Monday, 16 July 2018

We study the inverse problem of determining vector and scalar potentials appearing in the relativistic Schrödinger equation in space dimension 3 or higher from information about the solution on a suitable subset of the boundary. We prove unique determination of these potentials modulo a natural gauge invariance. This is joint work with Manmohan Vashisth.

Tuesday, 17 July 2018

We shall talk about the Witten conjecture which roughly says that the generating function of certain intersection numbers on the moduli space of curves satisfies the KdV equation. We shall also discuss a proof of the celebrated conjecture using the ESLV formula which relates them to Hurwitz numbers. Surprisingly the generating function for Hurwitz numbers satisfies the KP hierarchy.

Friday, 20 July 2018

After introducing controllability (reaching a desired state in finite time) and stabilizability (reaching a steady state as time tends to infinity), for ODE systems, I will discuss these issues for some PDE systems, arising in fluid models.

Laplacian growth is a problem of two-dimensional pattern formation, that has drawn a lot of attention in theoretical and experimental physics due to the spontaneous formation of fractal patterns in a non-equilibri um system. Although the problem has been shown to be integrable, albeit it in a somewhat unexpected and unconventional way, an explanation of the fractal properties of the growth has eluded theoretical study. Attempting to be methodical, I will first review the problem and its relation to integrable systems, then I will undoubtedly add to the confusion by suggesting tentative and speculative directions of research.

Wednesday, 25 July 2018

I will describe three models of physical interest in which Integrability and Supersymmetry are entwined. We will start with two elementary examples and end with a model which is still a topic of current research.

We present results on a combinatorial problem which was solved recently using techniques from integrable systems. Specifically, we introduce a new class of square-ice configurations (also known as six-vertex model configurations) on triangular subsets of the square lattice, which we call Alternating Sign Triangles (ASTs). The proof of the enumeration of ASTs uses the integrability of the six-vertex (or square-ice) model. We will explain the origin of this problem and the ideas involved in the proof. Time permitting, we will also give product formulas for other classes of square-ice configurations using similar ideas. This is joint work with Ilse Fischer and Roger Behrend.

Thursday, 26 July 2018

The sine-Gordon equation is a semi-linear wave equation used to model many physical phenomenon like seismic events that includes earthquakes, slow slip and after-slip processes, dislocation in solids etc. Solution of homogeneous sine- Gordon equation exhibit soliton like structure that propagates without change in its shape and structure. The question whether solution of sine- Gordon equation still exhibit soliton like behaviour under an external forcing has been challenging as it is extremely difficult to obtain an exact solution even under simple forcing like constant. In this study solution to an inhomogeneous sine-Gordon equation with Heaviside forcing function is analysed. Various one- dimensional test cases like kink and breather with no flux and non reflecting boundary conditions are studied.

Friday, 27 July 2018

Let S be a closed oriented surface of genus at least two. The Teichmüller space T(S) is the universal cover of the moduli space of hyperbolic structures on S. For any choice of a complex structure on S, a theorem of M. Wolf, and independently N. Hitchin, identifies T(S) with the vector space of holomorphic quadratic differentials on the resulting Riemann surface X, that we denote by Q(X). An earlier result of Hubbard and Masur identifies Q(X) with the space of certain topological objects called measured foliations on S. I shall discuss these results, the relation between them, and their generalizations to the case of meromorphic quadratic differentials. All these spaces have some natural symplectic structures, and I shall mention some open questions concerning them.

We shall first talk about Kostant-Souriau's method of geometric quantization of coadjoint orbits. Adler had showed that the Toda system can be given a coadjoint orbit description. We shall talk about quantization of the Toda system by viewing it as a single orbit of a multiplicative group of lower triangular matrices of determinant one with positive diagonal entries. We get a unitary representation of the group with square integrable polarized sections of the quantization as the module . We find the Rawnsley coherent states after completion of the above space of sections. Finally we give an expression for the quantum Hamiltonian for the system.This is joint work with Dr. Saibal Ganguli.

Monday, 30 July 2018

These talks review various approaches to non-equilibrium processes in integrable models. Examples include the increase in understanding since 2011 of Drude weight and semiclassical kinetic theory in integrable models, and obtaining exact far-from-equilibrium results for some quantities in the XXZ model through expansion potentials. In many cases the predictions of theoretical approaches based on integrability can be confirmed due to progress in DMRG and other matrix product state algorithms, and the essentials of such algorithms will be reviewed.

Tuesday, 31 July 2018

These talks review various approaches to non-equilibrium processes in integrable models. Examples include the increase in understanding since 2011 of Drude weight and semiclassical kinetic theory in integrable models, and obtaining exact far-from-equilibrium results for some quantities in the XXZ model through expansion potentials. In many cases the predictions of theoretical approaches based on integrability can be confirmed due to progress in DMRG and other matrix product state algorithms, and the essentials of such algorithms will be reviewed.

Wednesday, 01 August 2018

We address the Yang-Lee formalism for Integrable Quantum Field Theories, discussing various features of this approach to statistical physics and focusing the attention on the zeros of the grand canonical partition in the fugacity variable of the soliton sector of the Sine-Gordon model and of the simplest integrable quantum field theory, the so called Yang-Lee model.

These talks review various approaches to non-equilibrium processes in integrable models. Examples include the increase in understanding since 2011 of Drude weight and semiclassical kinetic theory in integrable models, and obtaining exact far-from-equilibrium results for some quantities in the XXZ model through expansion potentials. In many cases the predictions of theoretical approaches based on integrability can be confirmed due to progress in DMRG and other matrix product state algorithms, and the essentials of such algorithms will be reviewed.

Thursday, 02 August 2018

We study non-equilibrium dynamics of integrable and non-integrable closed quantum systems whose unitary evolution is interrupted with stochastic resets, characterized by a reset rate $r$, that project the system to its initial state. We show that the steady state density matrix of a non-integrable system, averaged over the reset distribution, retains its off-diagonal elements for any finite $r$. Consequently a generic observable $\hat O$, whose expectation value receives contribution from these off-diagonal elements, never thermalizes under such dynamics for any finite $r$. We demonstrate this phenomenon by exact numerical studies of experimentally realizable models of ultracold bosonic atoms in a tilted optical lattice. For integrable Dirac-like fermionic models driven periodically between such resets, the reset-averaged steady state is found to be described by a family of generalized Gibbs ensembles (GGE s) characterized by $r$. We also study the spread of particle density of a non-interacting one-dimensional fermionic chain, starting from an initial state where all fermions occupy the left half of the sample, while the right half is empty. When driven by resetting dynamics, the density profile approaches at long times to a nonequilibrium stationary profile that we compute exactly. We suggest concrete experiments that can possibly test our theory.

Tuesday, 07 August 2018

"We consider anti-ferromagnetic spin chains with a weak frustration -just one bond in a large chain-, such as systems with an odd number of spins with periodic boundary conditions. We show that, in certain cases, a new quantum phase of matter arises in these systems. Such phase is extended, gapless, but not relativistic. The low-energy excitations have a quadratic (Galilean) spectrum. Locally, the correlation functions on the ground state do not show significant deviations compared to the not-frustrated case, but correlators involving a number of sites (or distances) scaling like the system size display new behaviors. In particular, the Von Neumann entanglement entropy is found to follow new rules, for which neither area law applies, nor one has a divergence of the entropy with the system size. Such very long range correlations are novel and of potential technological interest. We display such new phase in a few prototypical chains using numerical simulations and we study analytically the paradigmatic example of the Ising chain. Through these examples we argue that this phase emerges generally in (weakly) frustrated systems with discrete symmetries.

[1] S.M. Giampaolo, F. Ramos, and F. Franchini, In Preparation"