General information
The School of Mathematics will launch an Experimental Mathematics Lab starting in Michaelmas Term 2025. The lab offers an opportunity for undergraduate students to engage in mathematical exploration beyond the standard curriculum. Students can apply to work on projects under supervision from September 2025 until April 2026. A list of offered projects may be found below.
​
Projects
The following projects are offered in 2025:​

• Title: Exploration in Topological Data Analysis
  Faculty member: Tommaso Cremaschi
  Level: Intermediate
  Group size: 1-3
  Key words: Geometry, topology, data analysis
  Course prerequisites: Linear Algebra (recommended), algebraic topology 
  Coding prerequisites: None
  Duration: 1-2 semesters
  Description: From wikipedia “In applied mathematics, topological data analysis (TDA) is an approach to the analysis of datasets using techniques from topology. Extraction of information from datasets that are high-dimensional, incomplete and noisy is generally challenging. TDA provides a general framework to analyze such data in a manner that is insensitive to the particular metric chosen and provides dimensionality reduction and robustness to noise. Beyond this, it inherits functoriality, a fundamental concept of modern mathematics, from its topological nature, which allows it to adapt to new mathematical tools.”
  The aim of this project is to explore the mathematics behind TDA (filtration complexes and homology) and try to understand some applications via the Gudhi python library (https://gudhi.inria.fr/). In particular we can try to look at the paper: https://arxiv.org/pdf/2211.03808 and try to emulate on other data sets.
   

• Title: Realising 27 lines on a cubic surface
  Faculty member: Marvin Anas Hahn
  Level: Intermediate
  Group size: 1-3
  Key words: Algebraic Geometry, cubic surfaces, enumerative invariants
  Course prerequisites: Fields, rings and modules (required), commutative algebra or algebraic geometry (recommended)
  Coding prerequisites: None
  Duration: 1-2 semesters
  Description: One of the most beautiful results of 19th century algebraic geometry is the fact that any (smooth) cubic surface has exactly 27 lines. This theorem was proved by Cayley and Salmon (yes, the one from the lecture theatre!). Not all 27 lines have to be real. In fact, the situation is slightly more pathological since not all numbers between 0 and 27 can occur as the real count. The only counts that can occur over the real numbers are 3,7,15 or 27. The goal of this project is to find examples for each of these occurences and 3D print them.
   

• Title: Tilings of the plane
  Faculty Member: Tommaso Cremaschi
  Level: Intermediate
  Group size: 1-3
  Key words: Geometry, tilings of the plane
  Course prerequisites: Linear Algebra (recommended)
  Coding prerequisites: None
  Duration: 1-2 semesters
  Description: It is a classical result that there are only 17 ways of tiling the euclidean plane by regular shape, in fact there are quite stringent restrictions on the tiles allowed. These patterns are called Wallpaper patterns and have been made famous by work of Escher (https://en.wikipedia.org/wiki/M._C._Escher) and also the Moorish tilings at Alhambra palace. The scope of this project is to 3D print these patterns and to study recent work on Aperiodic Tilings with only one tile (https://arxiv.org/abs/2303.10798). These tiling configurations were unknown until 2023.
   

• Title: The geometry of holding patterns
  Faculty member: Nicolas Mascot
  Level: Easy - intermediate
  Group size: 1-3
  Key words: Geometry, trigonometry, aviation, holding
  Course prerequisites: Plane geometry, trigonometry, Taylor series
  Coding prerequisites: None
  Duration: 1-2 semesters
  Description: As Wikipedia puts it (https://en.wikipedia.org/wiki/Holding_(aeronautics), also seehttps://skybrary.aero/articles/holding-pattern), "In aviation, holding (or flying a hold) is a maneuver designed to delay an aircraft already in flight while keeping it within a specified airspace; i.e. going in circles." As shown in this Wikipedia article, a holding pattern looks like a racetrack pattern consisting of two parallel straight 1-minute legs connected by 1-minute 180˚ turns, positioned with respect to some radionavigational fix.
  The airspace inside this "racetrack" is protected, but there may be mountains of other planes outside of it, so the pilots must ensure they remain within the pattern until Air Traffic Control clears them to leave it. For this reason, pilots have the choice between 3 different strategies to enter the holding pattern. These 3 entries are known are direct, parallel, and teardrop (a.k.a. offset) entries, as shown on the articles mentioned above. As per aviation rules, which of these 3 entries is flown is dictated by the angle at which the plane approaches the holding fix before it enters the hold (see the "entry sectors" picture on the Skybrary article). The first goal of this project is to understand where all these rules come from, and to study how optimal they are in view of keeping the aircraft inside of the holding pattern.
  Later on, if time permits, we also study how the holding pattern must be deformed to take crosswind and head/tailwind into account. Again, pilots follow some specific rules in order to compensate for wind drift in the holding pattern, and we will study how optimal these rules are.
   

​
​
Application
Any interested students may apply until May 23rd, 2025, via the following form: https://forms.office.com/e/5apEj4JPXk
​