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.

In a point pattern, let the variance associated with the number of points contained in a spherical window of radius $R$ be denoted by $\sigma^2(R)$. Hyperuniform point patterns in $d$ dimensions do not possess infinite-wavelength fluctuations or, equivalently, possess a number variance $\sigma^2(R)$ that grows more slowly than the window volume, i.e., $R^d$, and thus the corresponding static structure factor $S(k)$ vanishes as $k \to 0$. Hyperuniform systems include all infinite periodic structures, aperiodic quasicrystals, and some special disordered systems. Previous investigations showed that the characteristic small-wavelength scaling of the structure factor and the number variance for large $R$ in hyperuniform systems can be thought of as a useful tool to rank order systems according to the degree to which large-scale density fluctuations are suppressed. In this talk, we investigate the hyperuniformity in one- and two-dimensional quasicrystals with a variety of different scaling factors and rotational symmetries. First, we deliver a proof of the hyperuniformity of quasicrystals. We further study how the small-wavelength scaling and the number variance depends on the intrinsic properties of the quasicrystal such as incommensurate length scales. We find that, unlike crystals, quasicrystals' small-wavelength scattering behavior crucially depends on the dimension. We compare these results to a number of different periodic systems as well as disordered hyperuniform systems.

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, 165413 (2012).
[3] A.P. Dmitriev, I.V. Gornyi, A.D. Mirlin, I.V. Protopopov, in preparation.

Single photon emitters coupled to a complex photonic network hold great potential for next generation quantum information at the nanoscale. I will present our approach to photonic networks based on multiple scattering and how we experimentally couple a set of distributed single emitters with disordered electro-spun dielectric nanofibers, percolating plasmonic networks, and hybrid antenna-waveguide systems. By measuring individual quantum dots and organic emitters at room temperature we record both real and momentum spectroscopy of the fluorescence and we construct the distribution of the Purcell effect, i.e. the fluorescence modification by local mode density engineering. I will discuss the role of global long-range interactions linked to the network connectivity, degree of percolation and fractality, as well as the local near-field contribution.

I will discuss the theory of hard and soft spheres in the limit of infinite dimensions. At low temperature or high pressure, it predicts the existence of a new phase transition, the Gardner transition, beyond which minima of the free energy landscape become marginally stable. This landscape marginal stability merges naturally in the mechanical marginal stability that characterize the jamming point and is crucial to obtain new insights on the criticality of jammed packings. I will introduce some protocols to detect the Gardner point and I will show what are the main properties of the marginal phase. I will discuss what are the consequences of the Gardner transition on the elastic properties of glasses. Then I will focus on the vibrational spectrum showing how marginal stability provides a new scenario for the anomalous excess of soft modes.

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.

The high energy physics community is in the process of deciding which machines to propose for fundings. It is then particularly important to study the discovery prospects of different models at such machines. In this talk, I will do so for two motivated cases: i) new physics is almost natural, and an extra singlet-like Higgs is the lightest new particle around (conceivable e.g. in the NMSSM and in Twin Higgs); ii) independently of naturalness, Dark Matter is an electroweak multiplet, the prototype of a WIMP.
I will mainly focus on the reach of colliders in the first case, and of telescopes in the second one.