We develop a theory of twisted actions of categorical groups using the notion of semidirect product of categories. I will present many examples of semi-direct product of categories. If time permits I will also work-out an example of twisted action involving the Poincaré 2-group. Specializing to the case of representations, where the the category on which categorical group acts has some kind of a vector space structure, we will establish a categorical analogue of Schur's lemma.

This is a joint work with A. Lahiri and A. Sengupta.

This series of lectures is intended to give an elementary introduction to the topic of the canonical quantization of the gravitational field, in the context of the Loop Quantum Gravity approach.

The first lecture will be devoted to the formulation of the problem of quantization of the gravitational field. We will give an overview of perturbative quantization, discuss the issue of nonrenormalizability, and provide a general classification of most prominent approaches to constructing a theory of quantum gravity. One such approach is Loop Quantum Gravity, which will be studied in more detail in subsequent lectures.

This series of lectures is intended to give an elementary introduction to the topic of the canonical quantization of the gravitational field, in the context of the Loop Quantum Gravity approach.

The second lecture is devoted to the canonical quantization procedure within the LQG framework. We will begin by a short introduction to the notion of background independence, and differences between perturbative and nonperturbative quantization. We will then rewrite general relativity in the canonical space+time formulation and introduce Ashtekar variables, as preparation for the canonical quantization. Then the main step is the quantization itself, and the construction of the appropriate Hilbert space of the theory based on the notions of spin networks and spin-knots.

This series of lectures is intended to give an elementary introduction to the topic of the canonical quantization of the gravitational field, in the context of the Loop Quantum Gravity approach.

In the third lecture we will finish the construction of the spin-knot space and introduce the loop transform. Then we move on to the analysis of geometric observables (distance, area and volume) and the structure of the scalar constraint. Finally, matter coupling will be introduced. If time permits, we will also give a short review of two applications of the formalism: calculation of the black hole entropy, and the Big Bounce model of Loop Quantum Cosmology.

After briefly introducing the main ingredients of the loop quantum gravity approach, I show how it can applied to the calculation of black hole entropy. I review some well known results and open issues resulting from the interplay with Chern-Simons theory techniques. I then introduce a new analysis of the horizon degrees of freedom in terms of purely LQG methods, which turns out to be dual to a CFT description. I show how this unifying framework allows us to recover the semiclassical Bekenstein-Hawking entropy formula.

I will discuss the heat equation on the unitary group $U(N)$ in the limit as $N$ tends to infinity, with an eye toward study of the Segal-Bargmann transform. On certain classes of functions on $U(N)$, the Laplacian greatly simplifies as $N$ gets large. Indeed, the large-$N$ limit of the Laplacian satisfies a first-order product rule, meaning that the cross terms in the usual product rule for the Laplacian become negligible. As a result, we are able to obtain a limiting Segal-Bargmann transform on this class of functions. These results are joint work with Bruce Driver and Todd Kemp and were motivated by earlier work of Philippe Bianne.

Theoretical advances in PT symmetry have led to beautiful experimental results in optics, lasers, superconducting wires, atomic diffusion, NMR, microwave cavities, and electronic and mechanical simulations. Experiments on BECs are being planned and new PT-symmetric synthetic metamaterials are being developed. At an abstract level, PT-symmetric classical and quantum mechanics are complex generalizations of conventional classical and quantum mechanics. The advantage of extending physics into the complex domain is that theories that are unstable and physically unacceptable from the narrow perspective of real analysis may become stable and physically viable in the complex domain. This is because the conventional way to determine instability is based on energy inequalities; a theory is thought to be unstable if the potential is unbounded below. In generalizing from the real axis to the complex plane, the notion of ordering is lost; one can say that $x < y$ if $x$ and $y$ are real but not if they are complex. Thus, potentials that are unbounded below on the real axis may not be a problem.

A classical theory that is unstable on the real axis but stable in the complex domain is the $-x^4$ upside-down potential. Quantum theories such as the Lee model, the Pais-Uhlenbeck model, the double-scaling limit of $\phi^4$ theory, and time-like Liouville theory, were believed for many decades to suffer from instabilities (which give rise to ghost states and nonunitarity). These are all physically acceptable PT-symmetric theories. The powerful methods of PT-symmetric quantum theory may resolve even more timely problems such as the stability the Higgs vacuum in the Standard Model.

In this talk we will discuss some topics related to the interaction of physical and mathematical theories that have led to new points of view and new results in mathematics. The area where this is most evident is that of geometric topology of low dimensional manifolds. I coined the term Physical Mathematics to describe this new and fast growing area of research and used it in the title of my paper in Springer's book Mathematics Unlimited: 2001 and beyond.

We will discuss some recent developments in this area. General reference for this talk is my book Topics in Physical Mathematics, Springer (2010).

During the talk I will present a recent construction of observables for General Relativity invariant under spatial diffeomorphisms. The construction involves introducing a local structure representing the "observer" (based on arXiv:1403.8062). I will also present how those observables can be used to reduce the phase space of canonical General Relativity (based on arXiv:1506.09164).

If time permits, I will argue that the construction is particularly useful in spherically symmetric situations. This realization lead to a proposal of a scheme of reducing Loop Quantum Gravity to its spherically symmetric sector, which completes the standard, midisuperspace approach (based on arXiv:1410.5609).

Orthosymplectic $\operatorname{osp}(1|2n)$ superalgebras are being considered as alternatives to $d$-dimensional Poincaré/conformal superalgebras and thus have significant potential relevance in various subfields of High Energy Physics and Astrophysics. Yet, due to mathematical difficulties, even the classification of their unitary irreducible representations (UIR's) has not been entirely accomplished. This is also true for the physically most important subclass of positive energy UIR's.

In this talk I will first demonstrate this classification for the $n=4$ case (that corresponds to four dimensional space-time). The classification is obtained by careful analysis of the Verma module structure, which is particularly subtle due to the existence of subsingular vectors. Based on these results I will then conjecture their generalization to the case of arbitrary $n$ (thus also including cases relevant in the string/brane context). In addition, I will show an elegant explicit realization of these UIR's that exists for (half)integer values of the conformal energy and that makes manifest the mathematical connection existing between UIR's of orthogonal and orthosymplectic algebras. The existence of this realization, per se, proves a part of the conjecture.

This is an introductory talk about the homotopy theory of categories. I will present the classifying space of a category and the classical results of Thomason and Quillen for obtaining categorical descriptions of homotopy colimits and homotopy pullbacks.

In this continuation of my last talk I will give an introduction to the homotopy theory of bicategories. First I will present the way to convert bicategories to spaces, and then I will use this to describe homotopy pullbacks of homomorphisms of bicategories.

In the context of higher gauge theory (HGT) based on a 2-group, I will discuss how the language of double categories provides a natural description of 2-group actions on a category. One of the motivations is to understand moduli spaces of flat connections modulo gauge transformations in HGT, and this goal is achieved for some simple manifolds. I will also relate these ideas to a class of Topological Quantum Field Theories (TQFT's) for surfaces, obtained from finite groups and 2-groups.

This talk is based on callaborations with João Faria Martins, Jeffrey Morton and Diogo Bragança. It is also intended as preparation for the visit by Urs Schreiber, 15 Jan—14 Feb, 2016.

The theory of homological knot invariants - the categorification of polynomial knot invariants - appeared 15 years ago, and has been very active ever since. As in the case of the the quantum polynomial knot invariants, they turned out to be related with numerous different fields of mathematics (including topology, quantum groups, representation theory, homological algebra, von Neumann algebras, etc.). In this talk I'll present a basic overview of this categorification in the case of the HOMFLY-PT invariants - both concerning their definition and their properties. Finally, a particular recent application will be shown related to the physics interpretation via BPS invariants, which implies some surprising integrality properties of a pure number theoretical interest.