RESEARCH INTERESTS 

My research focuses on the study of quantum and classical integrable models. Such models have attracted a lot of research interest in recent years primarily because their study provides exact results, without recourse to perturbation theory or other approximation schemes as is the case when studying most physical problems. The main challenge when investigating a physical system is to develop exact, non-perturbative methods in order to resolve physically relevant questions. Integrability offers such an exact framework. Beyond their physical significance, integrable models are also of great mathematical interest. Their investigation leads to intriguing mathematical structures, whose study has increasingly grown over the last decades. I have been studying both mathematical and physical aspects of integrable systems as described below.

Quantum Spin Chains and Bethe Ansatz. Many ideas in quantum integrability have their origins in statistical physics dating back in the seminal works of Bethe and Onsager, who solved the Heisenberg and Ising models respectively, as well as the celebrated solution of the anisotropic Heisenberg model by Baxter. A more recent technique on the resolution of the spectrum of one-dimensional statistical models is the Bethe ansatz approach within the Quantum Inverse Scattering Method (QISM), an elegant algebraic method developed by the St. Petersburg group that also leads to the formulation of certain deformed algebras called quantum groups.

Factorizable Scattering-Quantum Groups. Integrable models display particle-like excitations, whose interactions are described by the bulk scattering S-matrix satisfying a collection of algebraic constraints, known as the Yang-Baxter equation. Physically the Yang-Baxter equation describes the factorization of multi-particle scattering, a unique feature displayed by quantum integrable systems.  From a mathematical viewpoint Jimbo and Drinfeld established independently, that the algebraic structures underlying the Yang-Baxter equation may be seen as deformations of the usual Lie algebras or their infinite dimensional extensions, the Kac-Moody algebras. Such algebraic structures are endowed with a non trivial co-product (Hopf algebras) and are known as quantum groups. 

Braid Group, Hecke and Temperley-Lieb Algebras. Hecke and Temperley-Lieb algebras are quotients of the Artin braid group. These algebras in addition to their own mathematical interest turn out to play an important role in the context of quantum integrable systems. In particular, the structural similarity of the braid group to the Yang–Baxter and boundary Yang-Baxter (reflection) equations can be exploited to convert representations of these algebras into new solutions of the (boundary) Yang-Baxter equation. Moreover, tensor representations of the Hecke and Temperley-Lieb algebras are closely related to representations of quantum groups.

Set theoretic Yang-Baxter Equation and Braces. Drinfeld suggested the idea of set-theoretic solutions of the Yang-Baxter equation. Set theoretic solutions and Yang-Baxter maps have been primarily studied in the context of classical discrete integrable systems as discrete maps. More recently, set theoretic solutions of the Yang-Baxter equation have been investigated by employing the theory of braces (skew-braces). The theory of braces was established by Rump who developed a structure that generalizes nilpotent rings, called a brace to describe all finite, involutive, set-theoretic solutions of the Yang-Baxter equation. 

Hamiltonian Formulation-Lax pair. In the context of classical integrable systems Lax representation of classical dynamical evolution equations is one key ingredient together with the associated notion of classical r-matrix as suggested by Sklyanin and Semenov-Tian-Shansky. It takes the generic form of an isospectral evolution equation. The spectrum of the Lax matrix or its extension or equivalently the invariant coefficients of the characteristic determinant provide automatically candidates to realize the hierarchy of Poisson-commuting Hamiltonians required by Liouville's theorem. Existence of the classical r-matrix guarantees Poisson commutativity of these natural dynamical quantities taken as generators of the algebra of classical conserved charges. The existence of the Lax-pair leads to non-linear, integrable PDEs or ODEs (evolution equations) that can be exactly solved via dressing methods and they typically display soliton type solutions.