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LMS Summer School 2023:
Broad area(s): Computational number theory
Title: Some topics in computational number theory
Abstract: Number theory has been intertwined with computation since its inception. From Diophantus of Alexandria using (what we recognise today as) the group law on elliptic curves to construct complicated rational solutions, to Bernhard Riemann finding zeros of his zeta function, the use of explicit computation has long guided progress in number theory. In this mini-course, we will tour this intersection of theory and explicit methods throughout the history of number theory. Topics will include: the Riemann hypothesis, the Birch & Swinnerton-Dyer conjecture, Fermat's last theorem, and others.
Broad area(s): Applied probability
Title: Theorems that are not probability
Materials: A link to the course materials can be found here.
Abstract: I will explore various different ways in which probability can be used to prove important results in mathematics - results that, at first sight, have no obvious connection to probability.
Broad area(s): Algebraic geometry and combinatorics
Title: Toric surfaces
Abstract: I will follow Jessica Sidman’s The American Mathematical Monthly article ``An introduction to Algebraic Geometry: Polygons, Parameterizations, and Equations’’. We will study toric surfaces mainly focussing on 4 examples: the projective plane, embedded by its second Veronese embedding, the product of projective lines, and the blow up of the projective plane at a point. I will be starting by defining projective space and homogeneous polynomials. We will then define toric surfaces, and our four friends. After describing the ideal of a toric surface, we will study the Veronese surface and the Blow-up from a geometric point of view.
Broad area(s): Probability theory, statistics, and information theory
Title: An introduction to group testing
Materials: A link to the course materials can be found here.
Abstract: Group testing is a way of improving the efficiency of medical testing, and has been applied in practice at times when COVID tests were scarce. The idea is that if we put swabs from multiple people into the same lateral flow solution then the test will be positive if and only if one of them is infected. Hence we can potentially efficiently screen out large numbers of non-infected individuals at times of low prevalence, though more work is required to find out which people have the disease. I’ll describe some ways of how to do this, both in terms of choosing a test strategy and in terms of deciding on peoples’ infection status, see how close we get to fundamental performance bounds on how well we can do, and suggest some problems you can try.
Broad area(s): Mathematical physics, algebra and topology
Title: Interactions between algebra, geometry and quantum theory
Materials: A link to the course materials can be found here.
Abstract: Modern approaches to representation theory and quantum (field) theory encode algebraic structures in terms of n-dimensional geometric pictures, such as embeddings of multiple n-disks into a single bigger n-disk. This mini-course gives an elementary introduction to this subjects through the lens of factorization algebras. As concrete examples, we will see how quantum mechanics and also Lie algebra representations admit an interpretation in terms of 1-dimensional geometry. We will conclude with some comments on the much richer, but also more complicated, case of algebraic structures arising from (n > 1)-dimensional geometry.
Broad area(s): Mathematical finance, risk theory, and actuarial science
Title: Mathematics of social finance
Abstract: As I am spending my sabbatical year in the Social Finance programme of the International Labour Organization, I would like to highlight some of the new risk management tools and financial products developed in the field of social finance. Social finance is an approach to managing investments that generate financial returns while including measurable positive social and environmental impact. It includes microfinance and microinsurance (financial inclusion) which direct financial instruments to underserved communities.
Broad area(s): Uncertainty quantification for computer models
Title: Using statistics to evaluate uncertainty in complex models of the atmosphere and climate
Abstract: Large computer models are used to simulate our knowledge of complex systems like the atmosphere and climate, with the aim to better understand and predict the system’s behaviour, how it might evolve and how it could respond to future changes. However, these models are inherently uncertain with many uncertain inputs (parameters), and we need to understand how this uncertainty can affect our predictions.
In this talk, I’ll describe a statistical framework that brings together a variety of statistical techniques to examine the effects of parameter uncertainties on outputs from complex models. We will see examples of the application of this approach with real models from atmospheric science – In particular, we will consider the effect of parameter uncertainties on predictions from the Met Office’s climate model of how aerosols (tiny particles in the atmosphere) have affected the Earth's energy balance since pre-industrial times, known as the 'aerosol radiative forcing', and explore efforts to use observations to reduce our uncertainty in these predictions.
Broad area(s): Algebraic geometry
Title: A bestiary in the wild world of moduli spaces
Abstract: In mathematics, we aim to classify objects that satisfy certain properties. In geometry, what we want to classify often ends up with an infinite class of them, but we can arrange them in such a way that we can build a space where each point corresponds to a geometric object. These are called moduli spaces. An example of the above is the following: consider all (infinite) rays starting at the origin in the plane. Can you think of a geometric space parameterizing all such rays? In this talk, we will venture into the world of moduli spaces and explore examples of what a few of these spaces look like.
Broad area(s): Mathematical biology, specifically modelling immune responses
Title: A story of co-infection, co-transmission and co-feeding: how to compute an invasion reproduction number
Abstract: Co-infection of a single host by different pathogens is ubiquitous in nature [3]. We consider a population of hosts (e.g., small or large vertebrates) and a population of ticks, both of them susceptible to infection with two different strains of a given virus. We note that for the purposes of our models, we have Crimean-Congo hemorrhagic fever virus (a segmented Bunyavirus) in mind, as the application system.
First, we focus on the dynamics of a single infection, proposing a deterministic to understand the role of co-feeding in the spread of the virus. We then compute the basic reproduction number by making use of the next generation matrix approach [4].
When considering co-infection by two distinct strains (one resident and one invasive), we make use of differential equations to model the dynamics of susceptible, infected and co-infected species, and we compute the invasion reproduction number of the invasive strain [2]. I discuss some problems with the calculation, and the solution proposed by Samuel Alizon and Marc Lipsitch [2].
I conclude with a perspective on how the co-infection model can be applied to HIV, and plans for future work and work in progress.
In summary, we present and analyse a novel mathematical model of viral infection to identify the different contributions from co-infection, co-transmission and co-feeding, to the establishment of a viral reassortant in a population of ticks and hosts.
Full abstract including references, coauthor information and affiliations can he found here.
Broad area(s): Number theory
Title: Beyond Fermat's Last Theorem
Abstract: Diophantine equations are polynomial equations in two or more variables, for which one seeks integer or rational solutions. A famous example is the Fermat equation, solved in 1995 by Andrew Wiles with his celebrated proof of the Tani yama-Shimura conjecture (also called the modularity conjecture).
In this talk, we explore how combinations of classical and modern (post-Wiles) approaches prove to be fruitful in solving families of Diophantine equations. We will explore plenty of explicit examples to guide us through various methodologies.
Broad area(s): Integrable Systems and Geometry, and Mathematical Physics
Title: Bernoulli numbers in group theory and algebraic topology
Abstract: Bernoulli numbers are probably the most mysterious numbers in mathematics. They were introduced by Jacob Bernoulli in relation with the computation of the sums of powers of the first N natural numbers, but soon magically appeared in many other problems, including the special values of Riemann zeta-function and Euler-Maclaurin summation formula in analysis. I will discuss much less known but very significant role, which Bernoulli numbers play in the theory of formal groups and algebraic topology. This will provide also a fresh view on these very classical numbers.
Broad area(s): Applied mathematics, magnetohydrodynamic waves, eigenvalue problems
Title: Sunspots are fun spots!
Abstract: In this talk I will discuss how the mathematics of solving the wave equation as an eigenvalue problem can be extended and applied to understand the oscillations of sunspots. These are regions of intense magnetic field and can be observed in visible light on the Sun’s surface as dark spots. Recently, Stangalini et al. (Nature Comm. 2022) reported the detection of the largest-scale coherent oscillations observed in a sunspot with a frequency spectrum significantly different from the Sun’s global acoustic oscillations, incorporating a superposition of many resonant wave modes. No existing magnetohydrodynamic models were accurate enough to match the high-resolution observational data so we had to take a completely new approach which involved inputting the exact cross-sectional shape of the sunspot umbra into our computations. Our findings not only demonstrate the possible excitation of coherent oscillations over spatial scales as large as 30–40 Mm in extreme magnetic flux regions in the solar atmosphere, but also pave the way for their diagnostic applications in other astrophysical contexts.
Broad area(s): Dynamics of the early universe and black holes
Title: Quantum simulators for fundamental physics
Abstract: The dynamics of the early universe and black holes are fundamental reflections of the interplay between general relativity and quantum fields. The essential physical processes occur in situations that are difficult to observe and impossible to experiment with: when gravitational interactions are strong, quantum effects are important, and theoretical predictions for these regimes are based on major extrapolations of laboratory-tested physics.
We will discuss the possibility to study these processes in experiments by employing analogue classical/quantum simulators. Their high degree of tunability, in terms of dynamics, effective geometry, and field theoretical description, allows one to emulate a wide range of elusive physical phenomena in a controlled laboratory setting. We will discuss recent developments in this area of research.
Professor Dummigan is the School of Mathematics and Statistics' Director of Postgraduate Research (PhD research.) Neil will discuss the PhD process, what a day in the life of a PhD student might look like, and answering any questions you may have about the process.
I will lead a day and a half of problem solving sessions based around a catalogue of (short) published mathematical problems on an array of topics. Students will choose an existing problem and solution, modify the initial problem, solve it, and (if they like) present their solution to the class. I hope this to be a group activity, but students can work independently, if they wish.
Nathan Blacher (University of Sheffield)
Nathan Blacher (University of Sheffield)
Date/ Time: 25th July, 11:30 - 12:30
Title: Understanding the borderline between commutative and noncommutative algebra
Abstract: Rings appear naturally throughout mathematics and their study has helped develop many areas, notably algebraic geometry, number theory or functional analysis.
Commutative algebra is often considered the 'nice' case in ring theory, whereas the structure of noncommutative rings is more difficult to understand. It can therefore be useful to consider certain classes of noncommutative rings whose behaviour resembles commutative algebra, such as rings satisfying a polynomial identity (PI).
We will introduce PI rings together with major results illustrating how PI theory bridges the gap between the commutative and noncommutative worlds. We will also discuss recently introduced minimal noncommutative rings and their connection to PI algebras.
Alberto Cobos Rabano (University of Sheffield)
Alberto Cobos Rabano (University of Sheffield)
Date/ Time: 20th July, 11:30 - 12:30
Title: What does enumerative geometry have to say about planet orbits?
Abstract: How many observations do you need to determine the orbit of a planet? Enumerative geometry can be helpful since it deals with problems such as ''how many geometric objects are there satisfying certain conditions''? For example, we will try to understand how (and when) five points determine a plane conic. This means that we can determine the orbit of a planet with only five observations. But, can we actually do with less observations?
Poppy Jeffries (University of Sheffield)
Poppy Jeffries (University of Sheffield)
Date/ Time: 18th July, 11:30 - 12:30
Title: Be different to be better: the effect of personality on optimal foraging with incomplete knowledge in pelagic seabirds
Abstract: Many animal populations include a diversity of personalities, which are often linked to foraging strategy. However, it is not always clear why populations should evolve to have this diversity. When considering foraging strategy mathematically, optimal foraging theory typically seeks out a single optimal strategy for individuals in a population. So why do we see a variety of foraging strategies existing in a single population? We aim to provide insight by modelling foraging seabirds in a patchy environment with partial knowledge of their surroundings. We use mathematical modelling techniques to explore the success and risk of individual foraging strategies and explore how robust they are to environmental change and uncertainty. This environmental uncertainty results in individuals only having partial knowledge of their surroundings, creating a trade-off between exploring and exploiting the environment whilst foraging. These contrasting foraging strategies have been observed in seabirds with differing personalities: bolder birds are explorers, moving quickly between patches, whilst shyer birds are exploiters, moving slowly between higher-quality patches. Using trade-off curves, we explore the success of individuals within populations, and we show that it is optimal to be shy (resp. bold) when living in a population of bold (resp. shy) birds. This observation gives a plausible mechanism behind the emergence of diverse personalities. We also show that environmental degradation will likely favour shyer birds and cause a decrease in diversity of personality over time.
Lisa Mickel (University of Sheffield)
Lisa Mickel (University of Sheffield)
Date/ Time: 27th July, 11:30 - 12:30
Title: Quantum gravity and cosmology
Abstract: This talk aims to give an introduction to the world of quantum gravity and how one of the approaches, namely group field theory (GFT), can be applied to cosmology.
We will begin with necessary background knowledge, which are the core concepts of general relativity, cosmology, and quantum theory. With the basics covered, we will motivate the study of quantum gravity and provide a short overview of different approaches to this problem. Here, we will focus on GFT to set the stage for the rest of the talk. In the second part, I will present some of my PhD research, which aims to find a description for cosmological perturbations in the early universe (the remnants of which we measure in the CMB) from quantum gravity. We will take a look at how modified early universe dynamics emerge from GFT and how these modifications affect cosmological perturbations.
Andrew Neate (University of Sheffield)
Andrew Neate (University of Sheffield)
Date/ Time: 25th July, 11:30 - 12:30
Title: The Unreasonable Effectiveness of Category Theory in Crumpled Paper
Abstract: The category, first written about in 1945 by Saunders and Mac Lane, is the modern language of mathematics for many mathematicians. In this talk I will introduce the notion of a category through the familiar notion of the integers. We will describe functors as ways to map the information between categories. Through the language of spaces, sets, and functors we will prove the Brouwer fixed point theorem. This theorem states that if I take two pieces of paper and crumple one up and place it above the other then there must be at least one point on the crumpled sheet that is in the same place as on the other piece of paper. While this theorem seems unconnected to the world of categories there exists a remarkable proof that uses only very core properties of functors.
Megan Oliver (University of Sheffield)
Megan Oliver (University of Sheffield)
Date/ Time: 18th July, 11:30 - 12:30
Title: Host Manipulation by Parasites to Facilitate Transmission
Materials: A link to the course materials can be found here.
Abstract: It may be a surprise to learn that a parasite can evolve to possess some control over a host. However, examples in nature show that to increase their chance of successful transmission, parasites can manipulate their host into changing their behaviour or appearance. These manipulation strategies tend to evolve in parasites that are trophically transmitted. An infected intermediate prey host will have increased vulnerability to predation, allowing the parasite to infect its definitive target host - the predator - more easily.
Mathematical models are used in many disciplines as an essential tool to study disease spread. Here we develop a compartment model based on the framework by Fenton and Rands (2006) to investigate when a parasite will most likely evolve these strategies. Manipulation is beneficial for the parasite since it facilitates movement through a stage of the life cycle from an intermediate host to the final host. However, increasing the likelihood of predation by manipulation is a costly approach for the parasite which we incorporate in a trade-off with spore production. These spores are vital for transmission since they are released into the environment to be consumed by susceptible prey causing infection or they may be left to decay.
Our results show that fluctuating dynamics are an outcome of this system and that the evolution of manipulation strategies is simultaneously governed by both the size of the susceptible prey population and the perceived threat of predation determined by the size of the entire predator population.
James Salsbury (University of Sheffield)
James Salsbury (University of Sheffield)
Date/ Time: 27th July, 11:30 - 12:30
Title: Assurance Methods for Adaptive Clinical Trials
Abstract: When designing a clinical trial, the number of patients recruited to the trial must be planned for carefully. If there are too few, there is a risk that the trial will not provide sufficient evidence that the treatment works. On the other hand, if there are too many, patients will be needlessly enrolled into a study. Traditionally, to choose the number of patients, the statistical concept of 'power' is used. However, this relies on the assumption that the treatment really does work as well as desired. We introduce an alternative method: ‘assurance’, which incorporates uncertainty into the design of clinical trials.
Jacob Saunders (University of Sheffield)
Jacob Saunders (University of Sheffield)
Date/ Time: 21st July, 13:30 - 14:30
Title: How many holes does a 91-dimensional straw have?
Abstract: Topology studies the properties of spaces which are not affected by deformation. Flattening a straw gives you a circle, so (to a topologist) a straw and a circle have the same number of holes. This is just one case of a more general problem: how many "holes" does the n-sphere (the unit sphere in (n + 1)-dimensional space) have? A topologist considers an n-dimensional hole in a space X to be an "interesting" map from the n-sphere into X, so this question boils down to understanding the "interesting" maps between spheres. These are infamously difficult to understand, and completely understood only for a small range of dimensions. In this talk, we will discuss what precisely makes such a map "interesting", and look at some of the "interesting" maps between spaces of low dimensions.
Yannik Schuler (University of Sheffield)
Yannik Schuler (University of Sheffield)
Date/ Time: 20th July, 11:30 - 12:30
Title: From Counting Curves to Strings and back
Abstract: When I started my studies, I was naively thinking that physics and mathematics were always going hand in hand… Well, and then I took a course in Quantum Field Theory. Even though the Standard Model which is entirely formulated in the language of Quantum Field Theory has proven highly successful, it not only features inconsistencies on the physics side but it is also extremely ill-defined on the mathematical front. I will take you on a rollercoaster ride explaining to you at which point in the last century physics and mathematics parted ways and how a certain limit of string theory brings them back together. More precisely, by the end of the talk you should get an idea why counting curves in algebraic geometry is related to the quantum mechanics of strings which is just the beginning of a beautiful story about how physics and mathematics inspired each other in the last decades.
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