My research in Mathematics is mostly in Geometric Analysis and its applications to Geometry and Topology. Right now I am interested in Vortex equations and instantons and their connection with the underlying geometric structures. I'm also interested in Ricci flow (Kaehler or not) and in the question of existence of special metrics on projective manifolds and its relationship with questions of various algebro-geometric notions of stability. Of late I have been studying Gauge theoretic questions -- in particular analytic and topological aspects of the Vortex equation-- and also the nature of minimal surfaces in manifolds with curvature conditions (e.g. PIC) and their topological consequences. My research in physics is mostly concentraded on the AdS/CFT correspondence, inparticular on the presence of non-localities in the boundary theory cause by tachyons in the bulk. The presence of such non-localities is related to the Firewall problem of Almheiri, Marolf, Polchinski and Sully.

Here are some of my papers

We show explicitly that the full structure of IIB string theory is needed to remove the non-localities that arise in boundary conformal theories that border hyperbolic spaces on AdS5. Specifically, using the Caffarelli/Silvestri\cite{caffarelli}, Graham/Zworski\cite{graham}, and Chang/Gonzalez\cite{chang:2010} extension theorems, we prove that the boundary operator conjugate to bulk p-forms with negative mass in geodesically complete metrics is inherently a non-local operator, specifically the fractional conformal Laplacian. The non-locality, which arises even in compact spaces, applies to any degree p-form such as a gauge field. We show that the boundary theory contains fractional derivatives of the longitudinal components of the gauge field if the gauge field in the bulk along the holographic direction acquires a mass via the Higgs mechanism. The non-locality is shown to vanish once the metric becomes incomplete, for example, either 1) asymptotically by adding N transversely stacked Dd-branes or 2) exactly by giving the boundary a brane structure and including a single transverse Dd-brane in the bulk. The original Maldacena conjecture within IIB string theory corresponds to the former. In either of these proposals, the location of the Dd-branes places an upper bound on the entanglement entropy because the minimal bulk surface in the AdS reduction is ill-defined at a brane interface. Since the brane singularities can be circumvented in the full 10-dimensional spacetime, we conjecture that the true entanglement entropy must be computed from the minimal surface in 10-dimensions, which is of course not minimal in the AdS5 reduction

Here we prove a regularity theorem for the Monge-Ampere equation with
right hand side of the that has a singularity in the form of poles. The equation naturally arizes in the study of conic Kahler-Einstein equations, when the Ricci potential is in Donaldson's space of functions with conic Laplacian which is H\"older continuous. Our result can be used to address a question of Donaldson's on the singularity of conic Kahler Einstein metrics along divisors which are split.

We investigate how to obtain various flows of K\"ahler metrics on a fixed manifold as variations of K\"ahler reductions of a metric satisfying a given static equation on a higher dimensional manifold. We identify static equations that induce the geodesic equation for the Mabuchi's metric, the Calabi flow, the pseudo-Calabi flow of Chen-Zheng and the K\"ahler-Ricci flow. In the latter case we re-derive the V-soliton equation of La Nave-Tian

Here we prove that the scalar V-soliton equation- introduced by myself and G. Tian in a previous paper- and which is an elliptic equation of degenerate type, has a solution (unique up to addition of constants) with bounded complex Hessian and which have derivatives which are Holder-continous of any exponent less than 1. We outline an argument to the effect that the equivalent theorem in the boundary case with Dirichlet condition, would have strong convergence consequences for the Gromov-Hausdorff limits of
Ricci flow that develop singularities in finite type.

We prove a conjecture of Gromov's to the effect that manifolds with
(uniformly) positive isotropic curvature and with a lower bound on the Ricci curvature,
are macroscopically 1-dimensional on the scale of the isotropic curvature. As a consequence we prove that compact manifolds with
positive isotropic curvature have virtually free fundamental groups. Our main technique builds on Donaldson's version of H\"ormander techniques to construct destabilizing sections with controlled sup and L-p norms.

In this note we point out a simple application of a result by the authors in Singularities, test configurations and constant scalar curvature Kaehler metrics. We show how to construct many families of strictly K-unstable polarized manifolds, destabilized by test configurations with smooth central fiber. The effect of resolving singularities of the central fiber of a given test configuration is studied, providing many new examples of manifolds which do not admit Kähler constant scalar curvature metrics in some classes.

In this paper we extend the notion of Futaki invariant to big and nef classes in such a way that it defines a continuous function on the Kaehler cone up to the boundary. We apply this concept to prove that reduced normal crossing singularities are sufficient to check K-semistability. This is effectively achieved by performing a version of semistable reduction for flat families with control over the Futaki invariant. A similar improvement on Donaldson's lower bound for Calabi energy is given

In this paper, we first show an interpretation of the Kaehler-Ricci flow on a manifold X as an exact elliptic equation of Einstein type on a manifold M of which X is one of the (K\"ahler) symplectic reductions via a (non-trivial) torus action. There are plenty of such manifolds (e.g. any line bundle on X will do). We call such equation the V-soliton equation, which can be regarded as a generalization of Kaehler-Einstein equations or Kaehler-Ricci solitons. As in the case of Kaehler-Einstein metrics, we can also reduce the V-soliton equation to a scalar equation on Kaehler potentials, which is of Monge-Ampere type. We then prove some preliminary results towards establishing existence of solutions for such a scalar equation on a compact Kaehler manifold M. One of our motivations is to apply the interpretation to studying finite time singularities of the Kaehler-Ricci flow.

In this paper we discuss some classification results for Ricci solitons, that is, self similar solutions of the Ricci Flow. Some simple proofs of known results are presented. In detail, we take the (static/elliptic) equation point of view, trying to avoid the tools provided by considering the dynamic properties of the Ricci flow (and in particular parabolic techniques). As much as we can, we also avoid Perelman's dynamical view-point, but do use the first eigenvalue of the F-functional of Perelman's (we also present the first rigourous proof of the regularity of the eigenfunctions).

In this paper we show the existence of stable symplectic non-holomorphic two-spheres in Kähler manifolds of positive constant scalar curvature of real dimension four and in Kähler–Einstein Fano manifolds of real dimension six. Some of the techniques used involve deformation theory of algebraic cycles.

In this note we show that the K\"ahler-Ricci flow fits naturally within the context of the Minimal Model Program for projective varieties. In particular we show that the flow detects, in finite time, the contraction theorem of any extremal ray and we analyze the singularities of the metric in the case of divisorial contractions for varieties of general type. In case one has a smooth minimal model of general type (i.e., the canonical bundle is nef and big), we show infinite time existence and analyze the singularities.

In this note we show how to find the stable model of a one-parameter family of elliptic surfaces with sections. More specifically, we perform the log Minimal Model Program in an explicit manner by means of toric geometry, in each such one parameter family. This way we obtain an explicit combinatorial description of the surfaces that may occur at the boundary of moduli (as well as a new proof of the completeness of the moduli space)