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Topology has many different manifestations in condensed matter physics. Real space examples include topological defects such as vortices, while momentum space ones include topological band structures and singularities in the electronic dispersion. In this talk, I will focus on two examples. The first is that of a vortex in a topological insulator that is doped into the superconducting state. This system, we find, has Majorana zero modes and thus, is a particularly simple way of obtaining these states. We derive general existence criteria for vortex Majorana modes and find that existing systems like Cu-doped Bi2Se3 fulfill them. In the process, we discover a rare example of a topological phase transition within a topological defect (the vortex) at the point when the criteria are violated. The second example is that of Weyl semimetals, which are three-dimensional analogs of graphene. Interestingly, the Dirac nodes here are topological objects in momentum space and are associated with peculiar Fermi-arc surface states. We discuss charge transport in these materials in the presence of interactions or disorder, and find encouraging agreement with existing experimental data.

A non-trivial test of the string vs. integrability correspondence is suggested: exact equivalence is established between strings in AdS4 x CP3 and the Gromov-Vieira all-loop integrable chain. To do that, the complete one- and two-magnon sector of each respective theory are calculated. In the single-magnon sector a direct perturbative one-loop calculation proves the validity of the dispersion law coming from the Bethe Ansatz, rather than of the one coming from the semiclassical analysis. In the two-magnon sector the full spectrum of the finite-size corrections has been calculated on the string side by us for the first time, that proves to be identical to the integrable chain spectrum. These results are interpreted by us as a confirmation of the exactness of the conjectured Gromov-Vieira Bethe Ansatz.

This talk will deal with a new connection formulation for higher-dimensional (Super)gravity theories and its applications. We will start by reviewing the basic ideas of loop quantum gravity. Next, the derivation of the new connection formulation will be discussed and it will be shown that the quantization methods developed in the context of loop quantum gravity apply. We comment on applications of the framework, focusing on making contact with String theory.

Cold atomic gases in optical lattices are emerging as excellent laboratories for testing models of strongly interacting particles in condensed matter physics. It is possible to tune the interactions, dimensionality, spin, statistics and a host of other variables in a completely disorder free environment. This has opened up unique possibilities of mapping out phase diagrams of quantum models and observing quantum phase transitions for the very first time. I will discuss some of the challenges in this field.

Dualities in physics are well known for their conceptual depth and quantitative predictive power in contexts where perturbation theory is unreliable. They are also remarkable for the staggering arrange of physical problems that exploit them, ranging from the study of confinement and unconventional phases in statistical mechanics and field theory to the unification of the string theory landscape. In this talk I will present a new, completely general approach to dualities that affords a systematic theory of quantum dualities and incorporates classical dualities as well into one unified framework. This new algebraic approach is remarkably successful in extending powerful duality techniques to the context of topologically quantum ordered systems and quantum information processing, and affords a compelling foundation for a general theory of exact dimensional reduction or holographic correspondences. Many systems however display only approximate dimensional reduction, and so I will present general inequalities -some of them based on entanglement- linking quantum systems of different spatial dimensionality. These inequalities provide bounds on the expectation values of observables and correlators that can enforce an effective dimensional reduction. In closing I will discuss some implications for (topological) quantum memories.