Our proposal builds on the successful start made by Cambridge Centre for Analysis (CCA), a current EPSRC Centre for Doctoral Training. We propose to develop further our activity in two important and rapidly evolving areas of analysis, namely mathematics of information and statistics of complex systems. Beginning with Newton, for whom the development of calculus and the mathematical understanding of bodies in motion were closely intertwined, the mathematics used to describe real phenomena consistently involves notions of continuity, rate of change, average value, and basic challenges such as the relationship between discrete and continuum objects. This is the domain of analysis, encompassing modelling by partial differential equations and by random processes, and the mathematical theory which guides effective computation for such models. The centrality of mathematical analysis in the relationship between mathematics and its applications has been acknowledged by successive International Reviews of Mathematics, as has the need to increase the capacity of UK PhD training in analysis. Mathematical Analysis and its Applications is an EPSRC Priority Area. Beyond the established and important uses of analysis in modelling physical phenomena, digital technology has created new areas where mathematical analysis, in guiding the extraction of knowledge from massive discrete systems, plays an essential role. These include the fields of high-dimensional statistics and the mathematics of information, including compressed sensing. In each of these, one is looking for a reliable means to interpret massive high-dimensional data. Already several CCA students are working in these areas. Big Data is one of the Eight Great Technologies championed by the Minister for Universities and Science. Statistics and Data to Knowledge are EPSRC Priority Areas. We propose a first year training programme based on our current successful model, now expanded by two further core courses, one in Statistics of Complex Systems and one in Mathematics of Information. These new courses will be paired with postgraduate level courses from the existing Cambridge Masters' (MASt), which students can use to consolidate their understanding. The core courses themselves are based on supervised student team assignments leading to student presentations. The other main components of the first year are research mini-projects (often the route to a PhD project) and an industry workshop. Years two to four are devoted mainly to the PhD thesis. First year training establishes a collaborative ethos in the cohort and, by mixing students with different prior skills, encourages cross-fertilization of ideas across the different threads of analysis. This is sustained in later years through a programme of seminars, workshops and training in transferable skills. The students appreciate that their collective understanding of a given problem using different skills will often exceed each individual's understanding. This makes cohort-based training especially valuable in analysis. We already expose all our students to the role of mathematics and the opportunities for mathematicians in industry and society, and we encourage first-hand engagement with applications through mini-projects, industrial seminars and study weeks, and, for some, PhD projects with industrial partners. The development of core skills and eventually the ability to generate new ideas is the hardest and crucial part of training as a research mathematician. This is necessarily our overriding task, in which we seek synergy and inspiration from user engagement. In the new CDT, our network of industrial connections will be further enhanced, along with our collaborations with Cambridge engineering colleagues, and our links with the Smith Institute for Industrial Mathematics.

How liquids wet solid surfaces is of fundamental importance to a wide-range of scientific disciplines and technological applications from creating thin films on semiconductor wafers, through adhesion and coating of surfaces, to effective droplet deposition and mixing on DNA microarrays. Electrostatic fields can alter how effectively a liquid wets a solid surface. In recent years uniform electric fields have been used to control and manipulate droplets of conducting (ion containing) liquids, typically a salt solution, by using the liquid-solid contact area as one electrode in a capacitive structure - so called electrowetting. This has led to new voltage controlled variable focus liquid lenses, liquid-based electronic paper and droplet-based microfluidic systems. The key to electrowetting is the ability of an applied voltage to reversibly increase the effective hydrophilicity of a solid surface and reduce the contact angle of the droplet without altering the surface chemistry. However, many liquids of interest are not conducting and the need for a sandwich-style capacitive structure and direct physical contact to the liquid limits its range of applicability. In this project we create a new method of controlling hydrophilicity and oleophilicity of materials by using the dielectric properties of liquids, but with the effects localized to an interface. Unllike electrowetting which focuses on the ions, our method focuses on the dipoles in a liquid. Using a non-uniform electric field generates unequal forces on the two ends of the dipole. The resulting dielectrophoretic force can result in movement and redistribution of the liquid into the areas of highest field gradient. The basis of our project is the understanding that when the liquid has solid-liquid, liquid-vapor or liquid-liquid interfaces, dielectric energy changes can be coupled to surface free energy changes. With a suitable decaying electric field, the effects of liquid dielectrophoresis can be confined to either the solid-liquid interface or to the liquid-vapor (or liquid-liquid) interface and can be used with a non-conducting liquid. By using microfabricated interdigitated electrodes a decaying, and hence non uniform, electric field can be created above a solid surface. For a droplet thicker than the decay length of the electric field, the major change of the surface energy compensating liquid dielectrophoretic energy changes is via a change in the contact area with a solid and so this can be a method of reversibly controlling the contact angle and, hence, the hydro- and oleo- philicity of a surface. For a thin liquid film the major change of the surface energy compensating liquid dielectrophoretic energy changes is via a change in the shape of the liquid-vapor (or liquid-liquid) interface and so, in this case, it becomes a method for shaping a liquid surface. In this method of localizing the effects of liquid dielectrophoresis to an interface the contrast to electrowetting is that, 1. the electric fields are non-uniform; 2. the electric fields are generated by surface microfabricated co-planar rather than sandwich electrode structures; 3. the forces act upon the dipoles in the liquids, which can therefore be non-conducting (or conducting), rather than upon ions of conducting liquids; 4. the method does not suffer from saturation of the contact angle and so can be used to produce liquid films. The research in this project seeks to establish an approach to wetting that allows conducting and non-conducting liquids to be manipulated using electric fields in a manner complementary to electrowetting. The project will provide the understanding needed to allow future development of novel droplet microfluidic, liquid microactuation, liquid-based optics and displays. The project includes industrial partners who have expertise in the development and commercialisation of microfluidic liquid handling, lab-on-chip devices, display devices and optofluidic systems.

The security of many cryptographic protocols in use today relies on the computational hardness of mathematical problems such as integer factorization. These problems can be solved using quantum computers, and therefore most of our security infrastructures will become completely insecure once quantum computers are built. Post-quantum cryptography aims at developing security protocols that will remain secure even after quantum computers are built. The biggest security agencies in the world including GCHQ and the NSA (the American National Security Agency) have recommended a move towards post-quantum protocols, and the new generation of cryptographic standards will aim at post-quantum security. Driven by the need to upgrade our cybersecurity infrastructures, many cryptographic algorithms have recently been developed which are claimed to offer post-quantum security. These proposals are based on a few distinct mathematical problems which are hoped to remain difficult for quantum computers, including lattice problems, multivariate polynomial system solving, coding theory problems, isogeny problems, and the security of cryptographic hash functions. Unfortunately, many of these problems, and more importantly the cryptographic algorithms that are built on top of them, have not been subject to a thorough security analysis yet, therefore leaving us with a risk to oversee major weaknesses in algorithms to be deployed in security applications. In this fellowship, we will develop breakthrough cryptanalysis techniques to analyse the security of post-quantum cryptography candidate algorithms, and determine which algorithms may or may not be further considered for digital security applications. Using the insight gained through cryptanalysis, we will then develop new post-quantum cryptographic algorithms offering better security, efficiency and functionality properties in applications.

This programme will lay the scientific foundations for a new generation of sensor systems that will be mounted in vehicles to enhance the safety and efficiency. The sensors, small enough to be mounted unobtrusively on vehicles, will allow high resolution images to be produced in real time, that can be read and interpreted by intelligent vehicle systems to determine appropriate actions in hazardous circumstances and to dynamically control the vehicle to reduce fuel consumption. Sharing the images, or the information obtained from them, with the infrastructure and with other vehicles, will also make it possible to enhance safety and efficiency collectively within whole cohorts of vehicles. Sensors based on this technology will impact on future integrated automotive transport systems, supporting an intelligent transport philosophy with efficient use of renewable energy sources, low carbon emissions and enhanced safety for all road users. The new sensors will exploit the technology of circuits and devices in the 0.3 THz to 1 THz frequency range. Although this range, falling in between the upper end of the radio spectrum and the lower end of the infra-red, is currently not widely used, the device and circuit technology will mature over the next decade. There are several potential advantages in the use of this frequency band, as opposed to the lower frequency microwave and mm-wave bands or the infra-red and optical bands. The antennas required in the low THz band are smaller than those in the microwave and mm-wave bands, in proportion to the wavelength. The image resolution achievable is improved. There are two reasons for this. Firstly, narrower beams can be produced while using reasonably small antennas, when the wavelength is so short (less than 1 mm). Secondly, the high bandwidths available when using such high frequencies make it possible to distinguish between more closely spaced features in the reflected signal. At the same time, waves in this band are not susceptible to complete obscuration by road dirt or precipitation, as infra-red and optical systems would be. Before and during this work, there will be a strong focus on vehicle system applications, with input from automotive industry experts, to identify the specific requirements of future vehicle systems. To generate the required images, low THz waves must be transmitted from the vehicle, propagate through the surrounding environment and be scattered from objects and surfaces. Scattered waves propagating back to the vehicle and received by the sensor antenna provide the information required to form an image. The main research work activities in this project all relate to these physical aspects of the imaging systems. Firstly, the properties of the road environment will be determined to find the specific frequency windows in which low THz signals can propagate through air, precipitation, vehicle exhaust gases, road spray and airborne particles such as dirt and grit. This will involve a combination of measurements in controlled, enclosed artificial environments created in the laboratory, and real road trials. Then, the scattering properties of typical road scenes and surfaces will be analysed to determine the most appropriate frequencies and waveforms to use for imaging. A major part of the research will involve the study of the antennas and beamforming networks that will be required to implement low THz imaging systems on vehicles. Working at the boundary between the radio frequency spectrum and the optical spectrum provides opportunities to exploit and merge transmitter concepts based on both lenses and antennas. The system requirements will be studied to arrive at recommendations for transmitter and receiver architectures that could be realised using the emerging circuit and device technologies.

The security of many cryptographic protocols in use today relies on the computational hardness of mathematical problems such as integer factorization. These problems can be solved using quantum computers, and therefore most of our security infrastructures will become completely insecure once quantum computers are built. Post-quantum cryptography aims at developing security protocols that will remain secure even after quantum computers are built. The biggest security agencies in the world including GCHQ and the NSA have recommended a move towards post-quantum protocols, and the new generation of cryptographic standards will aim at post-quantum security. This project will consider cryptography based on isogeny problems, a particular family of protocols that are considered for post-quantum security. Isogeny-based protocols are particularly appealing for three reasons 1) they require very small keys compared to other post-quantum cryptography candidates, saving on bandwidth and storage 2) there exists an isogeny-based version of the widely used Diffie-Hellman protocol, which can be used as a direct replacement of current instantiations 3) their mathematical grounding has a lot in common with currently used elliptic curve protocols, which will accelerate implementations in a wide range of devices. Isogeny-based cryptography protocols have only been invented recently, and like many other protocols currently investigated for post-quantum security they yet have to survive the "test of time". As they have not been investigated as thoroughly as currently deployed solutions, they may be more vulnerable to unanticipated weaknesses. Moreover, the protocols are still at the stage of theoretical papers and remain to be evaluated against the specific constraints of real-life applications. This project will advance the field of isogeny-based cryptography, from its mathematical grounding right up to the application of protocols in the real world. We will develop new protocols, new analysis techniques, and determine the suitability of isogeny-based cryptography for selected applications.