RISE — Research Internship Program
Welcome
RISE is a four-week mathematics research program for A-level students. The program covers graph theory, network controllability, combinatorial sequences, and discrete geometry — topics drawn from active mathematical research that are accessible with only A-level background.
Program Materials
📋 Coordinator Guide
Full week-by-week plan →
Day-by-day schedule, facilitator notes, problem sets, and student host assignments for all 16 sessions.
📄 Student Handouts
| Handout | When | |
|---|---|---|
| 1 | Background Survey | Day 1 |
| 2 | Introduction to Graphs and Networks | Days 1–2 |
| 3 | Zero Forcing Sets and the Controllability Game | Days 5–7 |
| 4 | Sequences, Subsequences, and PMI | Days 9–10 |
| 5 | The Centerpoint Theorem | Days 11–12 |
| 6 | How to Give a Research Presentation | Days 13–16 |
🎮 Interactive Demos
Run in your browser — no installation needed.
| Demo | Description |
|---|---|
| Zero Forcing Game | Click vertices to color them blue, then simulate the zero forcing rule on various graphs |
| PMI Grid Explorer | Explore zero forcing propagation on grid graphs and visualize the 2D sequence structure |
| Centerpoint Visualizer | Click to place points, then find the coordinatewise median, true centerpoint, and Tukey depth |
| Sequences & Patience Sort | Animate patience sorting on any sequence to find the longest increasing subsequence |
🐍 Python Notebooks
Open in Google Colab (free, no installation) or download for Jupyter.
| Notebook | Topics | |
| ————————————————————- | ————————————————- | ———————————————————————————————————————————————- |
| Zero Forcing | NetworkX, Z(G), propagation animation | 
|
| Centerpoint & Tukey Depth | Point clouds, depth computation, visualisation | |
| Sequences & Erdős–Szekeres | Patience sorting, tight examples, 2D subsequences | |
| Graph Theory Networking | Graph foundations, topologies, degree centrality | |
—
Open Problems
By the end of the program, students will be able to state and discuss these open problems:
- Zero Forcing Conjecture: Is Z(G) always equal to the maximum nullity M(G) for any graph G?
- Algorithmic Centerpoint: Can a centerpoint of n points in ℝ² be found in O(n) time?
- Colorful Centerpoint: If n points are colored with k colors (n/k of each), must there be a rainbow centerpoint?
- PMI Sequences: [Open problem from the professor’s research — to be stated in session.]
- Ramsey Numbers: What is R(5,5)? (Known to be between 43 and 48.)
Program hosted by Dr. Mudassir Shabbir at LUMS. Student coordinators: Danish, Uzayr, Nimra, Ahsan, Maaz, Basit.