Algebraic topology is the study of topological spaces via algebraic methods. The principal investigator will study various naturally-occurring topological objects from the perspective of algebraic geometry. A fundamental goal of the project is to place mathematical concepts in reach of computational methods. The investigator plans to use recent theoretical advances in algebraic geometry and topology in order to develop calculational tools to explore a relationship between quantum field theory and the elusive notion of elliptic object.
Generalized cohomology theories are arguably the most useful and important tool in modern algebraic topology. In addition to ordinary cohomology, examples such as K-theory, elliptic cohomology, and complex cobordism admit surprisingly close connections to algebraic geometry via the theory of formal groups. While the formal groups associated to ordinary cohomology and K-theory are additive and multiplicative, respectively, elliptic curves carry much more complicated group structures (in fact, their relative intractability has been exploited in real world applications such as public key cryptography). The advantage of formal groups that arise from global geometric objects, such as the multiplicative group or elliptic curves, is that their corresponding cohomology theories are very highly structured. For instance, K-theory is structurally similar to ordinary representation theory, whereas the analogous "elliptic representation theory," while related to diverse fields such as arithmetic geometry and mathematical physics, remains a mystery. Important work over the past few decades has led to a construction of a universal elliptic cohomology theory, a topological refinement of the classical theory of modular forms. Exploiting its structure allows one to associate to a topological stack an algebro-geometric object over the moduli stack of elliptic curves. While the derived ring of functions on this object is technically the elliptic cohomology of the original topological object, elliptic curves are not affine objects, meaning that this passage from geometry to algebra loses significant information. The more fundamental structure is that which exists on the algebro-geometric level itself, before affinization, and one retains considerably more conceptual and calculational control by manipulating these objects directly. An interesting twist is that, while there is no a priori understanding of elliptic cohomology classes (in stark contrast to K-theory, where cocycles correspond to formal differences of vector bundles), it may be possible to gain insight into this fundamentally important problem by interpreting calculations in several key examples.
