For species and populations to persist, individuals must reproduce. However, there are constraints on reproductive output, because without these, individuals would have limitless young. As such, pivotal models in evolutionary ecology demonstrate a cost to reproduction which manifests as a reduction in survival probability, and hence future reproductive output. However, although studies have modelled individual life-history pathways, they have rarely detected these trade-offs in the wild. Although the trade-off between current and future reproduction is widely accepted to exist, it continues to evade detection. We suggest there are two main reasons why this is the case: First, the trade-off between current versus future reproduction is driven by resources. These are generally limited so they must be divided between current and future reproductive effort. Attempts to estimate how these resources are allocated are hampered by the inability of most studies to measure actual resource values for each individual. If we don't know how many resources an individual has we can not understand how these are divided between life-history traits. Second, it is known that individuals can show differences in whether they use their resources for current or future reproduction. But when is the future? To individual A the future may be the next breeding attempt but to individual B the future may be much later in life. However, surprisingly studies fail to model such differences between individuals. Simulations have shown that fixing the temporal scale of trade-offs will fail to detect trade-offs that occur at another temporal scale and hence could be a major driver in masking trade-offs. Our model system provides data on the reproductive and foraging behaviour of albatrosses at four sites throughout the Southern Ocean. We have evidence from our study system of substantial variation in individual foraging behaviour, and hence the resources available for reproduction. We know that some individuals show variation in reproductive success over short time frames, and others over very long periods of their life. Hence, individual level analyses are required to measure the effects of changes in resources and consequences for reproduction. We will use existing long-term data on breeding behaviour (>100,000 breeding attempts; 63-year time series) and foraging (1305 individuals; 25-year time series), coupled with newly collected data (150 individuals), to examine how individuals vary in the resources they have available, and how they use them. We expect resource acquisition to be crucial to how many resources are allocated to reproduction, so that by capturing these measures, we will be able to detect previously hidden trade-offs between current and future reproduction. We expect that individuals will pay the cost of reproduction at different time points in the future, and that by allowing these differences to be modelled, we will be able to accurately detect reproductive trade-offs. The environment will change the resources available over time and we predict that some life-history strategies will be under selection as they enable individuals to maximise fitness in a changing climate. By modelling how fitness varies under future climate conditions we can predict how natural selection will act on individual life-history strategies.

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.

Future climate change is one of the most challenging issues facing humankind and an enormous research effort is directed at attempting to construct realistic projections of 21st century climate based on underlying assumptions about greenhouse gas emissions. Climate models now include many of the components of the earth system that influence climate over a range of timescales. Understanding and quantifying earth system processes is vital to projections of future climate change because many processes provide 'feedbacks' to climate change, either reinforcing upward trends in greenhouse gas concentrations and temperature (positive feedbacks) or sometimes damping them (negative feedbacks). One key feedback loop is formed by the global carbon cycle, part of which is the terrestrial carbon cycle. As carbon dioxide concentrations and temperatures rise, carbon sequestration by plants increases but at the same time, increasing temperatures lead to increased decay of dead plant material in soils. Carbon cycle models suggest that the balance between these two effects will lead to a strong positive feedback, but there is a very large uncertainty associated with this finding and this process represents one of the biggest unknowns in future climate change projections. In order to reduce these uncertainties, models need to be validated against data such as records for the past millennium. Furthermore, it is extremely important to make sure that the models are providing a realistic representation of the global carbon cycle and include all its major component parts. Current models exclude any consideration of the reaction of peatlands to climate change, even though these ecosystems contain almost as much carbon as the global atmosphere and are potentially sensitive to climate variability. On the one hand, increased warmth may increase respiration and decay of peat and on the other hand, even quite small increases in productivity may compensate for this or even exceed it in high latitude peatlands. A further complication is that peatlands emit quite large quantities of methane, another powerful greenhouse gas. Our proposed project aims to assess the contribution of peatlands to the global carbon cycle over the past 1000 years by linking together climate data and climate model output with models that simulate the distribution and growth of peatlands on a global scale. The models will also estimate changes in methane emissions from peatlands. In particular, we will test the hypotheses that warmth leads to lower rates of carbon accumulation and that this means that globally, peatlands will sequester less carbon in future than they do now. We will also test whether future climate changes lead to a positive or negative feedback from peatland methane emissions. To determine how well our models can simulate the peatland-climate links, we will test the model output for the last millennium against fossil data of peat growth rates and hydrological changes (related to methane emissions). To do this, we will assemble a large database of published information but also new data acquired in collaboration with partners from other research organisations around the world who are involved in collecting information and samples that we can make use of once we undertake some additional dating and analyses. Once the model has been evaluated against the last millennium data, we will make projections of the future changes in the global carbon cycle that may occur as a result of future climate change. This will provide a strong basis for making a decision on the need to incorporate peatland dynamics into the next generation of climate models. Ultimately we expect this to reduce uncertainty in future climate change predictions.

Predicting future climate change is one of the biggest scientific and societal challenges facing humankind. Whist carbon emissions from human activities are the main determinant of future climate change, the response of the earth system is also extremely important. Earth system processes provide 'feedbacks' to climate change, either reinforcing upward trends in greenhouse gas concentrations and temperature (positive feedbacks) or sometimes dampening them (negative feedbacks). A crucial feedback loop is formed by the terrestrial global carbon cycle and the climate. As carbon dioxide concentrations in the atmosphere and temperature rise, carbon fixation by plants increases due to the CO2 fertilisation effect and the lengthening of the growing season at high latitudes (this is a negative feedback). But at the same time, increasing temperatures lead to increased decomposition of the carbon stored in soils and this results in more carbon dioxide being released back to the atmosphere (this is a positive feedback). The balance of these competing processes is especially important for peatlands because they are very large carbon stores. Northern Hemisphere peatlands hold about the same amount of carbon that is stored in all the world's living vegetation including forests, so determining the response of this large carbon store to future climate change is especially critical. One hypothesis is that warming will increase decomposition rates in peatland soils to such an extent that large amounts of carbon will be released in the future. However, the vast majority of peatlands are in relatively cold and wet areas and evidence from past changes in accumulation rates suggest that for these regions, warming may lead to increased productivity that more than compensates for any increase in decay rates, leading to increased carbon sequestration overall. Furthermore, in the northernmost areas of the Arctic, there is potential for further lateral expansion of peatlands, increasing the total area over which peat accumulates. We intend to answer the question of whether changes in accumulation in Arctic peatlands plus the increased spread of peatlands in cold regions will lead to an overall increase in their carbon storage capacity. Our approach will be to use a novel combination of data from the fossil record stored in peatlands together with satellite data to test a global model that simulates changes in both carbon accumulation rates and the extent of peatland vegetation over Arctic regions. If we can demonstrate that the model performs well in simulations of past changes, we can then confidently use it to make projections of future changes in response to warming for several hundred years into the future. We know that fluctuations in Arctic climate over the past 1000 years should have been sufficient to drive changes in peat accumulation rates and lateral spread, so we are focusing our analyses on this period. In particular, we know there were increases in temperature over the last 150-200 years and especially over the last 30-40 years. If our hypothesis that increased temperature leads to increasing accumulation and spread of Arctic peatlands is correct, we expect to see the evidence for this in the fossil record of peat accumulation and spread, and also in satellite data of vegetation change. Our previous work and our new pilot studies show that we can reconstruct accumulation rate changes and also that our proposed remote sensing techniques can detect peatland vegetation increases since the mid-1980s, so we are confident in our methodology. The model will provide estimates of northern peatland carbon storage change for different climate change scenarios over the next century and longer term to the year 2300. If we can show that there is a potential increase or even no change in carbon storage in Arctic peatlands, it will radically change our perception of the role of the Arctic terrestrial carbon store in mediating climate change.

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.