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:
- Basic notions: the compact-open topology continuous, piecewise linear, smooth, H^1, Moore-loops, fibrations
2. Loop spaces in algebraic topology I: long exact sequence in homotopy groups, classifying spaces, Serre spectral sequence
3. Loop spaces in algebraic topology II: operads, iterated loop spaces, the recognition principle, infinite loop spaces and stable homotopy theory
4. Combinatorial models for loop spaces: simplicial and cubical models, Kan loop group/classifying space adjunction, polytopes inspired by loop spaces
5. Loop spaces and algebra: Hochschild an cyclic homology of algebras and coalgebras and their relevance to loop spaces, Chen’ iterated integrals, the signature of a path
6. String topology: structure, significance, and computation of geometric intersection type operations on loop spaces of manifolds
7. Geometry of loop spaces (time permitting)
- infinite dimensional manifolds, Riemannian metrics, length and energy functional, Morse theory
- closed geodesics, the Gromoll-Mayer Theorem, Bott’s iteration of the index formulas, Vigué-Poirrier-Sullivan Theorem
- relationship between loop space homology and the symplectic homology of the cotangent bundle
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