What is a Good Learning Roadmap for Topological Modular Forms?

Starting your journey into the world of topological modular forms (TMFs) can feel overwhelming, but with a strategic learning roadmap, you can navigate through the complex web of algebraic topology and homotopy theory. This guide is tailored for individuals who already possess a solid PhD-level background in algebraic topology, including an understanding of stable homotopy theory, structured ring spectra, the Adams-Novikov spectral sequence, and basic knowledge of model categories and even infinity categories. Here’s a comprehensive roadmap to get you started on your journey.

Step 1: Refine Your Algebraic Topology Infrastructure

Before delving into TMFs, it is crucial to solidify your understanding of the foundational concepts in algebraic topology. This involves revisiting and deepening your knowledge of stable homotopy theory, structured ring spectra, and the Adams-Novikov spectral sequence. Familiarizing yourself with these tools will be essential as you progress to more advanced topics.

Step 2: Dive into Derived Algebraic Geometry

Next, you need to learn about derived algebraic geometry, which provides a framework for studying moduli problems in a homotopical setting. Key concepts to explore include:

Derived stacks and their applications to moduli problems Obstruction theory for E-infinity ring structures Higher category theory, focusing on infinity categories Derived de Rham cohomology and derived algebraic geometry

Books such as Derived Algebraic Geometry by Jacob Lurie and From Maps to Modules: A Systematic Review of Modern Homotopy Theory by John Moore can be valuable resources. Additionally, online courses and lecture notes from leading mathematicians can provide a more practical and interactive learning experience.

Step 3: Explore the Construction and Calculations of Topological Modular Forms

Once you have a solid foundation in derived algebraic geometry and related areas, you can begin to focus on the construction and calculations of TMFs. The groundbreaking work on TMFs often involves intricate spectral sequences and advanced homotopical techniques. Key concepts to explore include:

Construction of TMF via the complex cobordism spectrum and the chromatic filtration Understanding the role of elliptic curves and their relation to TMFs Calculations using the Adams-Novikov spectral sequence and other advanced spectral sequences Applications of TMFs to the study of stable homotopy groups

For detailed information, consult the following references:

Cohomology Operations and Applications in Homotopy Theory by Robert E. Mosher and Martin C. Tangora Algebraic Topology: Homotopy and Homology by Robert M. Switzer Lecture Notes on TMFs and Their Applications by Mark Behrens and Pete Bousfield

Additional Resources and Further Reading

For a comprehensive understanding of TMFs, you should also explore the following resources:

The Topology Wiki (_Page) provides a wealth of information and references on a wide range of topics in algebraic topology and homotopy theory.

Books and lecture notes by leading experts in the field, such as Jacob Lurie's Higher Topos Theory and his work on Derived Algebraic Geometry.

Research papers and preprints available on arXiv () that delve into specific aspects of TMFs and their applications.

Conclusion

By following this detailed learning roadmap, you can build a strong foundation in the necessary concepts of algebraic topology and homotopy theory, and ultimately gain a deep understanding of topological modular forms. This journey requires dedication and a blend of theoretical and practical knowledge, but the rewards are significant in both theoretical insights and potential applications.

Keywords

Topological Modular Forms Algebraic Topology Homotopy Theory Derived Algebraic Geometry