The PI, Dev Sinha, proposes to investigate a wide range of topics in algebraic topology. He will make further calculations in the cohomology of symmetric groups. He will compute equivariant cohomology and K-theory of divided powers constructions and then extend them to orbifold cohomology and K-theory. He will develop and compute Hopf ring structures on cohomology, representation theory and suitable invariants of series of groups. He will unify integrals from Chern-Simons with the Lie coalgebraic model of homotopy theory to develop complete rational homotopy invariants of maps. He will generalize the Magnus expansion and use that to model non-simply connected spaces.
Topology is a fundamental study of shape, and thus has its roots in geometry. As a subject it has roughly split into point-set topology which considers foundational questions, geometric topology which studies particular shapes such as that of our universe, and algebraic topology which ultimately relates shape to numerical data. It is not typical for researchers to bridge thees communities. In his previously funded research, the PI applied methods from algebraic topology to knot theory, which is squarely in the geometric realm. In this proposal, the PI is using insight from the geometric study of configuration spaces (collections of particles) to better understand algebraic structures such as symmetric groups and symmetric functions. He also plans to connect algebraic topology with Chern-Simons theory from mathematical physics, and to develop a theory of "linking of letters" to answer basic questions in group theory.
The project outcomes revolved around better connecting algebraic and geometric reasoning. In algebraic topology, we study shapes - geometric objects - by finding numbers and other algebraic structures which help describe features. These features are meant to be robust under deformation, so geometric angle is not such a feature because angles can be slightly bent further or less, but how many components (parts) of a shape is because moving those parts by a sufficiently small amount doesn't increase or decrease the number of parts. This robustness has led to the field of topology being critical to new appoaches to data analysis and machine learning (aka "big data"). Algebraic topology has developed in more and more specialized ways, which is often more formal and less geometric. In this project, we brought new basic geometry to bear on questions central to algebraic topology. And we used algebraic topology for geometic questions in new ways. A central organizing theme was the rich structure of configuration spaces, which catalog all possible collections of particles in a box or other geometric shape. By further developing configuration spaces, we make their structre more accessible for physicists, neuroscientists and others who may want to use them for mathematical modeling. The connection between geometry and algebra is a key component of mathematical training - for postdocs and graduate students and then down through kindergarten. At the higher levels, it is key to carefully bring in geometric models which can point out new structures and directions for inquiry. Formal argument serves well to establish results but less well in pointing out where new results will be. At K-12, geometric underpinnings as a source for numerical insight is a key ingredient of the Common Core State Standards in Mathematics. For example, students understand addition of fractions on a numer line, or mulitplication of numbers or algebraic patterns through geometric arrays. The PI has been supporting implementation of these standards at the national, state and local levels. The consistent interplay of geometry and algebra throughout K-20 education strengthens efforts at all levels.
