We study the deformed Hermitian-Yang-Mills (dHYM) equation, which is mirror to the special Lagrangian equation, from the variational point of view via an infinite dimensional GIT problem mirror to Thomas' GIT picture for special Lagrangians. This gives rise to infinite dimensional manifold $\mathcal{H}$ mirror to Solomon's space of positive Lagrangians. In the hypercritical phase case we prove the existence of smooth approximate geodesics, and weak geodesics with $C^{1,\alpha}$ regularity. This is accomplished by proving sharp with respect to scale estimates for the Lagrangian phase operator on collapsing manifolds with boundary. We apply these results to the infinite dimensional GIT problem for deformed Hermitian-Yang-Mills. We associate algebraic invariants to certain birational models of $X\times \Delta$, where $\Delta \subset \mathbb{C}$ is a disk. Using the existence of regular weak geodesics we prove that these invariants give rise to obstructions to the existence of solutions to the dHYM equation. Furthermore, we show that these invariants fit into a stability framework closely related to Bridgeland stability. Finally, we use a Fourier-Mukai transform on toric K\"ahler manifolds to describe degenerations of Lagrangian sections of SYZ torus fibrations of Landau-Ginzburg models $(Y,W)$. We speculate on the resulting algebraic invariants, and discuss the implications for relating Bridgeland stability to the existence of special Lagrangian sections of $(Y,W)$.

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We propose a geometric inequality for two-dimensional spacelike surfaces in the Schwarzschild spacetime. This inequality implies the Penrose inequality for collapsing dust shells in general relativity, as proposed by Penrose and Gibbons. We prove that the inequality holds in several important cases.

We prove a lower bound for the first Steklov eigenvalue of embedded minimal hypersurfaces with free boundary in a compact n-dimensional Riemannian manifold which has nonnegative Ricci curvature and strictly convex boundary. When n=3, this implies an apriori curvature estimate for these minimal surfaces in terms of the geometry of the ambient manifold and the topology of the minimal surface. An important consequence of the estimate is a smooth compactness theorem for embedded minimal surfaces with free boundary when the topological type of these minimal surfaces is fixed.