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The term "higher structure" refers to a phenomenon in which natural algebraic identities hold only "up to homotopy". For example, in
topology, addition in the fundamental group is associative thanks to a homotopy that reparametrizes the domain when concatenating
three loops. This is a special case of a product operation on the chain complex of a based loop space (or analogously on the free loop space)
being only associative up to homotopy; the higher structure mainly comes from the domain reparametrization. Aside from this
reparametrization issue, the operation of cutting up loops is strictly co-associative on the nose, but to make things parametrized by
manifolds feasible for a homology theory, one has to achieve transversality, which makes the co-associativity identity hold only
up to homotopy. Higher structures can also naturally arise in moduli spaces with corners for elliptic geometric PDEs: for example, the
Floer cohomology ring acting on a Fukaya algebra at the chain level should be an E_2-algebra acting on a cyclic A-infinity algebra. The
combinatorial objects underlying the infinity-structures are also very interesting: these include Stasheff associahedra, permutohedra and
secondary polytopes.
Higher structures are much richer than numerical invariants and can in themselves be regarded as geometric objects, which are sometimes even
conjectured to faithfully represent the original geometric objects, e.g. in mirror symmetry and noncommutative geometry. At a more basic
level, they provide a framework for understanding what invariant information one has captured, and for parametrizing the inductive
constructions one hopes to carry out. Understanding and constructing higher structures is an active field of research and under rapid
development. I hope to cover some aspects of the above story.
The following will provide some concrete topics and key words: I hope to cover the classical higher Massey product (as a better way to
capture the ring structure on cohomology), and its "quantum" analogue in the Fukaya A-infinity algebra/category, the latter being an
important symplectic invariant that provides bridges to other vastly different subjects. As an A-infinity structure is the first basic
infinity structure, I will provide various viewpoints on it. I hope to cover the interaction of product and coproduct in the involutive
bi-Lie infinity algebra of Cieliebak-Fukaya-Latschev, which connects string topology with symplectic field theory and the higher genus
Fukaya category (and all three pictures can co-exist). Supplementary to this concrete description, I also hope to cover the recent
conceptual treatment by Campos-Merkulov-Willwacher of a similar topic: the Frobenius properad satisfies Koszul duality. As a last topic, when
a symplectic manifold is Kähler and equipped with a holomorphic Morse function, there is also a deep conjectural infinity-structure
uncovered by physicists Gaiotto-Moore-Witten and mathematicians Kapranov-Kontsevich-Soibelman on a Fukaya category.
I will provide background motivations and necessary definitions gradually, with useful results carefully stated and packaged into
independent units. Prerequisites should include Differential Geometry I+II and Topology I+II, but will otherwise be consciously kept to a
minimum, aside from curiosity and a certain willingness to favor “global and scenery pictures” over “complete details from axioms” (the
latter can still be furnished for motivated individuals with guided follow-up reading). The knowledge assumed in each lecture will be
summarized at the beginning of that lecture. Hopefully, the course should appeal to an audience with a variety of different interests and
tastes in geometry, topology and/or algebra. The examination format will be flexible, e.g, it may consist in explaining or following up on
a specific topic or result from the lecture. |
