Winners of the best student talks​
First place
Daria (Dasha) Barkova | Continuity and Its Application in Brouwer’s Fixed-Point Theorem
The concept of continuity in mathematical analysis is relatively intuitive: draw a curve without lifting your pen off the paper and it will be continuous. However, what happens if we abstract the notion of distance from this definition? At first glance, continuity might seem impossible to define without a metric. Yet, by shifting our perspective to topological spaces with the emphasis on ‘closeness’ rather than distance, we can develop a more general definition of continuity. One consequence of continuity in topological spaces is Brouwer’s Fixed-Point Theorem, proved by L. E. J. Brouwer in 1912. It states that any continuous function mapping a closed, bounded, and convex set onto itself must have at least one fixed point – a point that remains unchanged after the function is applied. This theorem is fundamental in proving the existence of solutions in mathematics, even when an exact solution is unknown. For example, there must always exist at least two antipodal points on Earth with identical temperature and pressure. This result demonstrates how topology extends our understanding of continuity through abstraction and provides powerful tools for solving problems across mathematics and beyond.​
Second place
Rebecca Hutchinson | Securing a Vault: The Mathematics of Secret Sharing
Some secrets are too critical to be entrusted to a single individual—such as the code to a bank vault. What if we could distribute the code among a group so that no individual possesses it entirely, yet the group, when combined, can reconstruct it? This is the foundation of secret sharing algorithms. However, as threats to cryptographic security evolve, several questions arise: How secure are these algorithms? Can they withstand the power of quantum computing? And are they efficient and practical for real-world implementation?​
Peyman Ramezanpour | Undecidability and Limits of Computation in Geometry
Given a set of geometric tiles and a few rules for attaching them together, is there a way to make sure whether we can cover the infinite plane or not ? The well-known “Tiling Problem” brings about many deep discussions about the nature and ability of computation and the extent to which computation, as we know it, can solve various mathematical problems that might arise in geometry (and elsewhere). In this talk, I will use examples from a variant of Wang’s “Domino” problem in plane geometry as well as Turing’s theory of computation, to deduce that the tiling problem goes beyond the scope of computation before briefly glimpsing into the generalisation of this proof in the extraordinary works of M.C. Escher and Roger Penrose. The real aim of this talk would be to scratch the surface of a deep connection between mathematical logic and geometry; something that anyone with a passion for beauty and puzzles can appreciate.
Third place
Amy Armitage | On the validity of classical logic within physics
Logical reasoning evolved from times long forgotten, acting as a crucial stepping stone in our ancestor’s survival by distinguishing danger from not danger. In the modern world, we use this same reasoning by assigning set values to statements -true or false. These come along with some basic rules: a statement cannot be true and false simultaneously; if a statement is not true, it must be false, and so on. This reasoning gave rise to the creation of classical logic; the default settings that mathematics operates under. This feels like the most natural way to do mathematics, with most mathematicians using classical proofs, which are typically simpler than constructive proofs that utilise intuitionistic logic. In general, any logical system relies on a foundation of assumptions that we cannot prove to be true. When applied to physics, it introduces a level of uncertainty as to the validity of our theories, which are usually mathematical in nature. This brings up a crucial question: is classical logic limiting our understanding of the universe? Or more importantly, does the universe abide by any logical system at all?
Rebecca Maver | Your Computer is a Girl: The History of Female Calculation
Before the development of computers to organise data or solve tedious equations, mathematicians had another tool at their disposal -rooms full of young women. In recent years light has been shed on some specific examples of this phenomenon, from NASA to Bletchley Park, but these were not isolated incidents. In fact, women were the largest technical workforce throughout the mid-twentieth century. Despite this, little has been discussed about the impact of women on the tech industry and, crucially, how they were pushed out. This talk will grapple with the rise and decline of female computers as a workforce, explore what their ‘mundane’ groundwork actually entailed and look at some examples based in Manchester.
Aiden Thomas | The Use of Three-Dimensional Nullclines to Visualise Three-Variable Differential Equations
Three-variable differential equations are fundamental in many scientific and engineering fields, including epidemiology, physics, and electrical engineering. While these equations can be solved analytically or numerically, interpreting their phase-space geometry remains a challenge. This talk explores the use of 3D nullclines as a visualisation tool to investigate the dynamics of three-variable differential equations. By constructing nullcline structures—where at least one variable’s derivative with respect to time is zero—we create a structured approach to analysing equilibrium points and phase space evolution. Phase spaces are three-dimensional diagrams that represent all the possible states of a system and how it evolves over time. This method offers an intuitive alternative to computationally expensive numerical simulations, providing a more efficient way to visualise system behaviour. Using the SIR model as a case study, we will create a generalised phase space representation to examine the influence of different initial conditions on the system’s evolution. The results from the case study demonstrate that this approach simplifies the interpretation of complex dynamical systems, enhancing both computational efficiency and the accuracy of system predictions. As a result, it serves as a valuable tool in fields such as epidemiology, physics, and electrical engineering.
Winners of the best student talks​
First place
Jakub ŠÅ¥avina | Introduction to the Lattice Boltzmann Method
In this talk, we will introduce the Lattice Boltzmann method as a powerful tool for solving partial differential equations involving transport of momentum or a substance. The Boltzmann transport equation will be used to motivate the method. We will discuss the fundamentals of this computational technique in a step-by-step breakdown. Considering the lattice structure and the collision rules that govern particle interactions, we illustrate how they mimic macroscopic fluid behaviour. Moreover, we will highlight the method's versatility in handling complex situations such as the thermal flows resulting from Navier-Stokes equations coupled to advection-diffusion equation. By the end of this talk, attendees will gain an appreciation of the method's capabilities and its potential applications in various fields, from engineering to biophysics.
Second place
Lluís Salvat Niell | Reinforcement Learning and Applications to Mobile Health
In this talk we outline the foundations and principles of reinforcement learning (RL), a branch of artificial intelligence (AI) that involves sequential decision-making under uncertainty. In particular, it sets up problems under the framework of an agent interacting with an environment, and it seeks to find a behaviour that maximises the agent’s reward. We cover the main model of an environment, a Markov decision process (MDP), as well as value functions and policies, which guide the agent in solving the problem, and also explain the main challenges. We briefly discuss the main applications, such as large language models (LLMs) and recommendation systems. Finally, we present a novel algorithm, Dyadic RL, to be deployed in a mobile health study to enhance medication adherence in cancer patients, designed using findings from domain science. It is based on establishing a hierarchy of states and actions to accelerate learning and reduce variance, and it exhibits superior empirical performance compared to existing baselines. This work was conducted during a research internship in the Statistical Reinforcement Learning Laboratory at Harvard University and can be accessed at https://arxiv.org/abs/2308.07843.
Third place
Rebecca Maver | The Witches of Mathematics: Representing Female Mathematicians of History
In this talk I explore a few historical female mathematicians, from the ancient Greek Hypatia to the 20th century ‘Rocket Girls,’ delving into their mathematical contributions and how their legacies were shaped - or neglected. Asking, in particular, who is the Witch of Agnesi? The portrayal of female mathematicians in art and text give us powerful tools to understand the shifting attitudes towards women in maths throughout history. Can we find the answers to the still prevalent gender gaps in mathematics in this history?
Best talk by a first year student
Jia Yin Wang | ARIMA: How can it reduce cost for an entrenched medical equipment company?
Previously, a medical instrument company relied on experiential judgement for sales promotion to hospitals due to the undeclared purchase period of a hospital. After speaking with experienced staff, I found that the purchase period might be seasonal and predictable, and it can be analyzed on the macro or micro level. Unlocking the potential of ARIMA analysis enables the company to streamline its sales efforts and significantly reduce costs. This talk will present the process of data collection, modeling, and evaluation.
Winners of the best student talks​
First place
Riccardo Ali | Different flavours of “sameness”: a categorical perspective Why do linear maps and matrices behave in exactly the same way?
How is studying the fundamental group of a space the same as studying the topology of the space itself? Category Theory offers a framework to make these intuitions precise by adjusting the strictness of the notion of “sameness”. In this talk, we will introduce the main concepts of Category Theory and many examples that showcase its power in formally encoding our “natural” intuition. Despite being traditionally used in pure Mathematics, the generality of its language made Category Theory popular in some applied domains as well, such as Machine Learning, for example in natural graph networks, or theoretical computer science and theory of programming languages.
Second place
Jakub ŠÅ¥avina | Spacetime as a mathematical structure
The concept of spacetime is central modern theoretical physics. In this talk, we will take an informal approach to understanding spacetime as a mathematical structure, drawing on our intuitions about Newtonian mechanics. As we explore the structure of spacetime, we will encounter fascinating areas of mathematics, including topology, differential geometry, and algebra. We will consider the philosophical implications of thinking about spacetime from the perspective of these fields of study. Throughout the talk, we will highlight the benefits of having precise mathematical formulation of seemingly vague concepts, and we will explore the extraordinary interplay between mathematics and physics which arises once effort is made to understand the suitable mathematical structure. This talk aims to provide attendees with a deeper appreciation of the role that mathematical physics plays in our understanding of the physical world
Third place
UgnÄ— MilašiÅ«naitÄ— | What can a collaboration network tell us?
Science is a social endeavour. Unsurprisingly, social scientists have been analysing how researchers work together to drive innovation and achieve success. They ask questions such as: how fractured or well-connected is the field? Who is the most influential scientist, and how do we measure this? Is she the most collaborative? Maybe someone who regularly works with other respected researchers? Perhaps he works with the most diverse set of people within the field? Although arguing which measure best approximates influence is outside the realm of computational sciences, we can try to put numbers to these questions. To do this, we will look at the collaboration graph - a type of social network. We will consider various measures proposed by network scientists, and by doing so, we will learn about complex network analysis, which has applications in many different areas, including but not limited to anthropology, environmental sciences, and biology.
Best talk by a first year student
Rebecca Maver | B-splines and solving the Schrödinger equation
Curve fitting has been an essential tool for centuries and now the draftsman’s spline has found its way into quantum mechanics. In this talk I introduce and define splines, a powerful mathematical tool that breaks down complex curves into simpler connected curves, making it more manageable and malleable. A particularly useful spline is the B-spline, with unique characteristics that make it ideal for developing a basis set from which functions can be approximated. We will explore the properties of B-splines and the recurrence relation that forms them. B-splines have a vast number of applications across mathematics, with this talk zooming in on uses in atomic physics. Specifically, we will use B-splines to approximate the radial wave function of atoms and ultimately provide a new method to solve Schrodinger’s equation.
MIMUC 2022
Winners of the best student talks​
First place
Arina Belova | Views on Probability
Probability is quite a confusing subject at first glance. It introduces a lot of concepts, and students (including me) are usually very confused from the beginning and it may lead to the loss of interest. In this talk, I will try to give a classical set-theoretical view on probability introduced by Kolmogorov in 1933. Towards the end, I would like to present another, some people may say, more "understandable" view on probability introduced by De Finetti, so-called "subjective probability" that developed further in Bayesian Statistics, so loved by a certain group of ML practitioners and statisticians.
Second place
Riccardo Ali | Theory and Praxis of Equivalence, a Philosophical and Mathematical Perspective
In this talk, we will explore the notion of equivalence, how it shaped mathematical thinking and how it is influencing contemporary machine learning research. Through equivalence, mathematicians usually try to represent the very essence of the objects they’re studying, to get rid of the unnecessary complications and uncover the uniqueness of such objects. For example, is the length of the radius a defining trait for a circle to be called a circle? We will start with very simple definitions and the ideas they encode, what equivalence is and what it represents. Then, Felix Klein’s Erlangen Program (1872) will be presented to explore how the notion of equivalence shaped the conception and development of geometry. Why do we say that, for example, a parabola and an ellipse are two different "shapes"? Are they really? Are there invariants that all equivalent shapes share with themselves? We will then enforce this concept of invariance through the lenses of Category Theory, which allows us to build bridges between different areas of Mathematics. Finally, we relate history to contemporaneity and present The Erlangen Program of Machine Learning (2021).
Third place
Jakub ŠÅ¥avina | Introducing Branched Flow and How to Simulate It
Branched flow is a wave phenomenon arising in physical systems from the propagation of tsunami and wave optics to quantum mechanics. It is characterized by the presence of branch-like caustics in the wavefront formed due to presence of a smooth randomly varying potential landscape. In this presentation, a motivating example of a physical system exhibiting the branched flow will be offered. A model of the branched flow will then be discussed, resulting in the connection of the phenomenon with the 1D Schrödinger equation. The quantum mechanical setting will be used to describe a pseudo-spectral method for the numerical solution of the governing partial differential equation. A simple implementation in Python will be outlined and finally, typical solutions exhibiting branched flow will be displayed and compared with their classical particle-based equivalents.