In this seminar I will give an introduction to integrability and the $q$KZ equation, with a focus on the integrable Temperly--Lieb loop model. The associated Temperley-Lieb algebra can be represented in graphical form, which in turn allows a graphical construction of the solutions of the $q$KZ equation. Using this graphical construction, we derive a factorised form for a generalised sum over components of the $q$KZ solution. Specialising this sum gives the normalisation of the ground state vector of the Temperley--Lieb loop model. But in its general form there are connections to certain Razumov--Stroganov conjectures and Macdonald and Koornwinder polynomials.

The geometry of supersymmetric gauge theories hides interesting algebraic structures responsible for the integrability of the theory. In this talk I will review how quantum algebraic structures such as affine Kac-Moody algebras, quantum affine groups, W-algebras and their relatives appear in the context of the supersymmetric gauge theories.

Quantum mechanics is the theory that describes the behavior of matter at the nano scale. Though routinely applied to the electrons in a molecular system, quantum theory is generally not applied to the atomic nuclei (except for very simple molecules), because such “exact quantum dynamics” (EQD) calculations are regarded to be too difficult. This is due to the “curse of dimensionality”: the computational (CPU) cost scales exponentially with system dimensionality. Nevertheless, quantum dynamical effects are known to be important in a wide range of molecular applications, and must therefore be accounted for at least approximately, if sensible numerical simulations are to be performed. In this presentation, we will discuss powerful new EQD methods developed in recent years, as well as applications. The methods are mostly based on so-called “quantum trajectories”—similar to those used in Bohmian mechanics, but arising from a completely new formulation of quantum mechanics for which the wavefunction Ψ(t, x) plays no role. Instead, quantum states are represented solely as ensembles of real-valued probabilistic trajectories, x(t, C), where C is a trajectory label. The applications discussed span a broad range, from models to highly accurate treatments of real molecules, and will be drawn from the following: (1) wavepacket barrier scattering; (2) extremely deep tunneling; (3) coupled 2D model chemical reaction system; (4) collinear H+H2 reaction; (5) hydrogen adsorbate dynamics on carbon nanotubes; (6) Fluxional H atom scrambling dynamics in Fe(H)2(H2)(PEtPh2)3 coordination complex.

Motivated by experimental observations of ripples in suspended graphene, we analyze elastic properties of two-dimensional crystalline membranes with various types of randomness [1]. We demonstrate that random fluctuations of the membrane curvature induced by static disorder lead to a non-monotonous scaling of the bending rigidity, causing the transition between the flat and crumpled phases. We also explore the effect of critical fluctuations of a membrane on electronic transport through the system in the presence of electron-phonon coupling [2,3]. [1] I.V. Gornyi, V.Yu. Kachorovskii, and A.D. Mirlin, Rippling and crumpling in disordered free-standing graphene, arXiv:1505.04483, to appear in Phys. Rev. B. (2015). [2] I.V. Gornyi, V.Yu. Kachorovskii, and A.D. Mirlin, Conductivity of suspended graphene at the Dirac point, Phys. Rev. B 86, 1654CR (2012). [3] A.P. Dmitriev, I.V. Gornyi, A.D. Mirlin, I.V. Protopopov, in preparation.

After introducing and reviewing some recent developments in discrete non-perturbative approaches to quantum gravity, I will present the construction of new vacua and representations for the quantum algebra of observables of canonical gravity. This will highlight the unsuspected richness of this class of theories, and make more transparent their interpretation as topological quantum field theories with defects. I will then explain the relevance of this construction for the study of the continuum limit via coarse-graining and renormalization.

We provide an interpretation of the Second Law of Thermodynamics in terms of the time it takes for an observer to decide on the direction of the arrow of time. Such a decision can be made efficiently with a prescribed reliability using the sequential probability ratio test. We show that the steady state entropy production rate is inversely proportional to the average decision time. Furthermore,the distribution of decision times obeys a fluctuation theorem which implies that the decision time distributions have the same shape for correct and wrong decisions. Our results are demonstrated by numerical simulations of two simple examples of nonequilibrium stochastic processes.