MA 59500TGA, Spring 2026, Prof. Manuel Rivera, Purdue University
Spaces of loops, paths, and strings in a background geometric space are ubiquitous across mathematics and physics. This course will explore both classical results and modern research directions concerning the structure of loop spaces, with an emphasis on their broad relevance to topology, geometry, algebra, and mathematical physics. The course is open to advanced undergraduates, graduates, faculty, and anyone with basic knowledge of algebraic topology and differential geometry.
While the exact trajectory will depend on the interests of participants, possible topics include:
1) The topology of loop spaces: continuous, piece-wise linear, smooth, H^1-loops, the compact-open topology, fibrations
2) The algebraic topology of loop spaces I: singular and simplicial (co)homology, Serre spectral sequence, homotopy groups, loop spaces and classifying spaces
3) The algebraic topology of loop spaces II: operads, iterated loop spaces, and recognition principle
4) Combinatorial models for loop spaces: simplicial and cubical constructions, polytopes inspired by loop spaces
5) Loop spaces and algebra: Hochschild an cyclic homology of algebras and coalgebras and their relevance to loop spaces in topology and geometry, Chen’ iterated integrals, the signature of a path
6) The geometry of loop spaces I: infinite dimensional manifolds, Riemannian metrics, length and energy functional, Morse theory
7) The geometry of loop spaces II: closed geodesics, the Gromoll-Mayer Theorem, Bott’s iteration of the index formulas, Vigué-Poirrier-Sullivan Theorem
8) The geometry of loop spaces III: quantitative topology
9) Loop spaces and symplectic topology: relationship between loop space homology and the symplectic homology of the cotangent bundle
10) String topology: structure, meaning, and computation of operations on loop spaces constructed through intersection theory
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