We discuss compactifications of moduli spaces of low rank Higgs bundles over a surface by synthetic objects. These objects are obtained by describing the limits of solutions to conformal variational problems on Riemann surfaces, as some of the data degenerates. We begin by describing the model case of compactifying classical Teichmuller space by real trees via limits of harmonic maps and then move from that PSL(2,R) case to PSL(2,R) x PSL(2,R) where minimal surfaces limit on 'mixed structures'. As time dwindles, we touch on PSL(2,C) where the limiting configurations of Mazzeo-Swoboda-Weiss-Witt are identified with shearings of pleated surfaces, on PSL(3,R) where we aim for affine spheres to limit to buildings, as well as on Sp(4,R) where we see the early part of a theory for maximal surfaces in the pseudo-Riemannian space H^{2,2}.

We study Dirac-type operators on stratified spacesendowed with incomplete Riemannian metrics which undergo iterated conic-type degeneration at the singular strata. Examples of such spaces include orbifolds and also the crossing incomplete cone-edge spaces studied in recent work on Käler-Einstein manifolds in dimension four. We construct a Fredholm problem for such operators and prove an index formula generalizing the Bismut-Cheeger index formulas on spaces with conical singularities. Along the way we develop the theory of elliptic operators on such spaces, in particular constructing Green's functions of some elliptic differential operators and their heat kernels. In the end we manage to generalize the Getzler rescaling of the heat kernel to this setting, which allows us to evaluate the supertrace of the heat kernel in the short time limit. This is joint work with Pierre Albin at UIUC.

We discuss singularities of Teichmüller harmonic map flow, which is a geometric flow that changes maps from surfaces into branched minimal immersions and discuss in particular how the topology of the domain surface can change in a well controlled way through singularities of the metric component.