local Noetherian ring R comes equipped with certain numerical invariants, defined homologically, that allow us to detect certain structural information about the module. For example, the projective dimension detects how close M is to being projective. Not only that, but projective dimension of modules can detect whether a ring is regular, as Auslander, Buchsbaum, and Serre proved in the 1950s, and this led to the solution of a famous open question about the localization properties for regular rings. In this course, I will discuss some of these ideas (there are many out there) focusing on a few favorites, including foundational topics.

Bibliography: *Abstract Algebra*, 3ed; Dummit and Foote (sections 7.1-9.5, 10.1-11.4,12.1, 13.1-13.2, and pp. 656-657, 706-709, 714-715, 718-719. *Algebra*; Hungerford, Algebra (Chapters 1-2 and sections V.1, VII.1, VIII.1, and Theorem VIII.4.9). *Introduction to Commutative Algebra*; Atiyah and MacDonald (Chapters 1-3 and 6).

## Algebraic K-Theory and connections to Topology

*By Mohamed Elhamdadi, University of South Florida.Language: English.*

We will give a foundational course on algebraic K-Theory. We will introduce the K-groups starting from the Grothendieck group K_{0} to higher K-groups including the Plus-construction of Quillen. Connections to Hochschild and cyclic homology will be given. If time allows we will discuss a variation of K-theory called Karoubi’s K-theory.

Biblography: *The K-book. An introduction to algebraic K-theory*; Weibel, Charles. Cyclic homology; Loday, Jean-Louis. A*lgebraic K-theory and its applications*; Rosenberg, Jonathan. *Approach to algebraic K-theory*. Research Notes in Mathematics, 56; Berrick, A. Jon.