Past Colloquia
Academic Year 2022/2023
Semester 2
- 16/06/2023 Nina Fischer (School of Mathematics, University of Edinburgh)
Time series emulation of carbon cycle model simulations
Simulators, complex physical models implemented in computer code, are a fundamental tool to assess climate change and terrestrial carbon dynamics. Coupled with observational data from satellites, models can infer unobserved ecosystem properties. However, their computational cost typically restricts the number of possible system runs. This limits our understanding of landscapes’ complex physical processes and associated uncertainties, hindering mitigation and adaptation strategies essential to meet climate change mitigation targets.To facilitate quick model calibration and inference, this talk presents statistical methodology to overcome computational constraints by building emulators, or statistical approximations of the simulators. We combined these emulators with history matching to find the set of input combinations for which the simulation gives acceptable matches to observed data.
In this work, we applied this to the carbon cycle model DALEC Crop, which models the effect of Nitrogen fertilisation on wheat yield in a field over a crop growing season. We applied dimension reduction techniques to emulate Leaf Area Index (LAI) time series and then history matched predictions to Sentinel-2 LAI observations. We then used the input space found in a DALEC Crop forward run to produce plausible yield predictions, skipping DALEC’s computationally costly model calibration process, to demonstrate the potential to reduce computational costs by using suitable statistical methodology.
- 22/03/2023 Theo Lavier (School of Mathematics, University of Edinburgh & Heriot-Watt University)
Optimal Transport and the Compressible Semi-Geostrophic Equations
The semi-geostrophic equations for a compressible fluid, first analyzed by Cullen and Maroofi [2003], provide a simplified model of the formation and evolution of atmospheric fronts. I will describe the use of semi-discrete optimal transport theory to construct a numerical particle method. This method is structure preserving in the sense that numerical solutions conserve energy. I will then present numerical results and discuss the challenges we faced in implementing this numerical method. Using this approach, we give a constructive proof of the existence of global-in-time weak solutions as the limit of spatially discrete approximations. This work directly extends the work of Bourne et al. [2022] from the incompressible to the compressible setting. This is joint work with David Bourne, Charlie Egan, and Beatrice Pelloni at Heriot-Watt University.
- 04/04/2023 Ryan Campbell (Lancaster University)
Statistical inference for multivariate extremes via a geometric approach
Multivariate extreme value theory is the study of the tail behaviour of joint random processes while taking into account the dependence between these processes. While a powerful tool for modeling several extremal processes simultaneously, multivariate extremes lacks a unified way to model these extremal processes when the underlying dependence structure varies. To remedy this, we make use of the so-called “gauge function”, which is intrinsically linked to the limit set obtained by appropriately scaling a random sample and letting the sample size grow arbitrarily large. The limit set, and thus the gauge function, enable us to determine the extremal dependence structure between random variables. In using the gauge function to perform statistical inference, we are able to estimate the probability of the occurence of an extreme event accurately when compared to existing methods on simulated and real datasets.
- 22/03/2023 Michelangelo Tirelli (Heriot-Watt University)
Introduction to non-relativistic geometry
In this talk the plan is to discuss about generalized instantons of noncommutative gauge theories in dimensions 4, with emphasis on their realizations in type II string theory, their geometric interpretations and their applications to the enumerative geometry of non-compact toric varieties.
- 10/02/2023 Raya Mancheva (School of Mathematics, University of Edinburgh)
Why Does the Ending of Interstellar Wrong Except on a set of Measure Zero?
In this short seminar we elaborate on the connection between the strong cosmic censorship conjecture and the failure of the ending of Interstellar to be physically realistic for almost all admissible initial data. For context, we begin by introducing key concepts in general relativity, including strong and weak cosmic censorship so that we can compare the two (logically independent) conjectures, [4]. We see examples of how the two forms of cosmic censorship manifest in simple black hole spacetimes, such as Reissner-Nordstrom and Kerr, and how these examples differ from the physically realistic scenario by being non-generic. We refine our statement of the strong cosmic censorship conjecture by specifying a degree of regularity of spacetime extensions. This leads us to discover that, while the original C 0 version of the conjecture, [5], fails in certain classes of spacetimes [3], there are higher regularity versions, which turn out to be more physically significant for the fate of Joseph Cooper in Interstellar, [1], [2]. This leads us to define the concept of a weak null singularity. Finally, I introduce a bit of my work and how it relates to the above picture. What happens to free particles when they approach the weak null singularity in the interior of a spherical black hole in a C 2 -inextendible spacetime? Are their worldlines extendible as C 1 curves beyond the singularity? Does the energy density of these particles remain finite? Answering these questions has stunning implications for Cooper’s fate.
REFERENCES:
[1] J. Sbierski, On holonomy singularities in general relativity the C 0,1 loc -inextendibility of spacetimes. https://doi.org/10.48550/arXiv.2007.12049[2] J. Luk, SJ. Oh. Strong cosmic censorship in spherical symmetry for two-ended asymptotically flat initial data I. The interior of the black hole region, https://doi.org/10.48550/arXiv.1702.05715
[3] M. Dafermos, The interior of charged black holes and the problem of uniqueness in general relativity, https://arxiv.org/pdf/gr-qc/0307013.pdf
[4] M. Dafermos, The mathematical analysis of black holes in general relativity, https://www.dpmms.cam.ac.uk/ md384/ICMarticleMihalis.pdf
[5] D. Christodolou The formation of black holes and singularities in spherically symmetric gravitational collapse, https://doi.org/10.1002/cpa.3160440305
Semester 1
- 09/12/2022 Leo Bidussi (Heriot-Watt University)
Introduction to non-relativistic geometry
In recent years the study of NR geometries and Physics has greatly increased as it shows potential for many applications and could further our understanding of quantum gravity. During the presentation we will cover the basic concepts behind Newton-Cartan geometry and NR Physics.
- 28/10/2022 Andrew Mair (School of Mathematics, University of Edinburgh & Heriot-Watt University)
Can root systems redistribute soil water to mitigate the effects of drought?
Plants combine a diverse range of morphological and physiological mechanisms to adapt to water deficit and drought. As an additional mechanism for plant drought resistance, this work considers how root-induced preferential flow redistributes soil water in a way that depends upon root system architecture. We develop a mathematical model for water transport through vegetated soil, which incorporates root-induced preferential flow, and used Bayesian optimisation to calibrate against experimental data. A finite element scheme was used to simulate the model, and assess how the fate of soil water is impacted by preferential flow strength, soil type, and root system architecture.
- 07/10/2022 Erik Sätterqvist (School of Mathematics, University of Edinburgh)
Elliptic Flavoured PDEs – the Laplacian and Beyond.
The study of PDEs is an incredibly diverse field, the questions asked and techniques used depend heavily on the type of PDE that is being studied. This talk will be focused on elliptic flavoured PDEs with the Laplacian as our jumping off point. We will explore what sets the Laplacian apart from the other fundamental PDEs like the heat and wave equation, introduce the simplest boundary value problem called the “Dirichlet problem” and see how understanding the solutions of this equation as a heat equilibrium naturally leads us to some of the fundamental properties of solutions. Finally, we apply a recent perturbation result by Erik, Martin Ulmer and Martin Dindos to see how solvability can be extended to amore general class of “elliptic” PDEs.Erik’s slides
Academic Year 2021/2022
Semester 2
- 18/3/2022 Isabella Deutsch (School of Mathematics, University of Edinburgh)
Uncovering Product Cannibalisation using Multivariate Hawkes Processes
When one product is bought instead of another, similar product, we talk about “product cannibalisation”. This is a relevant factor to understand customer behaviour and to improve product ranges. Together with a major fashion and retail company we investigate product cannibalisation based on their wholesale data. To uncover such product cannibalisation we use a multivariate Hawkes process model. Hawkes processes are typically used for point process data when events happen in clusters or bursts. They are popular in the earthquake literature, where the occurrence of one earthquake makes it more likely for another one to appear soon. For our application however, we extend this notion of the Hawkes process in the following. The occurrence of one event can make it not only more likely (like in the earthquake example), but also less likely that another event takes place afterwards. This translates well into the sales context and permits us to uncover product cannibalisation when the sale of one product makes it less likely that a different, similar product is bought. We discuss how such a model can be implemented and highlight challenges in the estimation procedure. Finally, we use Normal and sparsity-inducing priors in a Bayesian estimation procedure on our fashion wholesale data.Altogether, this allows for the quantification of product cannibalisation to both understand purchasing behaviour and aid product development in the future. talk slides
- 11/3/2022 Antoine Goldsborough (Heriott-Watt University)
Divergence in Groups
We will look at the notion of divergence in groups, a interesting idea in geometric group theory. This will be an elementary introduction to divergence, starting from the basics. Time permitting, we will also see a small proof of the divergence of hyperbolic groups.
- 4/3/2022 Gemma Crowe (Heriott-Watt University)
The maths behind the music
Mathematics and music have been known to be inextricably linked for centuries, from how strings vibrate at certain frequencies to the rhythmic structure of music. In this talk, I hope to offer a glimpse into this vast subject, as well as how a very abstract area of maths – group theory – can even appear in music. talk slides
- 25/2/2022 Hector Jardon Sanchez (Lancaster University)
The Aldous-Lyons conjecture and treeability
Graph limits appear naturally when studying large changing networks occurring in, for instance, the modelling of the internet. The aim of this talk is to introduce Benjamini-Schramm local convergence of a sequence of finite graphs and the Aldous-Lyons conjecture. As we will discuss, the local limit of such a sequence may always be described by a graphing on a standard Borel space. The Aldous-Lyons conjecture asks whether all graphings are local limits of sequences of finite graphs or not. This conjecture is related to other questions in Mathematics, such as the question of whether every group is sofic or not. The answer to the Aldous-Lyons conjecture is still unknown.The main class of graphings for which Aldous-Lyons is known to hold is the class of (Borel) treeable graphings, as shown by Elek-Lippner. Graphings, unlike finite graphs, may not always be treeable, and there are families of graphings for which it is not known it they are treeable or not. Time permitting, we will discuss the Elek-Lippner result and its applications on finding larger families of graphings for which the Aldous-Lyons conjecture holds.
- 18/2/2022 Axel Wings (RWTH Aachen University)
Knot energies of tangent-point-type and their generalization to higher dimensions
Knot energies of tangent-point-type are applied to (closed) space curves. They penalize high curvature and recognize self-intersections, i.e., the energy for smooth curves with self-intersections is infinite. That makes them useful for applications in computer graphics, physical simulations, as well as mathematical visualization.After a short introduction to knots and knots classes, the tangent-point radius is motivated and defined using basic geometry.This is the starting point for the definition of knot energies of tangent-point-type, whose useful properties are mentioned and illustrated by numerical simulations. Finally, the generalization to higher dimensions is presented and results obtained from numerical simulations for surfaces are shown. The simulations and the properties for the 1-D variants motivate further studies of the generalization to higher dimensions. Talk slides
Semester 1
- 10/12/2021 Erik Landstedt (KTH Royal Institute of Technology in Stockholm)
Easy to state, difficult to prove — some fun number theory conjectures
One of the most beautiful aspects of number theory is that it contains problems that are very easy to state, but extremely hard to solve. Such problems have stumped mankind for long time and during this talk we will take a look at some of them, e.g. the twin prime conjecture and the Collatz conjecture. We will also discuss a part of a conjecture of Erdős and Turán from 1936. This is related to Roth’s Theorem on Arithmetic Progressions, which deals with the question: “How large must a subset of the natural numbers be in order to contain a 3-term arithmetic progression?”.
- 26/11/2021 Jonna Roden (University of Edinburgh)
Simulating industrial processes – mathematical models and numerical solutions
How can we simulate industrial processes involving particles, such as filtering, printing or brewing? This question is of great interest to many industries because it can reduce the need for costly experiments, ensure quality standards of products and help cut cost and waste in the production of goods.In this talk, we will discuss how to derive models that describe the physics of such processes and how to solve these numerically. Unfortunately, real life seldom takes place in a box (or on a line, or on a donut). Therefore, we will focus on developing a numerical method that can deal with more complex shapes, such as funnels, vessels or curvy channels. At the end of the talk we will be rewarded by seeing some neat particle simulations (and by tea and biscuits afterwards…)!Below are some of the mathematical terms that make you think ‘I have no idea what this means – this talk is not for me’. Rest assured that it is certainly not a prerequisite for this talk to know any of them!! They are just there for people who think ‘Oh I know what this means and it sounds interesting!’.Key words: Pseudospectral and spectral element methods, (integro-)PDE models, dynamic density functional theory, multiscale particle dynamics.Talk Slides
- 12/11/2021 Martin Ulmer (University of Edinburgh)
The Dirichlet Problem for 2nd order linear elliptic PDEs – when can we solve it?
This talk is on the Dirichlet boundary value problem of second order linear elliptic PDEs. The presentation will describe the connection between the original problem Δu=0 and the more general PDE div(A grad(u))=0. We will discuss three different solvability criteria for this problem (t-independece, a Carleson norm condition and a pertubation result)and see how these criteria arise in a natural way from the solvability of Δu=0. The talk will end by highlighting some interesting open questions.
Talk slides
- 29/10/2021 Josh Fogg (University of Edinburgh)
Circumference of an ellipse? Can’t do it mate
At the 2021 Sutton Trust summer school a student asked us for help with an integral she’d been struggling with: I = ∫ₒᵃ (a⁴ + (b²−a²)x²)/(a⁴−a²x²) dx. What followed was a dive down the rabbit hole of history, travelling from work of the mathematical greats to the depths of Google Group archives, through Kepler, Maclaurin, Ramanujan, Ivory, Parker, and Gauss, all to answer one question: what’s the deal with the circumference of an ellipse? This talk gives a rundown of the different tools to hand and why Monte Carlo, though less efficient, less accurate, and more complicated, has a place among them.
Presentation slides Blog post
- 8/10/2021 Andreea Avramescu (University of Manchester)
Data-driven optimisation for the development and delivery of personalised medicine biopharmaceuticals.
The common drugs are only working for approximately 60% of the population. This is a direct result of the way the human body responds to different treatments as a result of genetic and social differences. It was this statement that led the medical research into discussing more dynamic and individualized treatments. These are known today as personalised medicines. While holding the promise to cure advanced stage and aggressive diseases, the delivery and development of such treatments are still problematic. Complex manufacturing processes, low demand, and high fragility of products have caused personalised medicine to exceed prices of £1 million and be used only as a last resort.Accommodating breakthrough therapies require the pharmaceutical industry to reinvent the means of scale-up and scale-out of these products.The traditional model of continuous manufacturing is no longer viable or sustainable, causing a necessary and imperative shift towards batch production. The overarching challenge addressed in our research is the development of a flexible decision-making framework able to support biopharmaceutical companies in the creation of a vein-to-vein supply chain, following the new model of production. The tool will identify the best organisational structure following the user’s objectives and requires a trade-off between economic and environmental output, coverage, and timely delivery. Hence, this talk will give an overview of the main challenges of personalised medicine supply chain, briefly describe the existent research, and introduce some of the possible research directions.
Presentation slides
- 1/10/2021 Andrew Beckett (University of Edinburgh)
Symmetry and supersymmetry: from JCM to BSM
Symmetry principles play a central role in modern physics, and much of the progress in theoretical physics in the 20th century was driven by the discovery and exploration of these principles. Starting with the symmetries in the theories of James Clerk Maxwell (who gives his name to our Grad School and one of our buildings), I will discuss some of these symmetries, how they have helped us to understand the world around us and how a hypothetical symmetry, supersymmetry, might help to resolve some of the biggest outstanding issues in physics today, taking us “Beyond the Standard Model”.
Presentation slides
Academic Year 2020/2021
Semester 2
- 30/4/2021 Benjamin Cox (UoE)
Parameter Estimation in Sparse Linear-Gaussian State-Space Models via Reversible Jump Markov Chain Monte Carlo
State-space models are ubiquitous for modelling complex systems that evolve over time. In such models, key parameters are usually unknown and must be estimated. In particular, linear systems can be parametrised by the transition matrix that encodes the dependencies among state dimensions. Due to physical and computational constraints, it is crucial to estimate this matrix by promoting sparsity in its components, in such a way that the interactions between dimensions are reduced. In this work, we propose a novel methodology to estimate model parameters and promote sparsity. The method is based on reversible jump Markov chain Monte Carlo and allows the exploration of the space of sparse matrices in an efficient manner by adapting the implicit model dimension. This novel methodology has strong theoretical guarantees and exhibits excellent performance in tricky numerical examples.
- 26/3/2021 Isabella Deutsch (UoE)
Earthquakes, Tweets, and Apparel Wholesale: Bayesian Estimation of Hawkes Processes
Hawkes Processes provide a flexible way to model point process data that occurs in clusters or bursts, such as earthquakes. After gently introducing these concepts, we take a closer look at their Bayesian estimation procedures. We sketch a model that allows self-influence within data and cross-influence across data sets and discuss its interpretation. This is then utilised for some real world applications using retweets from Twitter and apparel wholesale data.
- 12/3/2021 Andrew Mair (HW)
Modelling the influence of plant root systems on soil-moisture transport
Understanding the effect of vegetation on soil hydraulics is an important aspect of land management.
There is experimental evidence that plant root systems increase moisture transport through soil, by the creation of preferential flow channels. In this talk, I will describe a model for moisture flow in soil, which incorporates the preferential flow caused by plant roots. I will also show how this model can be calibrated using empirical data. The calibration results provide more evidence that preferential flow channels are a main cause of increased moisture transport through soil.
- 5/3/2021 Charlotte Summers (UoE)
TOASTA: a Talk On ASTrophysics Acronyms
Having a memorable, easy-to-google name for your project/instrument/code has become increasingly important in astrophysics, and this had led to the development of wide range of creative acronyms. In this talk, I’ll share some of the best (and worst) acronyms I’ve seen in astrophysics and discuss the mathematics and physics behind them. From the simple and well-known (LIGO, GAIA, VLT) to the eclectic and perhaps less well-known (BOOMERanG, GANDALF, BATMAN), I’ll give a brief look at some of the interesting physics behind these projects and some of the work done by them, with topics including numerical modelling of star and galaxy clusters, exoplanets and the first direct image of a black hole.
- 26/2/2021 Donald Hobson (MAC-MIGS)
Relationships between AI, Solomonoff induction, logical induction and nonstandard numbers
Prediction of arbitrary input data is a large part of AI. While various algorithms can do sequence prediction to some extent, the constraints and limits that all sequence predictors are restricted by are generally not well understood. In this talk I give an overview of Solomonoff induction, a sequence prediction algorithm that meets some formalizations of the notion of optimality. Solomonoff induction shows the limits of prediction achievable given limited data and unlimited compute (technically a halting oracle). Logical induction is an algorithm that assigns probabilities to mathematical statements, and which no polynomial time algorithm can out perform by more than a constant amount. Finally I will explore nonstandard numbers and see how they hint towards a deep relationship between mathematical and physical uncertainty. Any algorithm attempting sequence prediction that provably works (in ZFC for example) must work in all models of ZFC. This provides meaningful constraints on the solution, as the Kolmogorov complexity of data can vary between different models.
- 29/1/2021 SIAM-IMA Student Chapter joint colloquium: Connah Johnson (University of Warwick)
Why is life cellular? Using mathematical modelling to investigate the dynamics of early living systems.
From the most complex higher organisms to the simplest bacterium, cells are the basic structure of living systems. They act as containers for chemicals, sites for complex chemical reaction systems, and dynamically evolve through growth and division events. Rules govern cell behaviour through the presence of certain chemicals inhibiting or promoting different reactions. These behaviours are dependent on the local cell environment.
Cells interact with their microenvironment through consuming nutrients and excreting metabolic by-products. These chemicals then may react, alter, or diffuse through the environment, potentially being taken up into neighbouring cells. This sets up a feedback loop wherein the waste by-products from one cell may perturb the conditions of another. It is through the chaos of these chemical interactions that cells thrive.
Considering a cell in isolation is a research field in of itself. However, when we consider populations of simplistic cell models we begin to see the emergence of properties that we would normally associate with higher organisms. From these simple rules and our toolbox of mathematics we can begin to draw insights into how life began.
Semester 1
- 4/12/2020 Linden Disney-Hogg (UoE)
Topological Data Analysis
Topological Data Analysis (TDA) is a relatively new area of mathematics whose ethos is that, as topological invariants are invariant under ‘small’ perturbations, they are well placed to provide insight into noisy data. I will give a brief introduction to one aspect of TDA, persistent homology, with motivation from a real life application to cancer research. In that aspect, this talk should be approachable for those with a statistics background who hope to see a possible new methodology, but there is brilliant mathematical theory underpinning these data and I will explain some of this for those more topologically minded.
- 27/11/2020 Reuben Wheeler (UoE)
We (virtually) solved it!
Looking to both the past and the future, this talk will consider the subject of remote collaboration in mathematics. Practical insights into collaboration in virtual environments will be provided, taking lessons from epistolary collaborations of the past, the Polymath projects, and perspectives on scientific collaborations of the future.
Click here to download the talk slides
- 20/11/2020 Vadim Platonov (UoE)
Can an ordinary peasant survive the dark ages and become a nobleman?
In our recent work we consider the optimal investment-consumption problem for the generic agent under the competition environment (everyone wants to perform better than the others!) and solve it through the forward-utility and game-theoretic perspective. It is not surprising (or it is?) that the setting applies not to a fund-manager only, but to a wider narrative, up to minor discrepancies. We will try to reflect the life of the Medieval rustic and carefully monitor his welfare under the decisions he is to make.
- 6/11/2020 Ben Brown (UoE)
Algebraic Statistics or: Get a PostDoc or Become a Data Scientist* Trying
Henry Poincaré once said that “Mathematics is the Art of Giving the Same Name to Different Things”, and the purpose of this colloquium talk is to show that the fairly new field of algebraic statistics is no exception. To wit, we shall see how the independence model of two random variables is equivalent to the intersection of a probability simplex, along with the solution set to a system of quadratic polynomials. These are known as Segre varieties in the algebro-geometric literature. Then we can spice things up some more by extending our focus onto mixture models which lets us introduce hidden variables into the playing field. That is, we now consider cases when the marginal distribution of our known variable is a convex combination of the distributions of an unknown one, which again have an algebro-geometric interpretation. Namely this interpretation is that of secant varieties, thus further supporting the inalienable fact that varieties are the spices of life. Finally, we shall discuss how this construction can be applied to phylogenetic trees, and smoked copper river salmon, naturally.
*That is, a statistician with a MacBook.
Click here to access Ben’s slides on his GitHub page.
- 30/10/2020 Alex Evetts (Erwin Schrödinger International Institute for Mathematics and Physics, Vienna)
Growth in Virtually Nilpotent Groups
Growth and conjugacy growth are concepts that help us compare the ‘sizes’ of finitely generated infinite groups. They are easily defined but turn out to be connected to a lot of hard mathematics. I will define these concepts and discuss a few of their interesting aspects, focussing on the class of virtually abelian (and more generally virtually nilpotent) groups.
- 16/10/2020 Josh Fogg (UoE)
Improving Algorithms; Why Bother?
When creating an algorithm to solve a problem, finding one which works is far from the end of the story. The Critical Line Algorithm for financial portfolio optimization has existed for over 60 years and, despite rivals cropping up and modest improvements along the way, it still underpins much of the subject’s theory. As geneticists now look to apply CLA in their own work, a wide swath of problems has arisen requiring a new roster of solutions. In this talk we look at improving the algorithm (and algorithms more generally) across efficiency, accuracy, and applicability.
- 09/10/2020 Cob Bradley (UoE)
Weird Resolutions to the Fermi Paradox
Fermi’s Paradox posits a contradiction, via various formulations, between the nature of the observable universe and the lack of observable extraterrestrial life. It is well known that discussion of Fermi’s Paradox in ‘non-academic’ circles often leads to bizarre, unscientific, and occasionally disturbing hypotheses. Less well known is the extent to which this is also true in academia.
- 02/10/2020 Jonna Roden (UoE)
How do you brew perfect beer? I don’t know yet, but I am working on it!
What do beer brewing, bird flocking, printing, and nano-filtration have in common? Find out at the colloquium this Friday! There I will explain how to build a mathematical model that describes all the things I just mentioned, and more! Once we have this model, we will ask further questions: How do we get the yeast to sediment quicker, so the brewing process is sped up? How do we get ink to dry uniformly on paper when we print? Put in a broader context: What is the optimal problem setup that will get me the closest to some desired outcome?
In my research project I am working on exactly these questions and if you’re interested in the official description: I am working on optimal control for multiscale particle dynamics, which are modelled by integro-PDEs. But don’t worry; I promise you can definitely come along and learn something new without knowing anything (yet!) about my field!
Academic Year 2019/2020
Summer
- 24/07/2020 Charlie Egan (UoE)
Harmony in a Hypercube
The use of lattice diagrams to describe the structure of Western musical harmony goes back to Leonhard Euler who, in 1739, introduced the tonnetz, which literally translates as tone-network. In this talk, I will present a recent extension of Euler’s work which uses the geometry of a hypercube to clarify relationships between musical notes, chords and keys. This is joint work with musician Tom Glazebrook.
- 10/07/2020 Emine Atici Endes (HW)
Modelling Scratch Wound Healing Assay using an Improved Non-local Equation
- 03/07/2020 Ben Brown (UoE)
Decrypting Elliptic Curves…
Elliptic curves make up one of mathematics’ many red herrings, for they are neither ellipses nor are they topologically (real) curves. This misnomer stems from the rich and fascinating history of elliptic curves, spanning several centuries and foraying into many disciplines in mathematics; from the purer ones such as algebra and geometry to applied ones such as cryptography. Their influence stems largely from the fact that an elliptic curve forms an abelian group, whose structure is given by a basic geometry construction which can be defined over both infinite and finite fields.
In this talk I will explain how elliptic curves came to be, how one defines the group law and the geometric machinery behind it. I would like to discuss how elliptic curves over the complex numbers and over finite fields behave, and how the latter is used in cryptography.
- 26/06/2020 Tim Espin (UoE)
What’s in a name?
Have you had your fill of the Sandwich Theorem? Or wondered why on Earth the area of geometry is named so badly? While we’re on the topic, does the Hairy Ball Theorem just sound like a load of b******s to you? Maths is full of concepts and theorems with bizarre names. At this week’s PG colloquium, I’ll be digging up the history of some of the strangest (in my opinion) and giving some insight into whether the results match exactly what it says on the tin.
- 19/06/2020 Albert Solà Vilalta (UoE)
Mathematical Principles for Unlocking the Workforce
Can mathematics help in going back to work after lockdown? COVID-19 has forced many of us to work from home, but as the situation improves, we will gradually return to our workplaces. How should we return to work safely? In this talk, I will present some of the simple mathematical principles discussed in the Virtual Study Group – Mathematical Principles for Unlocking the Workforce. Given the short notice and duration of the study group, all principles should be taken with care.
- 12/06/2020 Alex Levine (HW)
Groups, regular languages and torsion
We define regular languages and discuss their application to group theory through Anisimov’s theorem. We also define the torsion problem of a finitely generated group; the language of all words representing finite order elements, and prove an analogue of Anisimov’s theorem fo`r these languages.
- 05/06/2020 Aswin Govindan (UoE)
Collatz Conjecture
Collatz problem, also known as 3x+1 problem, is one of the most notorious problems in Number theory. First proposed in 1937 by Lothar Collatz, a German mathematician, this problem can be stated in a very simple form. Consider the operation on positive integers x given by: if x is odd, multiply it by 3 and add 1; while if x is even, divide it by 2. The 3x+ 1 problem asks whether, starting from any positive integer x, repeating this operation over and over will eventually reach the number 1. The answer appears to be “yes” for all such x but this has never been proved. I will present some of the history behind the problem and its connections to other areas of mathematics. I will also focus on some of the recent developments (especially a significant result uploaded on Arxiv by T.Tao last year).
- 29/05/2020 Alec Cooper (HW)
Quantum Integrability and the Algebraic Bethe Ansatz
The Algebraic Bethe Ansatz (ABA) is a useful tool for diagnolising the Transfer Matrix (and hence the Hamiltonian) associated to Integrable Models (under suitable conditions). In this talk, I will introduce some of the concepts of Quantum Integrable Systems, with the goal of understanding the ABA construction, in the context of the XXZ Heisenberg Spin Chain with closed boundary conditions. If time permits, I will also discuss Baxter’s TQ-relations, which offer another perspective on the results of the ABA.
- 22/05/2020 Charlotte Summers (UoE)
Colliding black holes and neutron stars
Black hole and neutron star mergers have been the source of many recent discoveries in astrophysics, from the first direct detection of gravitational waves to the production of heavy elements in kilonovae. In this talk I will give a very short introduction to what black holes and neutron stars are, and then discuss why we study merging black holes and neutron stars. I will focus on the process of mathematically modelling these mergers before showing some results from simulations and discussing their implications on the theory of how they produce the effects we observe.
- 15/05/2020 Andrew Mair (HW)
Modelling and Simulating the Relationship between Soil Moisture and Root Abundance
Be it through carbon fixation or the provision of food and medicines, plants play a crucial role in supporting animal life on earth. Effective simulation of the relationship between plant roots and soil moisture is important for making predictions about the future abundance of above ground vegetation. The dominant model for water flow in soil is Richards’ equation and we use a transport equation to model root growth. In this talk, I will describe the numerical methods applied to both equations and show how we couple the models to simulate the influence of roots on soil water flow.
Semester 2
- 13/03/2020 Alina Kumukova – JCMB
What is Cryptography, or an introduction to the art of coding
Originated due to the desire of humans to send secret massages and protect valuable information from curious eyes, cryptography nowadays has become a powerful tool to secure states sovereignty and people’s welfare. When one thinks about cryptography, first things coming to their mind are advanced mathematics and sophisticated computer algorithms. However cryptography was not always about solving complex mathematical problems and creating efficient algorithms, it was relatively simple at the beginning. In this talk we will explore different techniques used to encrypt information and how they evolved over time.
- 06/03/2020 William Salkeld – Bayes Center
Royen’s Proof of the Gaussian Correlation Inequality
- 28/02/2020 Jo Kroese – Bayes Center
Playing in Someone’s Backyard with MrP: Life as a Freelance Data Scientist
- 14/07/2020 Tatiana Filatova -Bayes Center 5.02
- 07/02/2020 Vadim Platonov – Bayes Center 5.02
What is a randomness?
How to recognise a random “0-1” sequence among deterministic ones? It happened that this question is unsolvable within the probability theory, however surprisingly theory of algorithms can give us a hand! We will consider 4 intuistic candidates and discover which one suites better.
- 31/01/2020 Ben Brown – Bayes Center 5.02
What Is… Symplectic Geometry? Or: Why No Rich Man Will Enter the (Symplectic) Kingdom of God
The central concept in Euclidean geometry is that of distance, which can be captured algebraically through the notion of an inner-product and in turn gives rise to symmetries in the shape of isometries. Changing the inner-product however leads to different geometries and symmetries, and in choosing one to be a “volume form” instead leads the way into symplectic geometry. It was in 1939 when Hermann Weyl first used the adjective “sym-plectic” to describe the new group of symmetries that arose, with it being a root-for-root translation of “com-plex” — both etymologically meaning “together-braided”. In this talk I wish to discuss how symplectic geometry rose to prominence through classical mechanics before then moving onto more modern applications, of which a corollary is the justification of my sensationalistic title.
-
24/01/2020 Zoe Wyatt
How to track a cat: the maths and relativity behind GPS
General relativity is our most precise theory of gravity, and a fundamental building block of modern physics. Furthermore one of the most prominent applications of general (and special!) relativity is in ensuring correct GPS satellite communications. In this talk I will introduce and discuss some of the mathematics and relativity behind GPS and how this allows me to GPS track my furry flatmate, Amber. If time permits, I will also discuss a few other applications of relativity in our real world, such as why gold looks yellow and why the planet Mecury moves the way it does.
-
17/01/2020 Fabian Germ
An introduction to estimation for dynamical systems
Dynamical systems are arguably the most important tool in modeling phenomena that exhibit dynamic behavior. However, it is often necessary to incorporate uncertainty or unknown information in such a model. Methods to estimate the state of such a system, based on (a history of) observations, has hence been vastly researched and remain a topic that continues to receive interest from a variety of fields. In this talk, I aim to motivate and present some core ideas of estimation theory.
Semester 1
-
06/12/2019 Yiorgos Patsios (Hasselt University, Belgium)
Canard induced mixed-mode oscillations through piecewise-affine maps
Mixed mode oscillations (MMOs) is a phenomenon during which large amplitude oscillations of relaxation type (LAOs) alternate with small amplitude oscillations (SAOs). An important trigger for mixed mode oscillations are singularities known as folded nodes. We introduce a class of three dimensional systems that exhibit mixed mode oscillations (MMOs) without the presence of a folded node, thus not requiring the use of a (complicated) desingularisation technique. Moreover, we describe a technique to reduce the relevant three dimensional analysis to the study of one dimensional piecewise affine map (PAM), that shares the same oscillatory characteristics (signature). We also exhibit our findings with some relevant numerical simulations.
-
29/11/2019 Yvonne Calò
Asymptotic Flatness and Symmetries
Asymptotic Flatness (AF) represents a fundamental concept in the theory of General Relativity (GR). An asymptotically flat spacetime (AFS) can be considered as an isolated system, namely a system for which the influence of external objects can be ignored.
For such spacetimes the curvature vanishes at very large distance from the source, so the geometry at infinity becomes indistinguishable from that of Minkowski spacetime. Although the asymptotic geometry is minkowskian, (surprisingly!) the asymptotic symmetry group is not the Poincaré group but an infinite enhancement of it: the BMS group.
In this talk we naively review the concepts of spacetime, AFS (giving examples) and we explore the properties of the BMS group.
-
22/11/2019 Kajsa Møllersen (UiT Norway/ School of Informatics)
The blessing of dimensionality – a clustering approach for high-dimensional, sparse, binary data
In single-cell RNA sequencing (scRNAseq) the expression level of each gene in each cell is measured. Any cell activity – good or bad – is regulated by gene expression: fundamental biological mechanisms and complex diseases such as cancer. A tissue sample, from e.g. a cancer tumour, contains thousands of cells that are of different types and subtypes.
-
15/11/2019 Nestor Sanchez
Visualising extreme dependence structures
Extreme value distributions arise as the limit of normalised sample maxima; this property constrains the type of statistical dependence structures that can appear between them: in fact, such structures can be characterised as measures on a simplex set that fulfills some constraints. This talk is meant to explain this link and explore visually the different structures that can appear.
-
25/10/2019 Maria Lefter
Quick Overview of Stochastic Control
Stochastic control is a powerful tool for many applications to fields such as physics (landing on the moon) or finance (pricing financial products). This talk is meant to introduce the audience to the stochastic control theory through an illustration of the Merton Problem, meant to optimize a portfolio consisting of a bond and a risky asset.
-
04/10/2019 Nivedita Viswanathan
Climate Change: Can Mathematics be an answer?
Numerous studies are being conducted to either predict or arrest the adverse changes in climate. The sad truth is that ‘We have miles to go before we sleep’!
What is it that the mathematical fraternity can do to contribute? In this talk we try to answer this by exploring a few questions :
-
-
How does Mathematics help in predicting the future of this planet?
-
What are some of the Mathematical models used in the prediction of climate change?
-
What are the actions taken by different nations across the globe to deal with the current situation?
-
Are there any scientific solutions to this problem?
-
In this talk, I will start by describing the history of climate modelling and also try to answer some of these questions in detail. Hopefully at the end of the session I would like to discuss what we, the scientific community, can do that would help the current situation.
-
27/09/2019 Panagiotis Kaklamanos
What is slow-fast dynamics?
Systems of ordinary differential equations which have time as the independent variable are frequently referred to as “dynamical systems”, and such systems have been extensively used to model phenomena in various scientific disciplines. Dynamical systems that are characterized by the presence of two or more timescales, typically due to the existence of very small and/or very large parameters, are called “slow-fast” systems. In the last three decades, systems with two timescales have been studied via an approach known as “geometric singular perturbation theory”, which combines invariant manifold theory and resolution of singularities of vector fields (the “blow-up” technique); however, systems with three or more timescales have not been studied systematically yet. In this talk, we will recall some properties of slow-fast systems with two timescales and then we will discuss some recent developments in the theory of systems with three or more timescales; we will finally illustrate the effectiveness of this theory through some examples of slow-fast systems from applications.
