Algebraic Topology – Mathematical Association of America

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Complement (bottom-up) learning with top-down activities to develop mathematical thinking ability

By Keith Devlin @KeithDevlin@fediscience.org, @profkeithdevlin.bsky.social Sometime in the early 2000s, my primary care physician offered me the option of no longer having an annual health checkup in person. Rather, I could just call into my nearest walk-in testing lab and have a blood draw, followed up a few days later by a quick video-chat. (They also wanted...

Leveraging Data About Math Departments

By Dave Kung When I was department chair at St. Mary’s College of Maryland, we carefully tracked data about how we were serving students. Regularly updated spreadsheets showed how many students were in each course we offered and we used them to look for enrollment patterns and adjust our offerings as needed. We were particularly...

Differential and Low-Dimensional Topology

Differential and Low-Dimensional Topology by András Juhász is a useful guidebook written for aspiring topologists making the transition from “elementary” algebraic topology to the research frontier. A student of topology in the 21st-century begins with algebraic invariants: homology and cohomology, the fundamental group, and higher homotopy. With these tools as the foundation, a student proceeds to...

Topology and K-Theory: Lectures by Daniel Quillen

The book under review contains notes from a graduate student course on algebraic topology and K-theory taught by Daniel Quillen at the Massachusetts Institute of Technology during 1979-1980. Robert Penner does a fantastic job capturing the great mind of Quillen and preserving his work for future generations of mathematicians. The writing is true to Quillen's...

Algebraic Topology

There are several books aimed at undergraduate topology, including some focused on algebraic topology. For example, Algebraic Topology, An Intuitive Approach by Sato, and Topology Through Inquiry by Starbird and Su have been reviewed here. Does this book add to the literature? Yes, it does. Bray, Butscher, and Rubinstein-Salzedo’s volume Algebraic Topology is an outgrowth of a long-standing summer program at...

The Adams Spectral Sequence for Topological Modular Forms

One of the central problems of stable homotopy theory is the computation of the stable homotopy groups of spheres. This is a very hard problem for which there is no complete solution in sight. One way to formulate the problem is to define a stable homotopy category and to construct the "sphere spectrum" in this...

Geometry and Topology of Manifolds: Surfaces and Beyond

Geometry and Topology of Manifolds: Surfaces and Beyond is a self-descriptively titled textbook and reference. Though definitions are usually given for manifolds of arbitrary finite dimension, proofs and examples concentrate mostly on real two-manifolds. The book should be well-suited for several types of graduate-level geometry course: The material encompasses manifolds and categories; the basics of homotopy, singular homology, and de Rham cohomology; Riemannian metrics,...

Organized Collapse: An Introduction to Discrete Morse Theory

Discrete Morse Theory is the study of discrete spaces (simplicial complexes, polyhedral complexes, CW complexes) through the use of a Morse function defined on the space. In the conceptual setup of classical Morse theory, gradient flows on a manifold equipped with a differentiable function allows the simplification of the manifold to a CW-complex determined by critical points of the Morse...

Cover of the book "Organized Collapse: An Introduction to Discrete Morse Theory" by Dmitry N. Kozlov. The design is a minimalist teal background with a wide gold stripe in the middle. The title appears in yellow text over a dark teal box in the top left corner. In the bottom left corner are the words "AMS" and "American Mathematical Society" in dark blue font.