Abstract:
The troubles with causality in quantum theory were first unveiled in 1935 by Einstein, Podolsky and Rosen. The apparent paradox persisted for almost half a century until it was understood that although quantum correlation are non-local, they do not allow for any superluminal information transfer. On the other hand, in 1994 Popescu and Rohrlich realised that there could be an even 'more nonlocal' theory, which is still compatible with relativity. Since then a few different 'post-quantum' theories have been discovered. Although their physical significance is questionable, they provide a novel viewpoint 'from outside' on quantum mechanics. In my talk I will explain the basic philosophy behind 'post-quantum' theories and outline my own contributions to this field.

Abstract:
One of the most fruitful ideas in the study of phase transitions is Landau's theory, which classifies phases of matter according to their symmetries. Despite this success, it is currently known that there exist transitions not driven by spontaneous symmetry breaking. Distinct phases often may be distinguished by observables such as ground state degeneracy, which depends on topology of the material sample, or other topological invariants. Topological Quantum Field Theories (TQFTs) have been proposed as a framework for theoretical study of these phenomena. In this introductory talk I will present simple examples of TQFTs and an axiomatic definition due to Atiyah.

Abstract:
Both global symmetries and gauge transformations play an essential role in understanding various phenomena present in physical models. Sometimes even more crucial is the the lack of the global symmetry, the spontaneous symmetry breaking. On the other hand, the question about the possibility of gauging global symmetries leads to the concept of 't Hooft anomalies. All these aspects can be easily generalized to the case of so-called higher symmetries. In this talk, being the second one of the whole series, I will present the systematic treatment of global higher symmetries proposed by D.Gaiotto, A. Kapustin, N. Seiberg and B. Willett. Then the notion of higher gauge theory will be explored, together with the mathematical prelimaries essential for the forthcoming talks in this series.

Abstract:
In my presentation I will briefly explain how to construct lattice model whose (gauge) symmetry is described by two ``intertwined'', possibly non-abelian groups and try to motivate its physical relevance.

Abstract:
In the talk I will try to discuss the content of the recent proposal by Simon Catterall about Chiral Lattice Theories. I will explain its relevance and present previous historical attempts and numerical results.

Abstract:
In the fourth talk in this series I will discuss the notion of classifying spaces and their role in topology. I will concentate on those properties which will be useful to describe Dijkgraaf-Witten theories.

Abstract:
Dijkgraaf-Witten theory is a topological quantum field theory whose degrees of freedom are gauge fields with a finite gauge group. It is constructed in terms of a path integral, which can be defined rigorously in this case. The action is defined by integrating the pullback of a U(1)-valued cocycle from the classifying space of the gauge group. It is closely related to the Chern-Simons action. As shown by Dijkgraaf and Witten, these highly abstract notions can be translated to a concrete lattice model, which will be described in this talk.

Abstract:
In the second part of my talk I will finish the construction of the Dijkgraaf-Witten theory. The main part is the definition of its action and explanation how it relates to the so called group cohomology. An emphasis will be made on gauge-variance of this action on manifolds with boundary and consequences of this for the algebra of operators. Connection with anomalies will be discussed briefly.

Abstract:
The Skyrme model is a low energy effective field theory of strong interactions where nuclei and baryons appear as topological solitons, more concretely as collective excitations of pionic degrees of freedom. Proposed by Tony Skyrme in the sixties, his ideas received further support when it was discovered that in the limit of the large number of colours of QCD, an effective theory of mesons arises. In the last years, there has been a revival of Skyrme's ideas and new related models have been proposed to overcome two of the main drawbacks of the theory, namely, the too large binding energies and the lack of cluster structures. The aim of this talk is to successfully address both issues at the same time, something that has not been done before, by extending the standard Skyrme model with the inclusion of the rho meson, via dimensional deconstruction of pure Yang-Mills theory in one higher dimension.

Abstract:
Symmetry protected topological (or SPT) phases of matter are not described by Landau's paradigm of symmetry breaking. During my presentation I will give their definition, present a few examples and discuss some mathematical tools which are used in their classification. The talk is based on the literature and contains no new results.

Abstract:
Classical Dirac field is scattered in an external, time-dependent electromagnetic field, of the form expected to resemble that in the fully interacting, closed Maxwell-Dirac theory. We show that an appropriate choice of an electromagnetic gauge, and a Cauchy foliation of the spacetime solves the large time infrared problems of this system: clear identification of the free in/out fields, and asymptotic completeness.

Abstract:
Based on the assumption that infrared problems should disappear inside the future lightcone, Buchholz and Roberts proposed a certain class of representations of relativistic QED. The first part of the talk concerns a construction of asymptotic photon fields in such representations. It will also be shown that single-electron states, if they exist, are vacua of the asymptotic photon fields. This observation allows for a consistent construction of the outgoing wave-operator of Compton scattering. In the second part of the talk I will discuss the problem of velocity superselection in a non-relativistic model of QED. It will be shown that this aspect of the infrared problem indeed disappears after restriction to the future lightcone. (Based on joint works with S. Alazzawi and D. Cadamuro).