K-theoretic invariants for Lagrangian and Legendrian submanifolds via parametrized Morse theory

K-theoretic invariants for Lagrangian and Legendrian submanifolds via parametrized Morse theory

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Daniel Álvarez-Gavela, IAS
Fine Hall 214

The persistence theorem for generating families provides a deep connection between stable parametrized Morse theory and Lagrangian submanifolds of cotangent bundles / Legendrian submanifolds of 1-jet spaces. So far the bulk of the literature has been devoted to extracting homological information from this connection. However, all of Waldhausen's algebraic K-theory of spaces can be seen through the lens of stable parametrized Morse theory and hence much more subtle information should be available. In this talk we will explain how to produce K-theoretic invariants for Lagrangian and Legendrian submanifolds using generating families. For K_1 and K_2 this story is due to Y. Eliashberg and M. Gromov and can also be understood in terms of pseudo-holomorphic curves by work of M. Sullivan. For K_3 this story is current work in progress joint with K. Igusa and we obtain examples which so far do not have an obvious interpretation in terms of pseudo-holomorphic curves, although we have some guesses. For higher K-groups and A-theory the story is still unclear but there are some natural hopes and dreams which we will speculate about, time permitting.