One of the oldest topics in mathematics is the study of Diophantine equations (named after the 3rd century Hellenistic mathematician Diophantus of Alexandria). A Diophantine equation is simply a polynomial equation, usually with two or more unknowns, whose coefficients are whole numbers (integers) or fractions (rationals). Consider, for example, 5X+7Y=1, or Y=3X^2. The aim, given a Diophantine equation, is to find integer or rational values for the unknowns (X and Y in the above) so that the equation holds. Such a combination of values is referred to as an integral or rational solution to the equation. In general, the problem of finding integral or rational solutions to a Diophantine equation is extremely difficult and only the simplest cases can be handled explicitly. To take a more conceptual approach, one can think of a Diophantine equation as a geometric object (consider, for example, the parabola Y=X^2). In more modern times, it has become clear that this perspective has a key role to play, and some of the greatest mathematical advances of the 20th century have led to the striking discovery that geometry plays a governing role in arithmetic problems such as these. Indeed, this was profoundly demonstrated in 1983, when Faltings proved Mordell's conjecture, which states that a Diophantine equation in two variables satisfying certain geometric conditions has only finitely many rational solutions. In 1986, Faltings was awarded a Fields Medal for his proof. Mordell's conjecture gave rise to a number of other finiteness conjectures, due to Andre, Lang, Manin, Mumford, Oort, and others. However, although these conjectures shared obvious similarities, it was unclear how they related to one another. This was until the Zilber-Pink Conjecture came along; in a vast new conjecture, it simultaneously generalised all of the aforementioned conjectures. It achieved this in part by working within the rich mathematical objects known as (mixed) Shimura varieties. The Zilber-Pink Conjecture is a problem of Unlikely Intersections, which is a name derived from the simple principal that, in a space of dimension d, two geometric objects of dimensions n and m, respectively, are highly unlikely to meet if the sum of n and m is less than d. Consider, for example, two lasers fired from opposite corners of a laboratory; we expect them to miss each other because the lasers are lines (and, hence, of dimension 1) being fired in 3-dimensional space, and 1+1=2 is less than 3. Problems of Unlikely Intersections have produced a flurry of activity in recent years, in large part due to new tools coming from mathematical logic. These were first applied by Pila and Zannier to the so-called Manin-Mumford Conjecture, and their approach has inspired a general strategy, which has already had profound effects in the subject. The proposed research seeks to obtain new arithmetic results for Shimura varieties that, due to previous work of the author and his collaborator, are known to yield significant progress towards the Zilber-Pink Conjecture via extensions of the Pila-Zannier strategy. It also seeks to obtain effective results (in other words, results with numeric outputs that are in principal computable) in the setting of the Zilber-Pink Conjecture. It will achieve this latter aim using new tools from the geometry of differential equations that have already produced results in simpler settings.

Photosynthetic organisms are vital to our economy and survival, playing a critical role in the global carbon cycle and affecting the climate of our planet. Recent advances offer us the methods to understand the complex control of growth and activity in photosynthetic organisms. The 24-hour circadian clock is a key regulator in plants, and is also important in cyanobacteria, fungi and animals including humans. The UK teams have shown that rhythmic control of biological activity by the circadian clock increases growth and survival of Arabidopsis thaliana plants, probably because >15% of genes in Arabidopsis are clock-regulated. The Arabidopsis circadian clock is becoming a paradigm for systems biology. The clock mechanism is a small gene network with multiple feedback loops, comprising five pseudo-response regulators, three myb-related proteins, two F-box proteins, and additional plant-specific proteins. Millar's group has modelled a simplified Arabidopsis clock mechanism, with three interlocking feedback loops. Predictions of the models have been validated by new experiments, identifying an additional part of the clock network. This is still a rare achievement in any organism. Including the real complexity of the Arabidopsis clock, however, will greatly enlarge the models, making them more difficult to use and to understand. The dynamics of the clock system are complex. It can generate autonomous, 24-hour biological rhythms of gene expression but in nature the day/night cycle forces the system, resetting the clock. Light signals regulate four different components in the current clock model. In reality, these signals originate from at least eight photoreceptor proteins and probably control additional components. This complexity hampers circadian research in Arabidopsis. A simpler clock system that included only one of each protein type would enormously facilitate the experimental analysis of the clock mechanism. It would provide a natural test for the proposed benefits of complexity in the clock mechanism, giving general insight into other complex clocks for example in humans. If the whole organism were simple, it could reveal much more easily how correct timing of particular clock-regulated biochemical processes led to adaptive benefits. The French team has developed this ideal model. Ostreococcus tauri is the smallest free-living eukaryote, with a circadian system that is closely related to that of Arabidopsis. Crucially, each protein type is represented by only one gene in Ostreococcus. The Bouget lab has developed a unique set of experimental tools for functional genomics in this organism. Their recent results demonstrate that the Ostreococcus clock conserves the same mechanisms and gene interactions as the clock in Arabidopsis, but in a far simpler system. Modelling by the Lefranc group confirms that very simple mathematical models, which were invalidated by data in Arabidopsis, accurately describe the Ostreococcus clock. The world-leading results of the UK and French teams are naturally complementary, but in addition are supported by significant national and institutional investment on both sides. We are poised to make a major impact, gaining significant added value from these resources and opening up a new application area. We will combine the UK team's expertise in complex models, and the wealth of comparative data and models on Arabidopsis, with the French team's experimental system and expertise in nonlinear dynamics. Experimentally, we will generate biological materials to monitor and manipulate all the clock components in Ostreococcus, then use these materials to generate high-quality timeseries data for modelling. We will identify all clock-regulated transcripts and promoter sequences using RNA expression microarrays and promoter arrays. These results will form a case study for Plant Systems Biology, demonstrating the power of a unicellular system to accelerate understanding of core processes.

The applicant plans to combine mathematical tools with multi-resolution data to develop models and methods in a framework that considers both space and time. Due to the nonlinear nature of the real-world, understanding behaviors of the physical, biological and social fields require general and widely applicable mathematical frameworks. To do this, the applicant will focus on a particular biological problem of widespread interest: protein interactions and their ability to regulate cellular decision making. This is of paramount importance; for example, in cancer, a cell is unable to transmit a death signal through protein interactions, causing the cell to continue to proliferate when it should arrest. Often these signaling processes involve many agents, all interacting in nontrivial ways, in different locations and at different levels of organization. The development and analysis of mathematical models describing protein interactions will, crucially, allow us to understand their dynamics, predict molecular mechanisms, reveal their function, and guide cell decisions. This is an ongoing biologically and medically relevant problem of fundamental importance and mathematical/statistical approaches may provide insights into the role and dynamics in spatial organization in cells, but more generally, to other systems. Dynamical approaches will be used for model development since these consider the temporal deterministic evolution of the system and may provide mechanistic information about the system. Throughout the fellowship, analysis and methods will be developed with chemically resolved data of the protein interaction system. Multi-resolution data of these protein interactions will be collected from Supporting Partners at the Weizmann Institute, Princeton University and University of Tokyo to test our predictions and methods. A range of mathematical/statistical approaches will be used for analysis and method development. One particular problem we will focus on is how to determine which model could describe data generated from a system. This question of model selection, or model discrmination, has led the applicant to work and develop a range of methods to distinguish between models. Understanding the mechanisms responsible for the behavior of protein signaling in a spatio-temporal framework would advance the fields of mathematics and biology. The Supporting Partners and applicant have already contributed to this arena of studying complex systems; moreover, they share a mutual interest to advance the field through the construction of spatio-temporal models of protein interactions and develop novel mathematical techniques that may be applicable to other inherently spatial systems. More generally, the research has direct biological implications-the work may provide insights for dysfunction of protein signaling resulting in diseases such as cancer, and we can extend our framework to analyze other biological processes inside living organism. These types of investigations will benefit from the mathematical, statistical and computational protocols that are developed in this project. Furthermore, the nonparametric and statistical methods developed for different types of spatial models can be applied to other contexts.

In the area of probability, an increasingly important role has been played in recent years by random systems in which the randomness is observed in the spatial structure. Random systems defined on lattices have been introduced as discrete models that describe phase transitions for various phenomena, ranging from liquid in porous media to the spread of disease. Our understanding of some of these models, such as percolation and Ising model, has been improved greatly in the last decades, and works around it have led to Fields medals in 2006 and 2010. The aim of the proposed research is to open new directions for several long standing open questions in random systems on lattices. One circle of the questions concern the gradient Gibbs measures, which is a model of random surface introduced in the 1970s by Brascamp, Lebowitz and Lieb as a model for crystal interfaces. A long standing universality conjecture states that the large scale statistical properties of these random surfaces behave like a Gaussian free field. This has been partially confirmed by the work of Naddaf and Spencer (and others). The PI intends to improve the understanding of the gradient Gibbs measures, by quantifying the existing fluctuation theorems, settling the 20-year-old conjectures in surface tension (that describes the energy of a surface profile with a global tilt), and to establish some universality conjectures of the extremes of log-correlated fields. The second circle of questions concern the XY and the Villain models, which are mathematical models of liquid crystals, liquid helium and superconductors. Works around it have led to the Nobel Prize in Physics (Kosterlitz and Thouless) in 2016. Physicists predict that at low temperature the large scale property of these models are closely related to the Gaussian free field. This is known as the Gaussian spin wave conjecture. Some mathematical progress was made towards the conjecture in the 1970s and the early 1980s, building around the works of Frohlich, Simon and Spencer. However, methods developed in these papers (infrared bounds and Coulomb gas renormalization) were not sufficient to complete the proof of this conjecture. The PI intends to resolve this long-standing Gaussian spin wave conjecture for the XY and the Villain models in dimension three and higher. In doing so, the PI will develop a robust framework to study the scaling limits, fluctuations and large deviations of a large class of Gibbs measures. New bridges will be built between probability, statistical mechanics and mathematical analysis.

A living cell, e.g. a bacterium, is an information-processing machine. It is composed of a series of sub-systems that work in concert by sensing external stimuli, assessing its own internal states and making decisions through a network of complex and interlinked biological regulatory networks (BRN) motifs that act as the bacterium neural network. A bacterium's decision making processes often result in a variety of outputs, e.g. the creation of more cells, chemotaxis, bio-film formation, etc. It was recently shown that cells not only react to their environment but that they can even predict environmental changes. The emerging discipline of Synthetic Biology (SB), considers the cell to be a machine that can be built -from parts- in a manner similar to, e.g., electronic circuits, airplanes, etc. SB has sought to co-opt cells for nano-computation and nano-manufacturing purposes. During this leadership fellowship programme of research I will aim at making E.coli bacteria much more easily to program and hence harness for useful purposes. In order to achieve this, I plan to use the tools, methodologies and resources that computer science created for writing computer programs and find ways of making them useful in the microbiology laboratory.