Algebraic topology and modular forms

By M. Hopkins. 2002

147 citations recorded by the source corpus.

Modular forms appear in many facets of mathematics, and have played important roles in geometry, mathematical physics, number theory, representation theory, topology, and other areas. Around 1994, motivated by technical issues in homotopy theory, Mark Mahowald, Haynes Miller and I constructed a topological refinement of modular forms, which we call {\em topological modular forms}. At the Zurich ICM I sketched a program designed to relate topological modular forms to invariants of manifolds, homotopy groups of spheres, and ordinary modular forms. This program has recently been completed and new directions have emerged. In this talk I will describe this recent work and how it informs our understanding of both algebraic topology and modular forms.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/f9d49923e4af1527a86473a7ebe51360cea6bfb5

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Citations in the corpus

Central extensions of smooth 2–groups and a finite-dimensional string 2–group (2009, unknown): “The Witten genus has a refinement as a map of cohomology theories (see Ando, Hopkins and Strickland [2], Ando, Hopkins and Rezk [1] and Henriques [24]): MOh8i ! tmf : Here tmf is the theory constructed by Hopkins and Miller of topological modular forms [26; 27].”
The units of a ring spectrum and a logarithmic cohomology operation (2004, background): “ions, including the exterior power operations λk and the Adams operations ψk. Other theories of interest include some elliptic cohomology theories, including the spectrum of topological modular forms [Hop02]; bordism theories, including the spectrum MUof complex bordism. 1.3. Formal groups and isoge”
Derived Algebraic Geometry (2014, background): “The notion of topological modular forms lies at the interface between stable homotopy theory, algebraic geometry and number theory (see [68]).”
The first stable homotopy groups of motivic spheres (2016, methodology): “ion). Decades of research in algebraic topology have lead to beautiful mathematical structures unearthing the notoriously diﬃcult problem of classifying maps between spheres up to homotopy, see e.g., [17]. Our approach to the corresponding algebro-geometric problem employs the slice spectral sequenc”
Twists of K-theory and TMF (2010, background): “ite loop spaces K(Z,3) −→BString →BSpin −→λ K(Z,4). 23 Passing to Thom spectra, we get maps of commutative S-algebras Σ∞ + K(Z,3) →MString →MSpin. Let tmf be the spectrum of topological modular forms [Hop02]. Ando, Hopkins, and Rezk [AHR] have produced a map of commutative S-algebras MString −→σ tmf”
Units of ring spectra, orientations, and Thom spectra via rigid infinite loop space theory (2014, unknown): “As explained by Hopkins in his ICM address [14], the argument requires a new formulation of the obstruction theory for orientations of [27] in terms of the adjoint relationship between the units of a commutative ring spectrum and Σ+ Ω ∞.”