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Expressions of several information theoretic quantities involve an optimization over auxiliary quantum registers. Entanglement-assisted version of some classical communication problems provides examples of such expressions. Evaluating these expressions requires bounds on the dimension of these auxiliary registers. In the classical case such a bound can usually be obtained based on the
Caratheodory theorem, but we know almost no method to bound the dimension of auxiliary quantum registers. In this talk to compare the classical and quantum sides of the problem the notion of “quantum convexification” will be defined. It will be shown that quantum convexification is strictly richer that the usual classical convexification. Moreover some techniques will be discussed which might be useful for bounding the dimension of quantum auxiliary registers. 

By way of presenting some classic and many new results, my talk will indulge shamelessly in
advertising "Causal Dynamical Triangulations (CDT)" as a hands-on approach to nonperturbative quantum gravity that reaches where other approaches currently don't. After summarizing the rationale and basic ingredients of CDT quantum gravity and some of its key findings (like the emergence of a classical de Sitter space), I will focus on some very recent results: how we uncovered the presence of a second-order phase transition (so far unique in 4D quantum
gravity), news on the controversy between doing things the Lorentzian or the Euclidean way, and new results on the role of the preferred time slicing in CDT - all hopefully worth your while!

We discuss well-posed initial-boundary value formulations
in general relativity. These formulations allow us to construct solutions of
Einstein's field equations inside a cylindrical region, given suitable initial
and boundary data. We analyze the restrictions on the boundary data that result
from the requirement of constraint propagation and the minimization of spurious
reflections, and choosing harmonic coordinates we show how to cast the problem
into well-posed form. Then, we consider the particular case where the boundary
represents null infinity of an asymptotically flat spacetime. Here, the rôle of
the boundary conditions is to provide adequate regularity and gauge conditions
at infinity.

As an application of our setup we mention ongoing work on
the computation of quasi-stationary scalar field configurations on a
non-rotating supermassive black hole background.

Many of the topological insulators, in their naturally
available form are not insulating in the bulk. It has been  shown that some of these metallic compounds,
become superconductor at low enough temperature and the nature of their
superconducting phase is still widely debated. In this talk I show that even
the s-wave superconducting phase of doped topological insulators, at low
doping, is different from ordinary s-wave superconductors and goes through a
topological phase transition to an ordinary s-wave state by increasing the
doping. I show that the critical doping is determined using the
SU(2) Berry phase on the fermi surface of doped
topological insulator and can be modified by different tunable features of the
material. At the end I present the results of a recent experiment on the
Josephson junctions made of thin films of Bismuth selenide , which can be
explained using our theory of doping induced phase transition in topological
insulators.

Causal set quantum gravity is based on the marriage between the concept of causality as an organising principle more basic even than space or time and fundamental atomicity. Causal sets suggest novel possibilities for "dynamical laws" in which spacetime grows by the accumulation of new spacetime atoms, potentially realising within physics C.D. Broad's concept of a growing block universe
in which the past is real and the future is not. To do justice to relativity and general covariance, the atoms must accumulate in a partial order, exactly the order that the atoms have physically amongst themselves. That this is possible is demonstrated by the Rideout-Sorkin Classical Stochastic Growth models. This proof of concept -- of the compatibility of relativity and ``becoming'' -- is, however, classical and is challenged by the global character of the physical world within a path integral framework for quantum theory. Out of the struggle to reconcile the global and local natures of the physical world may arise a quantal dynamics for causal sets.

In this talk I will focus on two topics concerning compact binaries that involve a neutron star companion: a) the fate of and observable signatures from merging white dwarf-neutron star (WDNS) binaries, and b) electromagnetic signals from black hole - neutron star (BHNS) binaries.WDNS systems - the neglected child among compact binaries - generate detectable gravitational waves (GWs) and may also generate observable electromagnetic (EM) signals. One of the most fascinating aspects about these systems is that they are known to exist, but the final fate of massive, merged WDNSs remnants is currently work in progress. Determining the fate of WDNS remnants will be important for interpreting observations from future transient surveys. I will review recent work that provides insight into the physics of WDNSs remnants. Black hole - neutron star systems are among the most promising sources for gravitational waves, and at the same time also possible sources of detectable precursor and aftermath EM signals. I will present recent the results from general relativistic force-free simulations of binary BHNSs and a type of EM precursor signatures expected from these systems.

One of the main challenges that we face both as individual persons and as a species concerns the distribution and use of resources, such as water, time, capital, computing power or negatively valued
resources like nuclear waste. Also within theoretical physics, one frequently deals with resources like free energy or quantum entanglement. I will describe a mathematical theory of resources which makes quantitative predictions about how many resources are required for producing a certain commodity and outline some applications to information theory.
Joint work with Bob Coecke and Rob Spekkens.