This proposal falls into the general area of design and analysis of algorithms for discrete optimization problems. Such problems arise in Business Analytics, Management and Computer Sciences and in all Engineering subfields. The variety of models and problems arising in this area is astonishing. Nevertheless the method of choice to solve such problems in practice is some combination of mathematical programming solver (CPLEX, Gurobi, IPOPT) of a relaxed problem where some of the problem constraints (like integrality of decision variables) are relaxed or dropped and some rounding algorithm that converts a relaxed solution into a solution of the original problem. In many cases such practical algorithms work in multiple stages by slowly transforming the relaxed solution into an unrelaxed one while constantly monitoring the quality of the current solution. On the other hand it was long recognized in the Theoretical Computer Science, Mathematical Programming and Operations Research communities that understanding the performance of various methods to transform an optimal or near-optimal solution of an "easy" optimization problem into a high quality solution of a "hard" optimization problem is the key to understanding the performance of practical heuristics and design new algorithms to solve hard optimization problems. Such methods are usually called rounding algorithms since they usually transform a fractional solution into an integral one. In this project we would like to apply the modern methods of Probability Theory, Matroid and Polyhedral Theories to explain why such algorithms perform well in practice. We also would like to design new algorithms for transforming solutions of relaxed practically relevant optimization problems into solutions of original hard optimization problems. Along the way we would like to design new concentration inequalities of random processes associated with our probabilistic rounding algorithms. Such concentration inequalities are useful in explaining the quality of randomized rounding procedures and can lead to design of new rounding algorithms.

What don't we know? Recently, the journal Science selected 25 of the biggest unanswered questions facing scientists over the next 25 years. Number two on the list, right after figuring out the composition of the universe, is: What is the biological basis of consciousness? This indeed is a big question. Scientific descriptions of conscious experience, volition, and subjectivity will follow in the footsteps of Copernicus and Darwin by restructuring our relationship with each other and with nature, and many clinical and technological applications will follow.A scientific account of consciousness will not arrive fully formed in a 'Eureka' moment. What is needed is a multidisciplinary, integrative approach combining theory and experiment and exploiting the interchange between the information/computation sciences and the neural, psychological, and medical sciences. At the front-line of this interchange, computational neuroscience (CN) uses computational approaches to model intricate brain processes in much the same way that meteorology uses computers to forecast the weather. In this view and in contrast to early approaches to 'artificial intelligence' (AI), brains are not computers, and intelligent behavior and conscious experience arise from complex brain-body-environment interactions unfolding in temporally precise ways. Much current CN focuses on single levels of description of neural systems (e.g., how neural activity affects connections among neurons) and neglects the multi-scale relations that connect brains, bodies, and behavior. Moreover, current CN is also surprisingly silent with regard to consciousness itself. By targeting and overcoming these limitations, our research will deliver new insights into the neural mechanisms underlying adaptive behavior and conscious experience. We will follow three interacting themes: (i) design and analysis of large-scale CN models to explore how multi-scale neural interactions shape and are shaped by brain-body-environment interactions; (ii) development of new theory to identify causal interactions in complex networks (what we call 'causal network analysis'), and (iii) creation of CN models that account for functionally significant aspects of consciousness, for example that each conscious experience integrates diverse information sources into unified scenes. Theoretical work in the above themes will interact with experimental data from multiple sources. At a fine-grained level we will characterize causal interactions in the intact brain of a pond snail, shedding light on the integrated neural function of a simple (non-conscious) organism as it interacts with its environment. Zooming out, we will apply causal network analysis to brain-imaging data acquired from humans in various states of consciousness, to test predictions based on CN models, and to guide the design of new models. Insights at the fine-grained level will scaffold our understanding of the more complex mechanisms underlying consciousness, with causal networks cross-cutting brains, bodies and environments providing a common theoretical framework. Taken together, these research strands will catalyze an important shift from correlation to explanation in consciousness science.As well as advances in basic science, our research will have important practical benefits at the interface of the biological and information sciences. These will include new design principles for AI/robotic devices, new insights for the design and control of complex technological networks, and new tools for the management of large-scale datasets. A next-generation CN will also underpin new clinical approaches. Many brain-related health problems, from coma to depression to insomnia, can be understood as expressions of disordered consciousness, and many existing clinical approaches are palliative and lacking in theoretical foundation. Our research will provide a theoretical basis for a new generation of effective clinical interventions.

The terahertz (THz) frequency region within the electromagnetic spectrum, covers a frequency range of about one hundred times that currently occupied by all radio, television, cellular radio, Wi-Fi, radar and other users and has proven and potential applications ranging from molecular spectroscopy through to communications, high resolution imaging (e.g. in the medical and pharmaceutical sectors) and security screening. Yet, the underpinning technology for the generation and detection of radiation in this spectral range remains severely limited, being based principally on Ti:sapphire (femtosecond) pulsed laser and photoconductive detector technology, the THz equivalent of the spark transmitter and coherer receiver for radio signals. The THz frequency range therefore does not benefit from the coherent techniques routinely used at microwave/optical frequencies. Our programme grant will address this. We have recently demonstrated optical communications technology-based techniques for the generation of high spectral purity continuous wave THz signals at UCL, together with state-of-the-art THz quantum cascade laser (QCL) technology at Cambridge/Leeds. We will bring together these internationally-leading researchers to create coherent systems across the entire THz spectrum. These will be exploited both for fundamental science (e.g. the study of nanostructured and mesoscopic electron systems) and for applications including short-range high-data-rate wireless communications, information processing, materials detection and high resolution imaging in three dimensions.