Research Interests
Current Research
Numerical Methods in "ab initio" simulations

Hamiltonian formalism is ubiquitous in physics. In most cases all the available information is encoded in the Hamiltonian operator defined by the Schrödinger equation. Solving for this equation is, in general, very challenging already for simple atomic systems. Ab initio methods replace the complexity residing in the high dimensionality of the Schrödinger equation with a very effective approximation scheme based on a set of self-consistent equations. Among the many ab initio methods, Density Functional Theory (DFT) is widely used to understand the ground state of complex materials. A generalized eigenvalue problems usually arises in solving the Kohn-Sham equations for an effective Hamiltonian operator. From this brief introduction it is clear that understanding how to solve eigenvalue problems related to Hamiltonian systems is crucial.

My research efforts are mainly concerned (but not exclusively) in finding the eigenvectors and eigenvalues of the Hamiltonian operator numerically. Specifically, I aim at extending and improving the existing state of the art eigensolvers to take advantage of physical and mathematical information available to the physicist. This line of research is conducted keeping in mind the need to target emerging parallel architectures and supercomputers as our ultimate instrument of investigation.

Due to the interdisciplinary nature of the objective, it is important to draw inspiration from the tools and knowledge of two different disciplines, computer science and physics, combining them in a common language that can exploit their respective potential. The final outcome will be beneficial to both disciplines: on one hand it allows to devise high-quality algorithms, while on the other hand it permits to obtain effective numerical descriptions of complex structures. The overall philosophy can be simply expressed: the more information to be encoded into an algorithm, the better the quality of the algorithm, the more effective its solving potential. We follow this approach on the topics briefly described below:

  • Sequences of Eigenproblems - A DFT simulation is made of repeatedly executed self-consistent cycles, each requiring the solution of dozens of generalized eigenproblems. Recent evidence shows that these eigenproblems are correlated and suggest that treating them as part of a sequence could lead to many new algorithmic improvements.
  • Convergence accelleration - The positive outcome of a DFT simulation involve the minimization of the ground state energy through the computation of a one-particle electron density n(r) using a variational method. Optimizing the convergence is still an issue and there is still much room for improvement
  • "Reverse" simulation - We refer to this technique when using the outcome of a simulation to infer on the shortcoming of the mathematical paradigm used. The objective is to use the result of the simulation to improve the mathematical structure from which the algorithm derives.
  • Resolvent methods - A new interative eigensolver based on the resolvent of a generalized eigenproblem (in development)

Numerical Tensor Algebra

Tensors are increasingly ubiquitous in various areas of applied mathematics and computational physics because multilinear algebra arises naturally in many contexts. Consequently, there is a large demand for analysis and computational methods for multilinear objects. In recent years there have been substantial efforts to further develop the mathematical and computational aspects of tensor methods, with a goal of making them as easy to use as their matrix counterparts. At the beginning it seemed that reformulating a tensor problem in terms of multi-arrays would be beneficial. Unfortunately ignoring the multilinear structure of a tensorial object typically leads to inefficient algorithms and fails to adequately exploit its underlying geometrical structure. Furthermore, there are several applications in physics that are only tractable when we employ data approximations based on full tensors expressions (partial differential and integro-differential equations). In our research effort we would like to make numerically manifest the geometrical nature of tensors. In return this approach will allow us to treat them as objects quite distinct from multi-arrays. In many areas of applied physics this is the only possible modus operandi leading to correct results.

Past Research
Theoretical Physics

Theoretical physics encompass a wide variety of fields. One of the most complex, fascinating and varied field is String Theory and its satellite research fields: supersymmetry, algebraic geometry, theoretical cosmology, etc.. The following research topics encompass several of these satellite fields never involving directly String Theoretical research but touching on several of its developments as a source of inspiration.

  • De-Construction - Research on this topic was centered on 5-dimensional (5D) de-constructive quiver gauge theories that present interesting non-perturbative characteristics. From the gauge theory side, we showed how the non-perturbative 5D gauge coupling at low energy arises from the analysis of the 4D Seiberg-Witten spectral curve in the Coulomb branch. We also conducted a parallel geometrical analysis using webs of (p,q) branes realizing the same dynamical structure out of specific orbifolds in M-theory. This work was published in a long monographic journal paper.
  • Supersymmetric (SUSY) Gauge Theory - In the first part of this line of research we have throughly studied the chiral ring of the 5D quiver theory previously analyzed. Then inspired by in the early development of Dijkgraaf-Vafa matrix models and the following seminal papers of Witten et Al. on chiral rings, anomaly equations and random matrix models we showed that there is a correspondence between chiral bi-fundamentals and unitary matrices. This work was published in two massive papers and constituted the main topic of my Ph.D. dissertation.
  • Multi-Parametric Quantum Algebras - This line of research revolved around the basic idea of a generalized multi-parameter q-deformation of SO(4,1), the isometry group of the deSitter space-time solution of Einstein equation. We built a multiparametric deformation of the Drinfeld-Jimbo algebra for SO(5) and subsequently studied its *-structures so as to constrain the values of the deformation parameters in order to reproduce the deSitter isometry groups. At the end we produced rigorous results that demonstrated the incorrectness of what was currently believed to be the most promising solution to the problem of the finiteness of deSitter Hilbert space. The main result of the investigation was published in two separate papers
  • Inflation in M-Theory - In 2006 Acharya et al. have shown a very promising alternative to moduli stabilization in string theory without the use of fluxes. Their mechanism, based on hidden sector strong gauge dynamics, stabilizes the metric moduli in M-theory compactifications on G2 holonomy manifolds. The abundance of moduli can give rise to mechanisms similar to racetrack inflation or Roulette inflation. I explored scenarios in which a modified Kahler potential for the chiral hidden sector would drive the inflaton along the axion valleys. While working on this project I had the first exposure to rudiments of scientific computation as a means to achieve interesting new results in this line of research.

High Energy Physics Phenomenology

Phenomenology in high energy physics deals with the specific predictions produced by fine tuned models and compare their theoretical results with observational data. In recent years experimental results from accelerators have been scarce not very conclusive. On the other hand we had a series of breakthrough coming from cosmological measurements; dark matter, dark energy, accelerating universe are among the most glamorous. These results impose serious constraints on Grand Unified Theories (GUT). In other words specifying more precisely cosmological parameters has restricted the number of models that can describe the evolution of the universe until the current status. In my work I used this interplay between cosmology and high energy physics to analyze some of these models.

  • Dark Matter In GUT Models - Between 2005 and 2007 my research efforts focused on Particle Physics phenomenology in relation to Cosmology. I worked on anomaly cancellation in non-supersymmetric quivers and on dark matter candidates in Trinification GUT models. This work led to several short publications.
  • Dynamical Symmetry Breaking in the Standard Model (SM) - I examined the breaking of internal gauge symmetries from a different point of view. I showed that is possible to reproduce the electroweak phenomenological scenario of the traditional SM in a exhaustive and self consistent way without the use of the Higgs mechanism. The result is reached applying the main futures of the Nambu-Jona Lasinio (NJL) mechanism to an electroweak invariant Lagrangian. The use of functional formalism for composite operators naturally leads to a different dynamical approach. While the Higgs mechanism acts on the Lagrangian form, a NJL like model looks directly at the dynamical content hidden in the Green functions of the theory. The result of this original work was presented in my Laurea thesis.