Homotopy Theory and its Applications

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9977089Goerss Throughout different fields of mathematics, it is important tounderstand how various objects acquire some sort of multiplication.For example, in homotopy theory, which is the study of mathematicalphenomena that remain invariant under continuous deformation, onewants to decide if an object has a multiplication that is commutativeup to all ``higher homotopies'' -- this is the type of multiplicationthat is invariant under deformation. One way to discuss this issue iswith the device of opera, which make rigorous the phrase ``higherhomotopies.'' Goerss intends to develop and explore these ideas inthe context of homotopy theory; for example, he would like to discusswhen a spectrum has an operad multiplication, and to discuss theimplications of the fact that the cochains on a topological space havesuch multiplications. The broad goal of this project is to further understanding ofhomotopy theory, the branch of topology concerned with properties ofhigher-dimensional surfaces and other geometric objects that remaininvariant under continuous deformation. There are some surprisingand rather mysterious connections between homotopy theory and seeminglyunrelated fields of mathematics such as number theory and algebraicgeometry, and one hopes the project will ultimately illuminate these.***