An enormous amount of individuals' data is collected every day. These data could potentially be very valuable for scientific and medical research or for targeting business. Unfortunately, privacy concerns restrict the way this huge amount of information can be used and released. Several techniques have been proposed with the aim of making the data anonymous. These techniques however lose their effectiveness when attackers can exploit additional knowledge. Differential privacy is a promising approach to the privacy-preserving release of data: it offers a strong guaranteed bound on the increase in harm that a user I incurs as a result of participating in a differentially private data analysis, even under worst-case assumptions. A standard way to ensure differential privacy is by adding some statistical noise to the result of a data analysis. Differentially private mechanisms have been proposed for a wide range of interesting problems like statistical analysis, combinatorial optimization, machine learning, distributed computations, etc. Moreover, several programming language verification tools have been proposed with the goal of assisting a programmer in checking whether a given program is differentially private or not. These tools have been proved successful in checking differentially private programs that uses standard mechanisms. They offer however only a limited support for reasoning about differential privacy when this is obtained using non-standard mechanisms. One limitation comes from the simplified probabilistic models that are built-in to those tools. In particular, these simplified models provide no support (or only very limited support) for reasoning about explicit conditional distributions and probabilistic inference. From the verification point of view, dealing with explicit conditional distributions is difficult because it requires finding a manageable representation, in the internal logic of the verification tool, of events and probability measures. Moreover, it requires a set of primitives to handle them efficiently. In this project we aim at overcoming these limitations by extending the scope of verification tools for differential privacy to support explicit reasoning about conditional distributions and probabilistic inference. Support for conditional distributions and probabilistic inference is crucial for reasoning about machine learning algorithms. Those are essential tools for achieving efficient and accurate data analysis for massive collection of data. So, the goal of the project is to provide a novel programming language technology useful for enhancing privacy-preserving data analysis based on machine learning.

Programming is intrinsically based on the use of limited resources, such as memory and processing power of computers. Various abstractions of resources play an important role throughout computer science, but they are conceptualised in very different, and apparently unrelated ways. In particular, there is a big gap between studies focussing on precise quantitative issues of what we can do and how efficiently we can do it with limited resources, and those which concern more conceptual aspects, which underpin modern high-level programming languages, and application-oriented programming. In this project, building on some recent breakthrough developments which relate these different aspects, we aim to develop a unified theory of resources which will apply to all these aspects, and allow a flow of ideas between them. This will provide new tools and methods for computer scientists, and lead both to new kinds of results, and more general versions of existing ones.

Stochastic growth phenomena naturally emerge in a variety of physical and biological contexts, such as growth of combustion fronts or bacterial colonies, crystal growth on thin films, turbulent liquid crystals, etc. Even though all these phenomena might appear very diverse at a microscopic scale, they often have the same large-scale behaviour and are therefore said to belong to the same Universality Class. This in particular means that an in-depth analysis of those processes describing these large-scale behaviours is bound to give very accurate quantitative and qualitative predictions about the wide variety of extremely complicated real-world systems in the same class. Over the last 40 years, the Mathematics and Physics communities in a joint effort determined what were widely believed to be the only two universal processes presumed to capture the large-scale behaviour of random interfaces in one spatial-dimension, namely the Kardar-Parisi-Zhang and Edrwards-Wilkinson Fixed Points, and studied their Universality Classes. In a recent work, I established the existence of a third, new universality class, entirely missed by researchers, and rigorously constructed the universal process at its core, the Brownian Castle. The introduction of this novel class opens a number of new stimulating pathways and a host of exciting questions that this proposal aims at investigating and answering. The second pillar of this research programme focuses on two-dimensional random surfaces, which are particularly relevant from a physical viewpoint as they correspond to the growth of two-dimensional surfaces in a three-dimensional space. Despite their importance, two-dimensional growth phenomena are by far the most challenging and the least understood. Very little is known concerning their universal large-scale properties and the even harder quest for fluctuations has barely been explored. The present proposal's goal is to develop powerful and robust tools to rigorously address these questions and consequently lay the foundations for a systematic study of these systems and their features. The last theme of this research plan concerns the Anderson Hamiltonian, also known as random Schrödinger operator. The interest in such an operator is motivated by its ramified connections to a variety of different areas in Mathematics and Physics both from a theoretical and a more applied perspective. Indeed, the spectral properties of the Anderson Hamiltonian are related to the solution theory of (random) Schrödinger's equations or properties of the parabolic Anderson model, random motion in random media or branching processes in random environment. The Anderson Hamiltonian has attracted the attention of a wide number of researchers, driven by the ambition of fully understanding its universal features and the celebrated phenomenon Anderson localisation. This proposal will establish new breakthroughs and tackle long-standing conjectures in the field by complementing the existing literature with novel techniques.

When we begin to study mathematics, we learn that the operation of multiplication on numbers satisfies some basic rules. One of these rules, known as associativity, says that for any three numbers a, b and c, we get the same result if we multiply a and b and then multiply the result by c or if we multiply a by the result of multiplying b and c. This leads to the abstract algebraic notion of a monoid, which is a set (in this case the set of natural numbers) equipped with a binary operation (in this case multiplication) that is associative and has a unit (in this case the number 1). If we continue to study mathematics, we encounter a new kind of multiplication, no longer on numbers but on sets, which is known as Cartesian product. Given two sets A and B, their Cartesian product is the set A x B whose elements are the ordered pairs (a, b), where a is an element of A and b is an element of B. Pictorially, the Cartesian product of two sets is a grid with coordinates given by the elements of the two sets. This operation satisfies some rules, analogous to those for the multiplication of numbers, but a little more subtle. For example, if we are given three sets A, B and C, then the set A x (B x C) is isomorphic (rather than equal) to the set (A x B) x C. Here, being isomorphic means that we they are essentially the same by means of a one-to-one correspondence between the elements A x (B x C) and those of (A x B) x C. This construction leads to the notion of a monoidal category, which amounts to a collection of objects and maps between them (in this case the collection of all sets and functions between them) equipped with a multiplication (in this case the Cartesian product) that is associative and has a unit (in this case the one-element set) up to isomorphism. Monoidal categories, introduced in the '60s, have been extremely important in several areas of mathematics (including logic, algebra, and topology) and theoretical computer science. In logic and theoretical computer science, they connect to linear logic, in which one keeps track of the resources necessary to prove a statement. This project is about the next step in this sequence of abstract notions of multiplication, which is given by the notion of a monoidal bicategory. In a bicategory, we have not only objects and maps but also 2-maps, which can be thought of as "maps between maps" and allow us to capture how different maps relate to each other. In a monoidal bicategory, we have a way of multiplying their objects, maps and 2-maps, subject to complex axioms. Monoidal bicategories, introduced in the '90s, have potential for applications even greater than that of monoidal categories, as they allow us to keep track of even more information. We seek to realise this potential by advancing the theory of monoidal bicategories. We will prove fundamental theorems about them, develop new connections to linear logic and theoretical computer science and investigate examples that are of interest in algebra and topology. Our work connects to algebra via an important research programme known as "categorification", which is concerned with replacing set-based structures (like monoids) with category-based structures (like monoidal categories) in order to obtain more subtle invariants. Our work links to topology via the notion of an operad, which is a flexible tool used to describe algebraic structures in which axioms do not hold as equalities, but rather up to weak forms of isomorphism. Overall, this project will bring the theory of monoidal bicategories to a new level and promote interdisciplinary research within mathematics and with theoretical computer science.

Our everyday understanding of perception is that our sense organs enable us to see, touch, smell, taste and hear. The vocabulary of five distinct senses ramifies through descriptions of thought ("I see what you mean") emotion ("I was touched by her suffering") and aesthetics ("That's not to my taste"). Traditionally, philosophers have also thought that the five senses producing distinctive, separate conscious experiences. Equally, until recently, scientists have also studied each of the senses in isolation. But modern neuroscience is radically changing our understanding. Each sense organ contains many kinds of sensory receptors (think of all the different feelings from your skin). Everyday experiences - watching a film, eating a meal, walking along the street - involve different senses, working together. But most remarkable is a mass of recent research showing highly specific sensory interactions, in which one sense modifies the experience of another. Imagine listening to a syllable (say /ba/) spoken over and over, while watching a video of someone mouthing a different syllable (say /ga/), you actually hear the sound differently. Equally, the voice of a ventriloquist seems to come from the mouth of a doll some distance away. Somehow, what we see changes what we hear, presumably through processes that normally help us to associate sounds and sights correctly. Flavour provides the most surprising examples of sensory interaction, What we call the "taste" of food and drink is largely determined by smell rather than taste, but it also depends on the temperature and texture of food and drink, and its colour, and even the sounds that accompany eating. For instance, white noise reduces sensitivity to flavour (the so-called "aircraft food effect"). Equally, your sense of your own body can be changed by what you see and feel. If you look at a model hand being stroked with a brush, while your own hand, out of sight, is simultaneously stroked, you will soon feel that the model hand is part of you. The traditional view that information flows in one direction from basic sensation to perception, memory and action, has also been overturned. Recognising a spoken word, a familiar face, or a favourite piece of music draws on previous knowledge. Perception is influenced by memory, expectation, emotion and attention. Further, since our head, hands and eyes are constantly in motion, the brain must somehow stitch together perception from a sequence of sensory "snapshots". A comprehensive account of perception needs to begin with the relationships and interactions between the sensory modalities that produce our awareness of the world and of ourselves in it. Although the science of perception is moving very fast, it lacks the conceptual framework that philosophical thinking can bring to understanding the relationship between brain processes and experience. Our plan is for philosophers, psychologists and neuroscientists to work together in entirely new ways, including planning laboratory experiments together, to help us to understand how the brain puts together different sensory information, under the influence of past experience and expectation, to create the seamless flow of conscious experience, to identify objects and events in the world, to give us an sense of our own body, and to enable us to control our actions. We believe that our work will have wide impact beyond our university departments. It will help in the design of new forms of prosthetic devices to help deaf and blind people, and those who suffer untreatable pain, changes in body image or reduction in the sense of smell. It will inform the rapidly advancing technology of enhancement of sensory experience, cast light on the appreciation of the visual and performing arts, and stimulate new forms of preparation and presentation of food, and new understanding of the way in which people choose what products to buy, what works of art they prefer and what food they eat.