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The inner crust of neutron stars has a remarkable property that it is crystalline as well as superfluid. I will describe the low energy theory of systems with this property in general, and describe how to relate the low energy constants of the theory to derivatives of the free energy with respect to lattice shape and chemical potentials. As an application, I will discuss the mixing of lattice and superfluid modes in the neutron star inner crust.

Low-density neutron matter is relevant to the study of neutron-rich nuclei and neutron star crusts. Unpolarized neutron matter has been studied extensively over a number of decades, while experimental guidance has recently started to emerge from the field of ultracold atomic gases. We study population-imbalanced neutron matter (possibly relevant to magnetars and to density functionals of nuclei) applying a Quantum Monte Carlo method that has proven successful in the field of cold atoms. We report on the first ab initio simulations of superfluid low-density polarized neutron matter. For systems with small imbalances, we find a linear dependence of the energy on the polarization, the proportionality coefficient changing with the density. We also present results for the momentum and pair distributions of the two fermionic components.

We present new equations of state (EOS) of nuclear matter for a wide range of temperatures densities and proton fractions for use in supernova and neutron star merger simulations. We employ a full relativistic mean field (RMF) calculation for matter at intermediate density and high density and the Virial expansion of a nonideal gas for matter at low density. We tabulate the resulting EOS in the temperature range T = 0 - 80 MeV the density range nB = 10â

We construct the relativistic equation of state (EOS) of dense matter covering a wide range of temperature proton fraction and density for the use of core-collapse supernova simulations. The study is based on the relativistic mean-field (RMF) theory which can provide an excellent description of nuclear matter andfinite nuclei. The Thomas-Fermi approximation is adopted to describe the non-uniform matter which is composed of a lattice of heavy nuclei.We present two types of results. The first one takes into account only the nucleon degree of freedom while the second one considers additional contributions from Lambda hyperons. We tabulate the resulting EOS with an improved design of ranges and grids.

Recent discoveries, including a 2 solar mass pulsar, rapid cooling in the Cas A supernova remnant, and estimates of masses and radii from photospheric radius expansion bursts and thermal emissions from neutron stars, are able to constrain significantly the properties of dense matter. Implications for the pressure-density relation and properties of superfluids in neutron star interiors will be discussed.

Tensor network algorithms are a powerful technique for the study of quantum systems in condensed matter physics. In this short series of lectures, I will present an applied perspective on tensor network algorithms. Topics to be covered will include motivation and methodology, graphical notation, Matrix Product States (MPS) and the Time-Evolving Block Decimation (TEBD) algorithm, identifying the capabilities and limitations of tensor network algorithms, the Multi-scale Entanglement Renormalisation Ansatz (MERA) and the study of systems at criticality, and the exploitation of global internal symmetries. The intent of this lecture series is to provide attendees with the necessary theoretical background to be able to understand and implement the more common tensor network algorithms.

Cosmic inflation has given us a remarkably successful cosmological phenomenology. But the original goal of explaining why the cosmos is *likely* to take the form we observe has proven very difficult to realize. I review the status of "eternal inflation" with an eye on the roles various infinities have (both helpful and unhelpful) in our current understanding. I then discuss attempts to construct an alternative cosmological framework that is truly finite, using ideas about equilibrium and dark energy. I report some recent results that point to observable signatures.

Tensor network algorithms are a powerful technique for the study of quantum systems in condensed matter physics. In this short series of lectures, I will present an applied perspective on tensor network algorithms. Topics to be covered will include motivation and methodology, graphical notation, Matrix Product States (MPS) and the Time-Evolving Block Decimation (TEBD) algorithm, identifying the capabilities and limitations of tensor network algorithms, the Multi-scale Entanglement Renormalisation Ansatz (MERA) and the study of systems at criticality, and the exploitation of global internal symmetries. The intent of this lecture series is to provide attendees with the necessary theoretical background to be able to understand and implement the more common tensor network algorithms.

The Polchinski equations for the Wilsonian renormalization group in the $D$--dimensional matrix scalar field theory can be written at large $N$ in a Hamiltonian form. The Hamiltonian defines evolution along one extra holographic dimension (energy scale) and can be found exactly for the subsector of $Tr\phi^n$ (for all $n$) operators. We show that at low energies independently of the dimensionality $D$ the Hamiltonian system in question reduces to the {\it integrable} effective theory. The obtained Hamiltonian system describes large wavelength KdV type (Burger--Hopf) equation with an external potential and is related to the effective theory obtained by Das and Jevicki for the matrix quantum mechanics.