Teaching

Current Courses

• MATH 8301 – Algebraic Topology Fall 2024: Classification of compact surfaces, fundamental group/covering spaces. Homology group, basic cohomology. Application to degree of a map, invariance of domain/dimension.

Selected Past Courses

• Math 237 - Kan Seminar Winter 2021 (At UCLA): Students engage with classical texts in algebraic topology, presenting the major results and discussing them.

• Math 19 - Patterns and Symmetry in Art and Nature Winter 2020 (At UCLA): This Fiat Lux course looks at the ubiquity of patterns and symmetry in art and nature.

• Math 237 - Spectral Algebraic Geometry Fall 2020 (At UCLA): This topics course serves as an overview of spectral algebraic geometry.

• Math 121 - Topology Spring 2016 (At UCLA): This course introduces the foundations of point-set topology. Course materials can be found at the course website.

• Math 5651 - Advanced Linear Algebra (At UVA): A rigorous treatment of linear algebra, usually over an arbitrary base field. The course website includes homework and handouts.

• Math 885 - Computational Methods in Algebraic Topology (At UVA): 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.

Talks & Write-ups

Namboodiri Lectures: Evenness in Algebraic Topology | Talk 3 • Talk 2 • Talk 1

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

On the Non-existence of elements of Kervaire invariant one • PDF

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

G-Symmetric Monoidal Categories and Commutative Algebras • PDF

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.

Localizations of Equivariant Commutative Ring Spectra • MFO Report

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.

The slice filtration • Video

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.