Costs
The HCM covers the cost of the youth hostel. This includes breakfast and dinner. Travel expenses are not reimbursed, and participants must arrange their own transportation to and from the venue. In addition, participants must pay for their own lunch on site. Affordable options are available in the cafeteria, and all groups will be escorted there at lunchtime. Alternatively, students may arrange their own lunch elsewhere. Furthermore, participants must contribute 100 euros toward general expenses, which must be paid in advance via bank transfer. In cases of financial hardship, this 100-euro contribution may be exempted. This must be indicated and explained in the application.

The program begins on Monday, January 4, 2027, at 10 a.m. Participants should arrive on Sunday, January 3, 2027. Rooms at the youth hostel will be available starting Sunday night. The program ends on Friday, January 8, 2027, at 2 p.m. at the Hausdorff Research Institute for Mathematics (HIM). Luggage can also be stored there on Friday morning. The institute is located near Bonn Central Station, so departures from there starting at 2:30 p.m. are possible and can be planned. The exact program will be announced after the application period.

A lot of mathematics is about proofs. The Lean computer program supports us with proofs: it checks that each step is strictly correct and indicates which piece(s) are still missing. In this project, we embark on a mathematical search for fast-growing functions, accompanied by Lean. After defining "fast(er) growth" precisely, we aim to find faster and faster growing functions, and to prove their growth properties in Lean.

How does the brain encode and store information? When you see a face, hear a sound, or make a decision, large populations of neurons produce complex and variable patterns of activity, but how can we extract meaning from these signals? In this project, you will explore how mathematics helps neuroscientists analyze and understand brain activity. You will work with a neural dataset and use tools from linear algebra, probability and geometry to uncover patterns and make predictions about what the brain is representing. You could also explore how the brain manages to encode information reliably despite constant changes in neural activity and connectivity, or how neural connections get shaped in an energy-efficient way enabling the brain to learn. Note that this project involves working with data that can be obtained from animal experiments.

In this project, you will explore how to estimate the strength of players or teams based on the results of previous games and competitions. Using simulated data from fields such as chess, soccer, or online gaming, you will develop your own mathematical models, implement and test them on a computer, and compare them with established approaches such as the Elo system or Bayesian methods. In the process, you’ll experience the steps of applied mathematical research: Formulating questions, testing ideas, implementing them yourself, analyzing results, and improving your methods step by step. In the end, you’ll present your models and be able to predict who will be victorious in the next game.

In this project we explore combinatorics, the “art of counting”. We develop our own elementary proofs of classical combinatorial identities and also look for new identities, for example involving binomial coefficients, Fibonacci numbers, and other recursive sequences. To this end, we use methods such as double counting, counting via bijections, and arrow-chasing. As a simple example of the principle of double counting, we interpret Fibonacci numbers as tilings of a strip of n-1 cells with squares and dominoes, and from this perspective we obtain, among others, Honsberger’s identity F_m+n=F_m+1F_n+F_mF_n-1. In the first part we learn a range of tools, and in the second part we apply them to develop and present our own proof ideas.