Homotopy Theory and its Applications

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Homotopy Theory is a branch of pure mathematics which uses algebraic invariants (such as the dimension) to tell two geometric objects apart. In recent years, tools coming from Homotopy Theory have been applied in other areas, both within mathematics (for instance, in algebra) and outside, for example, in mathematical physics and theoretical computer science. This Discovery Grant proposal explores several such applications and helps develop the general theory. In brief, the proposed research falls under four main themes.******The first of these themes is Higher Category Theory, which aims to establish the general framework in which one can talk about homotopy theory, thus making the theory applicable to other areas. In this proposal, we explore possibilities of reshaping this framework in ways oriented towards computations and new applications, for instance in knot theory and geometric representation theory. The second theme, Homotopy Type Theory, investigates a newly discovered connection between homotopy theory and type theory, a logical system studied in theoretical computer science. This connection allows one to use dependent type theory to prove results in homotopy theory, while also use theorems from homotopy theory to suggest new principles of logic (such as Voevodsky's Univalence Axiom). Dependent type theories were previously studied due to their suitability for large scale computer formalization (and are currently used by many major corporations, including Intel and Toyota) and we can therefore use homotopy theory to enhance the existing software. The objectives of the third theme, Formalization of Mathematics, examine the resulting tools, as we will use them to formalize several results which had previously proven difficult. Finally, in the fourth theme, Cryptography, we will study applications of homotopy theory to cryptography. Specifically, we will attempt to use a particular algebraic invariant, the cohomology ring of a variety, to construct examples of cryptographically useful multilinear maps. Many applications of cryptographic multilinear maps, including broadcast encryption, internet voting, and indistiguishability obfuscation, have been known for the past 15 years, yet no one was able to construct an example of such a map.******Many of the problems proposed here are suitable for students at different levels and equip them with the skills and experience that can be used both in their further academic work and in industry. Many of our specific objectives involve collaboration between mathematicians and knowledge users, including software engineers and, for example, broadcast companies. Altogether the proposal takes techniques central to pure mathematics and investigates their applications outside this realm.