Ellipticity in geometric variational problems

 Ellipticity in geometric variational problems is a feature of a functional, defined on geometric objects (e.g. currents), which allows to prove existence and regularity of minimizers. The geometric objects we are dealing with are basically just d-dimensional subsets S of ℝⁿ and the functionals are defined by integrating certain integrand F over S with respect to the d-dimensional Hausdorff measure. The integrand may depend not only on the point in space but also on the tangent direction.  

 In my recent joint work with Antonio De Rosa (Courant Institute, NY) we compare two different notions of ellipticity. The first one, denoted AE, was introduced in 1960s by Frederick Almgren who also laid the foundations of regularity theory for minimizers (Ann. of Math. 87, 1968). The second one, called AC, is a new definition that appeared in the work of De Philippis, De Rosa, and Ghiraldin (Comm. Pure Appl. Math. 71(6), 2018) as a sufficient and necessary condition for a minimizer (more generally, for any critical point) to be rectifiable.  

 It is very hard, in practice, to verify that a given functional satisfies AE. Actually, there are essentially no non-trivial examples (at least in case n-d > 1). The condition AC is easier, more algebraic and, due to the work of De Philippis, De Rosa, and Ghiraldin, we know it is exactly the right condition.   

 In our joint work with De Rosa we prove that AC implies AE. Interestingly, in the course of the proof we had to employ some (classical) algebraic topology. In my talk, I will focus on the part of the proof were analysis meets topology.