In this paper, the first in a series, we study the deformed Hermitian-Yang-Mills (dHYM) equation from the variational point of view as an infinite dimensional GIT problem. The dHYM equation is mirror to the special Lagrangian equation, and our infinite dimensional GIT problem is mirror to Thomas' GIT picture for special Lagrangians. This gives rise to infinite dimensional manifold $\mathcal{H}$ closely related 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. As an application of our techniques we give a simplified proof of Chen's theorem on the existence of $C^{1,\alpha}$ geodesics in the space of K\"ahler metrics. In two follow up papers, these results will be used to examine algebraic obstructions to the existence of solutions to dHYM and special Lagrangians in Landau-Ginzburg models.

We prove that a general miraculous cancellation formula, the divisibility of certain characteristic numbers, and some other topological results related to the generalized Rochlin invariant, the -invariant and the holonomies of certain determinant line bundles, are consequences of the modular invariance of elliptic operators on loop space.

Two Dehn surgeries on a knot are called purely cosmetic, if they yield manifolds that are homeomorphic as oriented manifolds. Suppose there exist purely cosmetic surgeries on a knot in S^3, we show that the two surgery slopes must be the opposite of each other. One ingredient of our proof is a Dehn surgery formula for correction terms in Heegaard Floer homology.

Inversive distance circle packing metric was introduced by P Bowers and K Stephenson as a generalization of Thurston’s circle packing metric. They conjectured that the inversive distance circle packings are rigid. For nonnegative inversive distance, Guo proved the infinitesimal rigidity and then Luo proved the global rigidity. In this paper, based on an observation of Zhou, we prove this conjecture for inversive distance in (−1, +∞)by variational principles. We also study the global rigidity of a combinatorial curvature with respect to the inversive distance circle packing metrics where the inversive distance is in (−1, +∞).

The action of the mapping class group of the thrice-punctured projective plane on its GL(2,C) character variety produces an algorithm for generating the simple length spectra of quasi-Fuchsian thrice-punctured projective planes. We apply this algorithm to quasi-Fuchsian representations of the corresponding fundamental group to prove: a sharp upper-bound for the length its shortest geodesic, a McShane identity and the surprising result of non-polynomial growth for the number of simple closed geodesic lengths.