Algebraic topology and modular forms

Abstract

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.

References in corpus (1)

• Vanishing lines in generalized Adams spectral sequences are generic

Cited by in corpus (39)

• Galois extensions of structured ring spectra
• Units of ring spectra, orientations, and Thom spectra via rigid infinite loop space theory
• Units of ring spectra and Thom spectra
• Holomorphic SCFTs with small index
• The character map in (twisted differential) non-abelian cohomology
• Norm coherence for descent of level structures on formal deformations
• A p-adic Eisenstein measure for unitary groups
• Elliptic genera of Landau-Ginzburg models over nontrivial spaces
• On a ${\mathbb{Z}}_3$-valued discrete topological term in 10d heterotic string theories
• p-adic q-expansion principles on unitary Shimura varieties
• A p-adic Eisenstein measure for vector-weight automorphic forms
• Differential operators and families of automorphic forms on unitary groups of arbitrary signature
• The homotopy groups of the spectrum Tmf
• On the high-dimensional geography problem
• Topological modular forms (aftern Hopkins, Miller, and Lurie)
• Geometric representations of the formal affine Hecke algebra
• Cubic forms, anomaly cancellation and modularity
• A model for complex analytic equivariant elliptic cohomology from quantum field theory
• The Witten genus and vertex algebras
• Supersymmetric field theories and cohomology
• Analogs of Dirichlet $L$-functions in chromatic homotopy theory
• Equivariant stable stems for prime order groups
• On characteristic numbers of $24$ dimensional String manifolds
• T-Duality, Jacobi Forms and Witten Gerbe Modules
• Higher topological cyclic homology and the Segal conjecture for tori
• The Ando-Hopkins-Rezk orientation is surjective
• The Cayley Plane and the Witten Genus
• $TM{F}_0(3)$ Characteristic classes for String bundles
• Realizing Families of Landweber Exact Homology Theories
• An introduction to Eisenstein measures
• tmf Is Not a Ring Spectrum Quotient of String Bordism
• The Pseudo-hyperresolution and Applications
• OP2 bundles in M-theory
• Brave new local moduli for ordinary K3 surfaces
• The Adams-Novikov E_2-term for Behrens' spectrum Q(2) at the prime 3
• On the image of the unstable Boardman map
• String${}^{\mathbf{c}}$ Structures and Modular Invariants
• Cannibalistic Classes of String Bundles
• Mod 3 Congruence and Twisted Signature of 24 Dimensional String Manifolds