Two Brandeis professors have been awarded highly prestigious and competitive Simons Fellowships for 2017. Daniel Ruberman received a 2017 Simons Fellowship in Mathematics. Matthew Headrick was awarded a 2017 Simons Fellowship in Theoretical Physics. This is the second of two articles where each recipient describes their award-winning research.

Daniel Ruberman’s research asks “What is the large-scale structure of our world?” Einstein’s unification of physical space and time tells us that the universe is fundamentally 4-dimensional. Paradoxically, the large-scale structure, or topology, of 4-dimensional spaces, is much less understood than the topology in other dimensions. Surfaces (2-dimensional spaces) are completely classified, and the study of 3-dimensional spaces is largely dominated by geometry. In contrast, problems about spaces of dimension greater than 4 are translated, using the technique called surgery theory, into the abstract questions of algebra.

Ruberman will work on several projects studying the large-scale topology of 4-dimensional spaces. His work combines geometric techniques with the study of partial differential equations arising in physics. One major project, with Nikolai Saveliev (Miami) is to test a prediction of the high-dimensional surgery theory, that there should be `exotic’ manifolds that resemble a product of a circle and a 3-dimensional sphere. The proposed method, which would show that this prediction is incorrect, is to compare numerical invariants derived from the solutions to the Yang-Mills and Seiberg-Witten equations, by embedding both in a more complicated master equation. The study of the Seiberg-Witten invariants is complicated by their instability with respect to varying geometric parameters in the theory. A key step in their analysis is the introduction of the notion of end-periodic spectral flow, which compensates for that instability, as illustrated below.

Other projects for the year will apply techniques from 4-dimensional topology to classical problems of combinatorics and geometry about configurations of lines in projective space. In recent years, combinatorial methods have been used to decide if a specified incidence relation between certain objects (“lines”) and other objects (“points”) can be realized by actual points and lines in a projective plane. For the real and complex fields, one can weaken the condition to look for topologically embedded lines (circles in the real case, spheres in the complex case) that meet according to a specified incidence relation. Ruberman’s work with Laura Starkston (Stanford) gives new topological restrictions on the realization of configurations of spheres in the complex projective plane.

Caption: Solutions to the Seiberg-Witten equations of quantum field theory provide topological information about 4-dimensional spaces. However, the set of solutions, or moduli space, can undergo a phase transition as a parameter T is varied, making those solutions hard to count. This figure illustrates a key calculation: the phase transition is equal to the end-periodic spectral flow, a new concept introduced in work of Mrowka-Ruberman-Saveliev. In the figure, the spectral set, illustrated by the red curves, evolves with the parameter T. Every time the spectral set crosses the cylinder, the moduli space changes, gaining or losing points according to the direction of the crossing.