In this lecture a short introduction to basic observational facts about neutron stars is given. In their interior, the ultimate densities of baryon-rich matter in the Universe are realized at zero temperature (mean kinetic energy well below the Fermi energy).

It is explained how with measurements of masses and radii of pulsars (rotating neutron stars) strong constraints for the cold nuclear matter equation of state (EoS) can be obtained.

They will play a similar role for developing effective models of the cold, baryon-rich EoS as simulations of lattice QCD do for the the hot, baryon-poor EoS. It is suggested that radius measurements of high-mass pulsars might decide whether a strong first order deconfinement transition occurs in compact stars.

В лекции рассматриваются различные способы вывода аномалии с помощью методов, имеющих простую физическую интерпретацию. Обсуждается сокращение явного и аномального нарушения киральной инвариантности и его проявления для странных кварков. Анализируются киральные магнитный и вихревой эффекты и их теоретиковозмущенческие аналоги.

In this lecture, the principal difference of this approach to all previously discussed approaches will be explained.

The most successful application of Lattice Gauge Theory in general, which is a particular set of field thories formulated on discretized space-time, is Quantum Chromodynamics, considered as Statistical Field Theory which is amenable to direct "analog simulation", even beyond the reach of perurbation theory.

In lesson the natural transcription of the field degrees of freedom to a manageable amount of data that can be dealt with on computers will be explained, and the impressive possibilities given by "importance sampling" (as long as this method is applicable) .

The basic "observables" will be explained and the way they are nterpretated , which has made the lattice approach a good candidate to answer the basic problems of confinement and chiral symmetry breaking, which make QCD so different from other parts of the Standard Model.

• Lifshitz theories: Generalities, motivations, basic examples
• Detailed balance, entropic force and Onsager-Machlup theory
• Horava-Lifshitz gravity
• Instantons of Lifshitz theories, in general, as gradient flow equations; entropy functionals
• Instantons solutions of Horava-Lifshitz gravity: Construction and classification via geometric flows, comparison with Einstein gravity
• Quantum anomalies: axial and conformal anomalies in relativistic and Lifshitz theories, Dirac-Lifshitz operator, index computations, eta invariant
• Chiral symmetry breaking via instantons in Lifshitz models, level crossing, harmonic spinors and comparison of results with relativistic quantum field theories
• Cosmological solutions: mixmaster universe or BKL analysis for HL gravity
• Some open questions and overview