The third semester algebraic topology course, covering spectral sequences, K-theory, bordism, and stable homotopy theory.

This course will introduce you to quantitative literacy, critical thinking, and problem solving skills in socially relevant contexts. While students may be accustomed to thinking about mathematics as an abstract set of principles and proofs, this course will encourage thinking about mathematics in concrete contexts as a way to improve our communities and the world. Students will develop the ability and inclination to understand and develop realistic mathematical approaches to social, political, and economic issues. Examples of specific topics include: sea level change in an island community, student loans, and voting systems. The mathematical tools used will include basic statistics, modeling, and data analysis, among others. While most students in this class will be “good at math,” this class explores using math to do good and how some well-intentioned uses of math can have surprising implications.

Algebraic Topology II. This course covers cohomology, Poincare duality, homotopy groups, the Serre spectral sequence, and the basics of stable homotopy. Last updated: Fall 2017.

This Fiat Lux course looked at the ubiquity of patterns and symmetry in art and nature.

This course introduces the foundations of point-set topology. Course materials can be found at the course website.

A rigorous treatment of linear algebra, usually over an arbitrary base field. The course website includes homework and handouts.

This course is a self-contained introduction to spectral sequences with an emphasis on the spectral sequences in algebraic topology. The course website includes notes, homework sets, spectral sequence pictures, and some podcast classes.

I gave the 2017 Namboodiri Lectures at the University of Chicago.

• Grassmanians, Thom spectra, and Wilson spaces: classical constructions and $C_2$-equivariant analogs
• Extending to larger groups: the norm, $G$-equivariant Wilson spaces, and the equivariant Steenrod algebra
• Towards $RO(G)$-graded algebraic geometry: explorations of duality for Galois covers via equivariant homotopy

This is my ICM talk on my solution with Hopkins and Ravenel to the Kervaire invariant one problem.

This talk is about the evolving notion of a G-symmetric monoidal ctegory. basic properties are discussed, grounded in genuine equivariant spectra. At the end, several algebraic examples are presented.

This talk discusses joint work with Hopkins on localization of commutative rings. In particular, it sketches the proof of when localization preserves commutative ring objects in spectra.

This talk is my discussion of the slice filtration and its generalizations at the Hot Topics workshop for the Kervaire Invariant One problem at MSRI.