# Undergraduate Research Seminar

Develop your knowledge of Mathematics in a more informal setting and network with members of our School's research community.

### What is the Undergraduate Research Seminar?

The Undergraduate Research Seminar is a series of sessions exploring fun maths topics not covered in the lectures. These sessions are often led by PhD students, and we make sure to introduce every topic in an accessible manner. There are no prerequisites! All undergraduates and postgraduates students in the School of Mathematical Sciences - single honours and joint degrees - are welcome to join.

We also lead sessions where students can ask PhD students and alumni any questions they may have about research, applying for a Master's degree or a PhD program, including what kind of careers a PhD can lead to. All sessions are run in a relaxed environment where we always encourage discussions and provide snacks.

### When and where does the Undergraduate Research Seminar take place?

Sessions take place during term time on **Wednesdays at 3pm** **in the Maths Building, room MB 204**. The sessions and topics to be covered are updated below. Suggestions for session future topics and general ideas and feedback can be given here. To receive email reminders about each event, please fill out this form. Further resources, such as slides from previous talks, can also be found on the SMS EDI Committee's QMplus page.

### Contact

- To subscribe to email reminders about upcoming research seminars, please fill out this form.
- Please provide your ideas, suggestions and feedback via this (anonymous) web form.
- Further resources, such as slides from previous talks, are available here.
- This webpage will also be updated periodically with new events, so please check back regularly.

The research seminars are organised by the following PhD students from the School of Mathematical Sciences: Samuel Brevitt, Luka Ilic, and Adam Onus.

### Upcoming Events

**27th March 2024, 3pm, MB-204**

Dr Matthew Lewis *(title tbc)*

**20th March 2024, 3pm, MB-204**

*Congruent numbers and a Millennium Prize Problem* by James Kiln

Abstract: A congruent number is a rational number which is the area of some right-angled triangle with three rational side lengths. Tabulations of such numbers (which included 5 and 6) have been found in Arab manuscripts from the 10th century but the problem as to exactly which rational numbers are congruent still remains open. Only in the 13th century did Fibonacci discover that 7 is congruent, and it took until 1640 for Fermat to give the first accepted proof that 1 is not congruent. In this talk, I will outline how to relate the congruent number problem to objects known as elliptic curves, and mention a theorem of Tunnell which, conditional on the Birch and Swinnerton-Dyer conjecture, completely resolves the congruent number problem.

### Past Events (2023/24)

**13th March 2024, 3pm, MB-204**

*An informal introduction to fractals* by Adam Onus

Abstract: Let’s play the chaos game: start with a triangle and a random point in the plane, randomly pick a corner of the triangle, take the midpoint of the corner and your starting point, and repeat with this new point. The points you get will at first appear to have no pattern, but if you play for long enough then you will see some beautiful structure emerge. The chaos game is an example of an iterated function system (IFS), and in this talk I will summarise the maths behind how we can use IFS’s to generate a myriad of beautiful fractal shapes.

**28th February 2024, 3pm, MB-204**

*A History of Online Problems* by David Hannon

Abstract: I will give a basic overview on the history of online problems. I will cover the initial formations of the secretary problem and show who solved it. Along with this I will show some of the variant problems such as prophet inequalities, pandoras box and online matching problems, and some general method people have used to solve them. This talk will be pretty informal and will just present a rough overview of the field.

**21st February 2024, 3pm, MB-204**

*Differential Calculus over Finite Sets* by Julio Narciso Argota Quiroz

Abstract: Usually, differential calculus is applied over continuous spaces. It is an unnecessary restriction. A generalized definition of differential calculus is provided, that allows the definition of a first-order differential calculus over a finite set, which will produce a directed graph where the elements of the finite set are the vertices of such graph and the partial derivatives are related to the edges of the graph. Finally, we present a noncommutative Riemannian geometry constructed using the generalized differential calculus and some examples.

**14th February 2024, 3pm, MB-204**

*The importance of Bayesian statistics* by María Fernanda Pintado Serrano

Abstract: Our first encounter with statistics often comes from a frequentist viewpoint, with Bayes’ theorem representing our initial contact with Bayesian statistics. However, as we increasingly face problems that require complex modelling, Bayesian methods have been gaining popularity. In this talk, I will provide an introductory exploration of Bayesian statistics, and how it was used to solve the mysterious case of an airplane that went missing for two years.

**7th February 2024, 3pm, MB-204**

*Sets and Infinities* by Luka Ilic

Abstract: As mathematicians, we have the ability to talk about the nature of the word "infinity", and what its implications are on mathematics as a science. The journey towards the understanding of some aspects of infinity will take us through the disciplines of logic and set theory. Finally, we will discuss a few meanings of the word "infinite" and show that they are the same under the axiom of choice. We will show a few more results about infinite sets, and show that some others can never be proven, and consider what that means.

**31st January 2024, 3pm, MB-204**

*Inverse problems in medical imaging* by Alexandra Valavanis

Abstract: This talk will provide an introduction to inverse problems, focusing on what characterizes an inverse problem as ill-posed and how we can solve these with methods such as variational regularisation. We will explore how such ill posed inverse problems play a role in medical imaging, with specific emphasis on MRI. Specifically, we will discuss a proposed method to expedite the MRI data acquisition process which would reduce patient scanning time, thereby contributing to shorter patient waiting lists. The content of this talk includes topics typically covered in both mathematics and computer science degrees however, it is structured to be accessible to listeners who only have minimal background in either of those fields. Certainly, no medical knowledge is required. This presentation aims to connect theoretical concepts and their practical applications in medical imaging.

**13th December 2023, 3pm, MB-204**

*What on earth is topological data analysis?* by Adam Onus

Abstract: Topological data analysis (TDA) is a relatively new field of mathematics which uses tools from (algebraic) topology and geometry to perform a shape analysis of real-world data - think about morphing a coffee cup into a doughnut and applying this notion to a discrete set of points. In the first part of this talk, we will introduce pure mathematical tools used in TDA such as simplicial complexes and persistence diagrams. In the second part of this talk, we will see examples of how TDA has been applied in areas such neuroscience, ecology, LCD appliances, phylogenetics and crystallography. In the worst case scenario, we will just look at pretty pictures ¯\_(ツ)_/¯.

**29th November 2023, 3pm, MB-204**

*Central Limit Theorem and its analogue for Random Matrix Theory* by Svetlana Malysheva

Abstract: I will discuss the proof of Central Limit Theorem using the moment method, show the combinatorial method of the proof of semi-circle law and outline how the semi-circle law becomes the analogue of Central Limit Theorem for independent Wigner random matrices.

**22nd November 2023, 3pm, MB-204**

*Game theory and mechanism design* by David Hannon

Abstract: In this talk, I will gently introduce game theory and it's connection to the study of economic behaviour, after I will introduce mechanism design and discuss some of the relative developments in the field.

**15th November 2023, 3pm, MB-204**

*An introduction to Fourier analysis* by Sebastian del Bano Rollin

Abstract: We will introduce the notion of Fourier Series and discuss applications in physics, sound and image compression, pure maths and finance.

**1st November 2023, 3pm, MB-204**

*Q&A panel session* with Luka Ilic, María Fernanda Pintado Serrano, and Samuel Brevitt

**25th October 2023, 3pm, MB-204**

*The hidden secrets of 1/x* by Jordan Marajh

Abstract: The simple reciprocal function has many properties which are known but are not presented in an intuitive manner. In this talk we consider how one can derive the natural logarithm and e (Euler's constant) from 1/x by looking at area transformations. One can extend the knowledge of the trigonometric functions sine and cosine defined on the unit circle to derive the hyperbolic functions hidden in the geometry of 1/x. This talk requires a basic understanding of concepts encountered in calculus.

**18th October 2023, 3pm, MB-204**

*Logic Riddles* by Mehmet Sahin

Abstract: In my session I want to go through a selection of puzzles that I found interesting with everyone in the seminar. As I am familiar with solutions, I want to help everyone make their way to a solution.

Most of these will be logic puzzles; requiring out of the box thinking, different ways to approach problems and creative solutions. Additionally, I have some probability puzzles that exploit the fact that probability can be misleading. Lastly, some puzzles that have very cool gimmicks.

**11th October 2023, 3pm, MB-204**

*Chaos and Fractals in Complex Systems* by Dr Rainer Klages

Abstract: What are complex systems? What is chaos? And what are fractals? My talk will introduce these fundamental concepts of modern science by showing how they can be understood mathematically. An important part of my presentation will be mathematical modelling, which combines physics with mathematics. By means of a simple example, related to my own research, I will demonstrate how chaos in complex systems can generate fractal structures.

### Past Events (2022/23)

**5th April 2023, 1pm**

*An Introduction to Matroids and Tropical Goemetry* by Nicholas Anderson (Maths Building, Room MB 203)

Abstract: Every mathematician will encounter either matrices or polynomials over the course of their undergraduate studies. In addition to being a convenient way to perform computations and organize information, matrices encode the idea of linear systems and their properties, which are vital in much of mathematics. On the other hand, polynomials are the foundation of much of the study of abstract algebra and geometry and appear frequently in other fields of mathematics too. In this seminar we will explore a somewhat exotic departure from these familiar concepts through matroids and tropical geometry. Matroids were introduced as a generalization of linear spaces which are often found lurking deep in the background of more familiar disciplines. In a seemingly completely different direction we find tropical geometry, which is a relatively new field of mathematics that transforms the classical relationship between polynomials and their geometry into questions about collections of polyhedra— shapes with flat sides such as squares and triangles. In this talk we will see that there is a surprising and very deep connection between matroids and tropical geometry that is at the forefront of their study.

**29th March 2023, 1pm**

*All or Nothing or How to Play Billiards as a Mathematician* by Oscar Bandtlow (Maths Building, Room MB 203)

Abstract: Billiards are a class of dynamical systems given by a point particle moving inside a planar domain with elastic collisions at the boundary of the domain. In this talk, aimed at a general undergraduate audience, I will provide an overview of the case where the domain is a polygon, which, in spite of the fact that it has been studied in considerable detail for a long time, is still not completely understood. In particular, I will report on recent work with Katerina Zahradova, Julia Slipantschuk and Wolfram Just which settles a long-standing open problem in this area.

**22nd March 2023, 1pm**

*The Mathematics of Algorithm Design* by Theophile Thiery (Maths Building, Room MB 203)

Abstract: We will talk about the link between mathematics and algorithms. More precisely, how mathematics has emerged as an incredible tool in the design of __provably efficient algorithms__. To sort a pack of cards or to go rapidly from A to B, we all have our personal strategy. Is there a method that is more efficient than others? If so, can we prove it?

This talk will be an informal introduction to the mathematics of algorithm design. I'll mention an interesting conjecture that is worth 1 million dollars, surprising results for multiplying integers and matrices rapidly, and recent discoveries by DeepMind.

The talk will be light and no prerequisites are needed.

**1st March 2023, 1pm**

*H**ow Can We Understand Cancer Prognosis and Improve Therapies Using Maths?* by Elisa Scanu (Maths Building, Room MB 203)

Abstract: In recent years, there has been a growing interest in applying mathematical and computational techniques to real world situations. For example, Maths is largely used in the field of oncology. Mathematical models have the potential to aid in predicting the progression of tumours and developing new therapeutic strategies. In this talk, we will explore a mathematical and numerical model used to understand the evolution and progression of a genetic mutation, called extra-chromosomal DNA, including probabilistic and analytical techniques. Additionally, we will touch on some of the challenges that arise when applying mathematical models to cancer research, such as data limitations and model complexity. By combining mathematical and computational methods with experimental data, we hope to gain a better understanding of cancer progression and ultimately develop more effective treatments for patients.

**22nd February 2023, 1pm**

*What Does a Mathematician Look Like?* by Adam Onus (Maths Building, Room MB-203)

Abstract: Most people have the vision of a mathematician being an antisocial tormented genius who is typically an old white man with greying hair - The Imitation Game, X+Y & Good Will Hunting come to mind as examples of this trope. In fact, mathematicians, and what they study, come in all shapes and sizes, have all sorts of side interests, and are usually normal humans like you and me! Over the past Summer, I was a part of a group of PhD students who researched and wrote biographies of mathematicians from underrepresented groups (which hopefully you have seen examples of in your modules this year....), and in this talk I will talk about some of those people, who they are/were, and the sort of maths they do/did.

**15th February 2023, 1pm**

*Logic and Incompleteness Theorems* by Luka Ilic (Maths Building, Room MB-203)

What does it mean for something to be true and what does it mean to prove it? These are very important philosophical questions and in the world of mathematics they have clear answers. But with these answers come new questions like "can we prove everything that is true?" The incompleteness theorems by Gödel show that the answer to this question is "no". Whenever we do mathematics in a sensible way, we will never be able to prove everything that is true. In particular, our mathematics can never show that our mathematics is done the right way. I will discuss what all this means, how it is stated and proven and what the implications for all of mathematics are. All this will be done in a way that does not require any previous knowledge of the topics, but quite some imagination.

**8th February 2023, 1pm**

*Ricci Flow, the Poincare Conjecture, and Data Science* by Louis Yudowitz (Maths Building, Room MB-203)

Starting in the late 1900's, many seemingly impossible or hard problems in geometry have been solved using partial differential equations. Arguably the most famous case is Perelman's use of Ricci flow to resolve the Poincare conjecture, which is the only Millennium problem solved so far. I will give an overview of what Ricci flow is and why it is a powerful tool to classify certain shapes and spaces. Along the way I will introduce some basic information about differential geometry and PDEs, so no knowledge of either topic is expected. At the end of the session, we will also look at how Ricci flow can be applied to the real world!

**7th December 2022, 1pm**

*What on Earth is Topological Data Analysis?* by Adam Onus (Bancroft Building, Room 3.24)

Topological data analysis (TDA) is a relatively new field of mathematics which uses tools from (algebraic) topology and geometry to perform a shape analysis of real-world data - think about morphing a coffee cup into a doughnut and applying this notion to a discrete set of points. In the first part of this talk, we will introduce pure mathematical tools used in TDA such as simplicial complexes and persistence diagrams. In the second part of this talk, we will see examples of how TDA has been applied in areas such neuroscience, ecology, LCD appliances, phylogenetics and crystallography. In the worst case scenario, we will just look at pretty pictures ¯\_(ツ)_/¯.

**30th November 2022, 1pm**

*From PageRank to Biological Networks* by Anthony Baptista (Bancroft Building, Room 3.24)

The PageRank algorithm is a famous way to define centrality measures in networks. Since it was developed at the end of the 90s by the founder of Google, this algorithm and its extension have become a standard way to explore networks thanks to random walks. Moreover, amounts of available data, variety, and heterogeneity have been increasing drastically for several years, offering a unique opportunity to better understand complex systems. In this context, various extensions have been developed for exploring the whole topology of large-scale complex networks. However, the exploration of large multidimensional datasets remains a major challenge in many scientific fields. We will first introduce the basic notion of graph theory and the basic knowledge about Pagerank and its extensions. We will conclude with a recent extension of the Personalised PageRank algorithm (a.k.a Random Walk with Restart) to explore very general kinds of networks in the context of biological networks.

**23rd November 2022, 1pm**

*Neurohydrodynamics* by Tom Foteinos (Bancroft Building, Room 3.24)

Neurohydrodynamics is a general term for the field of scientific enquiry associated with the fluid dynamics of the brain and spinal cord. This talk will have a particular focus on the flow of cerebrospinal fluid (CSF) through the perivascular space (PVS), an annular region which surrounds most arteries in the brain.

**16th November 2022, 1pm**

*Abstract Maths and Thinking Outside the Box* by Arick Shao (Bancroft Building, Room 3.24)

From the modules you have taken (or will be taking soon), you may have noticed that a lot of the mathematical material is quite general and abstract! This is especially true if you are more focused on the pure mathematical side. Thus, you may (and should!) be wondering why modern maths is so abstract, and what is the point of it all.

In this talk, I will try to survey just a few answers to this question. I will take some simple, familiar concepts (e.g. limits, derivatives) and discuss some interesting ways to generalise and extend them. We will explore some of their applications, as well as how these can be used to build surprising connections among different areas of maths, or between maths and other fields.

**2nd November 2022, 1pm**

Q&A Session on Further Studies after Queen Mary (Bancroft Building, Room 3.24)

In this session we will help answer any questions or concerns you might have about applying and studying for a master’s or PhD after your time at Queen Mary. The panel will consist of the organisers and another PhD student (Zain Kapadia), who have each had a variety of experiences when applying and studying. We hope to see you there!

**26th October 2022, 1pm**

*Order and Disorder* by Robert Johnson (Bancroft Building, Room 3.24)

Do regular patterns exist even in the most disordered structures? This talk is about a branch of mathematics called Ramsay theory which deals with the existence of patterns in the most unlikely places.

One important facet of Ramsay theory concerns structures in graphs (or networks). The starting point is the following puzzle which you can think about before the talk: is it true that among any group of 6 people there are either 3 mutual friends or 3 mutual strangers?

Along the way we will see how graphs are popular objects of study for pure mathematicians as well as being important tools for understanding the world. We'll also get a glimpse of how graph theory has developed as a subject and some of the research going on in the Combinatorics group.

**19th October 2022, 1pm**

*1=2* by Luka Ilic (Bancroft Building, Room 3.24)

Wouldn't it be nice if mathematics would allow for a way to take a ball and duplicate it? Starting with one and ending up with two of the exact same thing, like cheating master balls in Pokémon back in the day.

Or would this be awful, because 1=2 and our whole world would collapse?

I will discuss the way mathematics allows us to duplicate our balls and the implications and precise statement of this. Keep an open mind, because serious mathematicians started long lasting mud-fights about this very topic 94 years ago and the splatters last until today.

**12th October 2022, 1pm**

*Hardly simple *by Tim Davis (Bancroft Building, Room 3.24)

In our first session, we’ll look at some problems in maths that may seem easy at first but are more subtle than they look and don’t often have any solutions!