RISE — Research Internship Program

Coordinator’s Guide · Theory Group, LUMS

Instructor: Dr. Mudassir Shabbir, Theory Group, Department of Computer Science, LUMS
Student Coordinators: Danish · Uzayr · Nimra · Ahsan · Maaz · Basit
Duration: 4 weeks × 4 days × 4 hours = 64 hours of contact time
Schedule: Monday–Thursday, 9:00–13:00 (with a 15-min break around 10:45)
Audience: 3–4 A-level students (ages 16–18)
Fridays: Free / independent reading / optional office hours

Program Architecture

The program has three tracks that build progressively:

Week	Theme	Core Content
1	Graphs and Networks	Graph theory fundamentals, special families, properties, puzzles
2	Controlling Networks	Controllability motivation, zero forcing sets, the game, propagation sequences
3	Sequences and Geometry	Erdős–Szekeres, PMI sequences, centerpoint theorem
4	Communicating Research	Scientific storytelling, slide and poster design, talk delivery, full rehearsal

Weeks 1–3 each end on Thursday with informal mini-presentations (whiteboard only, 5–8 min). Week 4 builds toward a polished external presentation — the deliverable that follows the program.

Daily Block Structure

Every session follows the same rhythm:

Time	Block	Purpose
0:00–0:30	Warm-up	Puzzle, recap of previous session, or short exploration
0:30–1:45	Main session	New content: definitions, examples, proofs, discussion
1:45–2:00	Break
2:00–3:00	Problem session	Students work problems individually or in pairs; coordinator circulates
3:00–4:00	Discussion and preview	Share solutions, discuss wrong turns, preview next session, assign homework

On Day 4 of each week (Thursday), the structure changes: the morning is a full open problem/exploration session, and the afternoon is mini-presentations.

Host Assignments

Each session is led by one student coordinator. Claim a day by replacing — with your name and committing. Students: Danish · Uzayr · Nimra · Ahsan · Maaz · Basit

Day	Week	Theme	Handout / Demo	Host
1	W1 Mon	Welcome, Survey, Graph Introduction	Networks Intro	Danish, Uzayr
2	W1 Tue	Graph Families, Trees, Bipartite	Networks Intro	Danish, Uzayr
3	W1 Wed	Coloring, Planarity, Euler’s Formula	Networks Intro	Danish, Uzayr
4	W1 Thu	Problem Session + Mini-Presentations	—	Danish, Uzayr
5	W2 Mon	Controllability Motivation + Zero Forcing Intro	Zero Forcing · Game	Uzayr, Danish
6	W2 Tue	Computing Z(G): Bounds and Examples	Zero Forcing · Notebook	Uzayr, Danish
7	W2 Wed	Zero Forcing Game + Propagation Sequences	PMI Grid	Uzayr, Danish
8	W2 Thu	Problem Session + Mini-Presentations	—	Uzayr, Danish
9	W3 Mon	Sequences, Patience Sorting, Erdős–Szekeres	Sequences & PMI · Demo	—
10	W3 Tue	PMI Sequences + Controllability Connection	Sequences & PMI · Notebook	—
11	W3 Wed	Centerpoint Theorem	Centerpoint · Demo	—
12	W3 Thu	Final Presentations + Open Problems	—	—
13	W4 Mon	What Is a Research Presentation?	Presentation Guide	—
14	W4 Tue	Structuring the Narrative	Presentation Guide	—
15	W4 Wed	Delivery and Rehearsal	Presentation Guide	—
16	W4 Thu	Final Presentations	—	—

Week 1 — Graphs and Networks

Week 1 Goals

Understand the formal definition of a graph and be comfortable with basic vocabulary.
Recognize graphs in real-world contexts.
Know the key families: paths, cycles, complete graphs, trees, bipartite graphs.
Be able to prove simple graph properties using the Handshaking Lemma and basic arguments.
Have encountered Euler’s theorem and a few coloring/planarity ideas.

Day 1 (Week 1, Monday) — Welcome, Survey, and the World of Networks

Theme: Why do we study graphs? What is a graph?

0:00–0:30 — Welcome and Ice-Breakers

Activity — Two Truths and a Maths Lie:
Each student states three things about themselves — two true and one plausible but false mathematical claim. The group guesses the lie. Encourage mathematical lies that sound reasonable (e.g., “the sum of interior angles of any polygon is 360°”).

Warm-up Puzzle on the whiteboard:

Let them struggle for 5 minutes. Reveal that the lines may extend outside the grid boundary. Moral: unexamined assumptions are the biggest obstacle in research.

0:30–1:00 — Motivational Talk

Key messages:

Research means asking questions that nobody — not even the professor — can currently answer.
The problems this week look playful. But the same mathematics underlies robot swarms, social media dynamics, cancer research, and internet security.
The goal this week is to build intuition slowly and carefully, not to rush to results.

Show or sketch three images:

A social network graph (e.g., six people, friendships drawn as lines)
A map of airline routes between cities
A biological neural network (neurons and synapses)

Ask: What do these three pictures have in common? Take several answers. The answer: they are all graphs — collections of things (vertices) and connections (edges).

1:00–1:45 — The Königsberg Bridge Problem

Tell the story of Königsberg (1736): four land masses, seven bridges, the residents’ puzzle.

Draw the map on the board, then redraw it as a graph (land masses → vertices, bridges → edges). Emphasize: the graph captures only the structure that matters for the question. Distance, shape, and color of bridges are irrelevant.

State and explore Euler’s theorem:

Verify for Königsberg: all four vertices have odd degree → no Eulerian path.

Discussion: Can you modify the Königsberg map (add or remove exactly one bridge) to make an Eulerian path possible? There are several ways; find them.

1:45–2:00 — Break
Distribute Handout: survey.md during break so students can fill it in during the problem session.

2:00–3:00 — Survey + First Problem Session

First 20 minutes: Students complete the background survey. Coordinator circulates and chats informally.

Remaining 40 minutes: First problem set. Students work in pairs.

Problem Set 1A:

Draw all non-isomorphic graphs on 4 vertices. (Two graphs are isomorphic if one can be relabeled to match the other.) How many are there?
A party has 10 people. Each person shakes hands with some others. Prove that at least two people shook the same number of hands.
(Hint: what are the possible numbers of handshakes? Can someone shake 0 hands and someone else shake 9 hands simultaneously?)
Draw a graph with 6 vertices where every vertex has degree exactly 2. How many non-isomorphic such graphs exist?
Can you draw a graph with 5 vertices where the degree sequence (list of degrees) is (1, 2, 3, 4, 4)? If not, why not?

3:00–4:00 — Discussion, Definitions, Preview

Go over solutions together. Introduce formal definitions while discussing the solutions:

Graph G = (V, E): V = vertices, E = edges (unordered pairs of vertices)
Degree d(v): number of edges touching v
Handshaking Lemma: Σ d(v) = 2 E (consequence: number of odd-degree vertices is even)
Path, cycle, connected graph (informal descriptions — formal definitions tomorrow)

Preview Day 2: “Tomorrow we’ll formalise these definitions and explore special families of graphs.”

Homework: Problems H1 and H2 from Handout networks_intro.md.

Day 2 (Week 1, Tuesday) — Formal Definitions and Special Graphs

Theme: Building the vocabulary of graph theory.

0:00–0:30 — Warm-up: Puzzle of the Day

Give 5 minutes. Then: K₁ has 0 edges, K₂ has 1, K₃ has 3, K₄ has 6. Pattern? Answer: n(n-1)/2.

Review homework from Day 1. Ask two students to present their solutions to H1 (non-isomorphic graphs on 4 vertices) and H2 (Handshaking Lemma proof).

0:30–1:45 — Main Session: Graph Vocabulary and Special Families

Distribute Handout: networks_intro.md.

Formal definitions (25 min):

Walk, path, cycle: careful distinction (a walk may revisit vertices; a path may not)
Connected: there is a path between every pair of vertices
Component: a maximal connected subgraph
Distance d(u,v): length of the shortest path between u and v
Diameter: maximum distance over all pairs of vertices
Subgraph: a graph formed by a subset of V and E

Special families (30 min):

Graph	Notation	Key properties
Path	Pₙ	n vertices, n-1 edges, two leaves
Cycle	Cₙ	n vertices, n edges, every vertex degree 2
Complete graph	Kₙ	every pair connected, n(n-1)/2 edges
Star	K₁,ₙ	one hub (degree n), n leaves (degree 1)
Complete bipartite	Kₘ,ₙ	two groups, all cross-edges
Grid	Pₘ × Pₙ	m×n lattice

Bipartite graphs (20 min):

A graph is bipartite if vertices can be 2-colored (red/blue) so every edge connects red to blue.

Examples: paths, even cycles, trees. Non-examples: odd cycles, K₃.

Theorem (state without proof): A graph is bipartite if and only if it contains no odd-length cycle.

Activity: Draw several graphs. Students identify which are bipartite and find a 2-coloring if so.

1:45–2:00 — Break

2:00–3:00 — Problem Session

Problem Set 1B:

Prove that a connected graph on n vertices has at least n-1 edges. When does equality hold?
A complete bipartite graph K₃,₃ has two groups of 3 vertices each, with every vertex in group 1 connected to every vertex in group 2. How many edges does it have? Draw it. Is it planar (can you draw it without edges crossing)? (This is the famous “houses and utilities” puzzle.)
Show that in any graph, the sum of all vertex degrees equals twice the number of edges. Use this to prove: the number of vertices with odd degree is always even.
A graph is called regular if every vertex has the same degree. Draw all non-isomorphic 3-regular (cubic) graphs on 6 vertices.
(Challenge) A graph on n vertices has more than n²/4 edges. Prove it must contain a triangle (K₃ as a subgraph). This is Turán’s theorem for r=3 — try to find the argument.

3:00–4:00 — Trees: Discussion and Introduction

Go over Problem 1 — this naturally introduces trees:

Definition: A tree is a connected graph with no cycles. A forest is a graph with no cycles (connected or not).

Key facts (prove or sketch proofs as appropriate):

A tree on n vertices has exactly n-1 edges.
Between any two vertices in a tree there is exactly one path.
Every tree with n ≥ 2 vertices has at least 2 leaves (degree-1 vertices).

Proof of the third fact: Take the longest path in the tree. Both endpoints must have degree 1 — otherwise there would be a longer path or a cycle. □

Preview: Trees turn out to be the “simplest” graphs and will reappear when we study zero forcing.

Homework: Problems H3 and H4 from networks_intro.md.

Day 3 (Week 1, Wednesday) — Graph Properties and Coloring

Theme: Deeper properties — chromatic number, planarity, and graph invariants.

0:00–0:30 — Warm-up: Map Coloring Puzzle

Show a map with 8 regions. Challenge: color the map so no two adjacent regions share the same color. What is the minimum number of colors needed?

This naturally motivates graph coloring: create a graph where regions are vertices and adjacency between regions becomes edges. Coloring the map = coloring the graph.

0:30–1:45 — Main Session: Graph Coloring and Planarity

Graph coloring (40 min):

Chromatic number χ(G): The minimum number of colors needed to color the vertices of G so no two adjacent vertices share the same color.

Examples:

Path Pₙ: χ = 2 (alternating colors)
Odd cycle C₅: χ = 3
Complete graph Kₙ: χ = n
Bipartite graph: χ = 2 (that’s the definition!)

Greedy coloring: Assign each vertex the smallest color not used by any already-colored neighbor. This gives at most Δ(G) + 1 colors, where Δ(G) is the maximum degree.

Theorem (Brook’s, 1941, state only): For any connected graph G that is not a complete graph or an odd cycle: χ(G) ≤ Δ(G).

The Four Color Theorem (state and discuss): Every planar graph (one that can be drawn without edge crossings) can be 4-colored. Proved by Appel and Haken in 1976 — the first major theorem proved with substantial computer assistance. This was controversial at the time. Discuss: what counts as a mathematical proof?

Planarity (35 min):

A graph is planar if it can be drawn in the plane without edges crossing.

Euler’s formula for planar graphs: For any connected planar graph drawn without crossings: \(V - E + F = 2\) where V = vertices, E = edges, F = faces (regions, including the outer unbounded region).

Verify with examples: triangle (V=3, E=3, F=2 → 3-3+2=2 ✓), cube graph (V=8, E=12, F=6 → 8-12+6=2 ✓).

Consequence (edge bound): For any simple planar graph with V ≥ 3: E ≤ 3V - 6.

Proof: Each face is bounded by at least 3 edges, each edge borders at most 2 faces, so 3F ≤ 2E. Substitute into Euler’s formula. □

Corollary: K₅ and K₃,₃ are non-planar (check: K₅ has V=5, E=10 > 3·5-6=9; K₃,₃ has V=6, E=9 > 2·6-4=8 using the bipartite version of the bound).

1:45–2:00 — Break

2:00–3:00 — Problem Session

Problem Set 1C:

Compute χ(G) for: (a) the Petersen graph, (b) the 3×3 grid, (c) the “wheel graph” W₅ (a 5-cycle with one central hub connected to all 5 cycle vertices).
Prove: χ(G) ≥ ω(G), where ω(G) is the clique number — the size of the largest complete subgraph. (A clique is a set of pairwise-connected vertices.)
A graph G has 10 vertices and 40 edges. Prove G is non-planar.
Use Euler’s formula to prove that K₃,₃ is non-planar. (Hint: in a bipartite graph, every face is bounded by at least 4 edges, since there are no triangles.)
(Challenge) The chromatic polynomial P(G, k) counts the number of proper k-colorings of G. For a tree Tₙ on n vertices, show P(Tₙ, k) = k(k-1)ⁿ⁻¹. (Hint: root the tree and color greedily from the root down.)

3:00–4:00 — Algorithms on Graphs + Preview of Week 2

Brief introduction to two famous algorithms (describe informally, no code required):

Breadth-First Search (BFS): Start at a vertex. Explore all neighbors, then neighbors-of-neighbors, etc. Gives shortest paths. Think of it as “rings spreading outward.”

Depth-First Search (DFS): Explore as far as possible down one branch before backtracking. Think of it as “always take the first new turn you see.”

Application: BFS gives the distance (shortest path length) between any two vertices. The diameter of the graph is the maximum BFS distance over all starting vertices.

Preview of Week 2: Next week we ask: if you want to control a network — steer its behavior to a target state — which vertices do you need direct access to, and how few can you get away with? This will lead us to the zero forcing game.

Homework: H3 and H4 from networks_intro.md, plus:
Bonus: Find a planar graph where χ(G) = 4 and ω(G) = 3 (a graph needing 4 colors but containing no K₄). (Hint: Grötzsch graph, or any odd-wheel.)

Day 4 (Week 1, Thursday) — Problem Session and Mini-Presentations

Theme: Consolidate Week 1; students present.

0:00–1:45 — Open Problem Exploration

Students choose one of the following investigation threads:

Thread A — Graph Isomorphism:
Two graphs are isomorphic if one can be relabeled to match the other. How do you tell quickly if two graphs are not isomorphic? (Compare degree sequences, number of triangles, diameter, etc.) How do you tell they are isomorphic? This is a famously hard problem — no efficient general algorithm is known.

Task: Find two non-isomorphic graphs with the same degree sequence. Try to find graphs that are hard to distinguish.

Thread B — Ramsey Theory:
Suppose you have 6 people. Must there be either 3 mutual friends or 3 mutual strangers? (This is R(3,3)=6, the smallest Ramsey number.) Try to prove it. Then ask: what about 4 mutual friends or 4 mutual strangers? R(4,4) = 18 — it is known, but much harder. R(5,5) is unknown.

Task: Prove R(3,3) ≤ 6 and demonstrate that 5 people are not enough.

Thread C — Degree Sequences:
The Erdős–Gallai theorem says exactly which lists of non-negative integers can be the degree sequence of a simple graph. Try to discover the condition yourself by exploring small cases. When can (4,3,3,2,2) be a degree sequence? What about (4,4,3,3,2)?

1:45–2:00 — Break

2:00–4:00 — Mini-Presentations

Each student presents one result, puzzle, or worked problem from Week 1 (5–8 minutes, whiteboard only). Possible topics:

Proof of the Handshaking Lemma
Why K₃,₃ is non-planar
A construction showing R(3,3) ≤ 6
The chromatic polynomial of a path

Coordinator and other students ask at least one question per presentation. Emphasis on clarity and correctness, not polish.

End of Week 1 discussion: What surprised you? What is still unclear? Write one remaining question on a sticky note — these go on the board and are answered during Week 2 warm-ups.

Week 2 — Controlling Networks: Zero Forcing

Week 2 Goals

Understand what network controllability means, physically and mathematically.
Define zero forcing sets and the zero forcing number Z(G).
Compute Z(G) for paths, cycles, complete graphs, stars, trees, and small grids.
Understand the two-player zero forcing game and think about strategy.
Understand propagation sequences and their structure.
See the connection between Z(G) and linear-algebraic controllability.

Day 5 (Week 2, Monday) — Controllability Motivation

Theme: What does it mean to control a network?

0:00–0:30 — Warm-up: Answer the Sticky-Note Questions

Address the open questions from Week 1 (collected on sticky notes on Day 4). Answer two or three; defer harder ones to the end of the week.

Puzzle of the Day:

This is the zero forcing intuition. Discuss informally.

0:30–1:45 — Main Session: Three Motivating Examples

Example 1 — Influencing a Social Network (30 min)

Consider a simplified social network of 12 users as a graph. Each user holds a binary opinion (0 or 1). The influence rule: a user flips their opinion to match the majority of their neighbors if that majority is strong enough.

A political campaign can “buy” 3 accounts to hold a fixed opinion (1). The goal: through social dynamics, eventually all users adopt opinion 1.

Question: Which 3 accounts gives maximum eventual influence?

Draw a sample graph on the board. Let students propose which 3 vertices to color. Simulate the dynamics. Notice that a badly chosen set of 3 gets “stuck” — the dynamics stall before everyone is converted. A well-chosen set propagates to everyone.

Key insight: The structure of the graph determines which sets work. This is a controllability question.

Example 2 — Swarm of Robots (20 min)

You have n drones in the air. Each drone adjusts its velocity to match the average velocity of its neighbors in the communication graph. You have direct radio control over only k drones.

Question: What is the minimum k such that you can steer the entire swarm to any target formation?

This is a linear dynamical system. The answer depends on Z(G), the zero forcing number — which we will define today.

Example 3 — The Mexican Wave (15 min)

In a football stadium, each person stands if both their immediate left and right neighbors are already standing. A coordinator seeds the wave by getting a group to stand simultaneously.

Draw the stadium as a cycle graph (seats in a ring). If the seeded group is S, simulate the wave.

Question: What is the minimum

so the wave goes all the way around? (Answer: 2 adjacent people. We will see this is Z(Cₙ) = 2.)

1:45–2:00 — Break

2:00–3:00 — Defining Zero Forcing

Distribute Handout: zero_forcing.md.

The Color Change Rule:
A blue vertex with exactly one white neighbor forces that neighbor to become blue.

Walk through examples: P₅ (starting from one endpoint), C₄, K₄, K₁,₃. Students follow along and predict the next step before the coordinator reveals it.

Definitions:

Zero forcing set S: Starting with S blue, repeated application of the color change rule eventually colors all vertices blue.
Zero forcing number Z(G): Minimum S over all zero forcing sets S.

3:00–4:00 — First Computations and Discussion

Students work in pairs:

Find Z(P₆), Z(C₆), Z(K₄), Z(K₁,₄), Z(K₂,₃).
Prove Z(Pₙ) = 1 for all n ≥ 1.
Prove Z(Cₙ) = 2 for all n ≥ 3.

Discuss solutions. Introduce the key question: how does the structure of G determine Z(G)?

Homework: H5 from zero_forcing.md (grid conjecture).

Day 6 (Week 2, Tuesday) — Computing Z(G): Examples and Bounds

Theme: How do we prove lower bounds on Z(G)?

0:00–0:30 — Warm-up: Grid Exploration

Students bring their Z(grid) computations from homework. Pool results:

Grid	Z
2×2	2
2×3	2
3×3	3
2×4	2
3×4	3
4×4	4

Conjecture together: Z(Pₘ × Pₙ) = min(m, n).

0:30–1:45 — Main Session: Lower Bounds

Why lower bounds are hard (10 min):
To show Z(G) ≤ k, we exhibit a zero forcing set of size k. To show Z(G) ≥ k, we must argue that every set of size k-1 fails to zero force G. This requires ruling out all possible sets.

Lower bound technique 1 — The Degree Argument (25 min):

If every vertex in G has degree ≥ 3, then after forcing begins, a blue vertex with one white neighbor means all its other d(v)-1 ≥ 2 neighbors are already blue. This constrains propagation heavily.

Formal bound: Z(G) ≥ δ(G) in some cases — but not always. Find an example where Z(G) < δ(G).

Lower bound technique 2 — Cut Vertices and Bridges (30 min):

A bridge is an edge whose removal disconnects the graph. A cut vertex (articulation point) is a vertex whose removal disconnects the graph.

Claim: If G has a bridge e = (u,v) and removing e splits G into components G_u and G_v, then any zero forcing set must contain at least one vertex from each component before the bridge is traversed.

This gives a recursive structure for computing Z(G) on trees.

Trees (20 min):

For trees, Z(T) equals the path cover number P(T) — the minimum number of vertex-disjoint paths that together cover all vertices of T.

Intuition: each path in the cover corresponds to one “chain” of forcing.

Example: A star K₁,₅ has path cover number 5 (each leaf is its own path), so Z(K₁,₅) = 5… wait, we said Z(K₁,ₙ) = n-1. Recheck! With 4 leaves blue, the hub has exactly 1 white neighbor. So Z(K₁,ₙ) = n-1.

This corrects the path cover formula — the centre is also a path endpoint. Work through the star carefully.

1:45–2:00 — Break

2:00–3:00 — Problem Session

Problem Set 2A:

Find Z(G) for the Petersen graph (10 vertices, 15 edges). Prove both the upper and lower bound.
Let T be the tree formed by a path P₄ with a pendant (leaf) attached to each of the middle two vertices. Draw T and find Z(T).
Prove: for any graph G, Z(G) ≤ n-1, where n = V(G) . When does equality hold?
Prove: if G is a connected graph and v is a cut vertex, then Z(G) ≥ 2. (Hint: consider what happens when propagation needs to “cross” through v.)
(Challenge) Show that the zero forcing number is not monotone with respect to subgraphs: find graphs H ⊆ G where Z(H) > Z(G). Why does this make zero forcing harder to analyze than graph coloring?

3:00–4:00 — The Linear Algebra Connection

This session connects zero forcing to the original controllability motivation.

The adjacency matrix: The adjacency matrix A of a graph G is the n×n matrix where Aᵢⱼ = 1 if (i,j) ∈ E and 0 otherwise.

Controllability of linear systems (on the board):

A linear dynamical system on a graph G with controlled set S:

\[\dot{x}(t) = Ax(t) + Bu(t)\]

is fully controllable iff the controllability matrix [B

A²B

…

Aⁿ⁻¹B] has rank n.

The connection (state as a theorem):

This is a deep result. Students do not need to prove it, but they should see why the zero forcing rule is a natural “simulation” of the matrix rank condition.

Discussion: Why does a combinatorial rule (coloring) capture an algebraic condition (rank)?

Homework: H6 from zero_forcing.md.

Day 7 (Week 2, Wednesday) — The Zero Forcing Game and Propagation Sequences

Theme: Playing the game; understanding propagation order.

0:00–0:30 — Warm-up: Directed Zero Forcing

Introduce the idea briefly: what if edges in G have a direction (directed graph / digraph)? The forcing rule can be adapted: a blue vertex forces its white out-neighbor only if it is the only white out-neighbor.

Does Z change for directed paths? Directed cycles? Explore on the board.

0:30–1:45 — The Two-Player Zero Forcing Game

Game definition:

Setter wants to zero force G with as few initial blue vertices as possible.
Blocker can, on each round, protect one vertex (that vertex cannot be forced this round even if it “should” be).

Play alternates: Setter picks a vertex to add to the initial set (or has already fixed S before the game starts — variants exist). Then Blocker picks a vertex to protect. Then the color change rule is applied (skipping protected vertices). Repeat until all vertices are blue (Setter wins) or propagation stalls forever (Blocker wins).

Play on a 4×4 grid (on the board):

Draw a large 4×4 grid. Assign Setter and Blocker roles to two students. Others observe and advise. Play two rounds (switch roles after the first).

Observe: Where does Blocker’s protection have the most impact? Blockers should target “bottleneck” vertices — places where propagation must pass through.

The game zero forcing number Z_B(G): The minimum

such that Setter has a winning strategy, assuming both play optimally.

Question: Is Z_B(G) always equal to Z(G)?

Answer: No — Z_B(G) ≥ Z(G) always, but strict inequality is possible. (Blocker can disrupt a zero forcing set that would work in the solo game.)

1:45–2:00 — Break

2:00–3:00 — Propagation Sequences

Every zero forcing process generates a propagation sequence: the order in which vertices turn blue.

Formal definition: Let S = {s₁, …, sₖ} be a zero forcing set. The propagation sequence is an ordering of all n vertices of G such that:

The first k elements are the elements of S.
Each subsequent vertex vᵢ was forced by some vertex vⱼ with j < i (the forcing vertex appears earlier in the sequence).

Example on P₅:

Vertices: 1, 2, 3, 4, 5 (path in order)
S = {1}
Propagation sequence: (1, 2, 3, 4, 5) — increasing sequence

Example on the 3×3 grid (vertices labeled (row, column)):

Start with top row blue: S = {(1,1), (1,2), (1,3)}. Propagation:

(1,1) forces (2,1) (only white neighbor)
(1,2) forces (2,2)
(1,3) forces (2,3)
(2,1) forces (3,1)
(2,2) forces (3,2)
(2,3) forces (3,3)

Sequence: (1,1),(1,2),(1,3),(2,1),(2,2),(2,3),(3,1),(3,2),(3,3).

Observation: Row indices are non-decreasing throughout. This is a 2D monotonicity condition — we will formalise it in Week 3.

Problem session extension: Find all valid propagation sequences for Z-forcing on a 2×3 grid, starting from the minimum zero forcing set. Are they all “2D monotone” in some sense?

3:00–4:00 — Discussion: What Makes a Good Propagation Sequence?

Group discussion around these questions:

Can two different zero forcing sets produce the same propagation sequence?
Can the same zero forcing set produce multiple different propagation sequences (different forcing orders)?
Is there always a propagation sequence that is “monotone” in some natural sense?

These questions motivate Week 3. The answers connect to the PMI sequences defined by the professor.

Homework: H7 from zero_forcing.md.

Day 8 (Week 2, Thursday) — Problem Session and Mini-Presentations

0:00–1:45 — Extended Problem Exploration

Students choose an investigation thread:

Thread A — Zero Forcing Bounds Race:
For each of the following graphs, find the exact Z(G) with proof: (i) the icosahedron graph, (ii) the 3-cube graph Q₃ (vertices = binary strings of length 3, edges between strings differing in one bit), (iii) a random graph on 8 vertices (each edge present with probability 1/2 — draw it, then find Z).

Thread B — Game Analysis:
For the zero forcing game on C₅ (5-cycle), determine Z_B(C₅). Write out the optimal strategy for both Setter and Blocker. Try the same for C₆. What pattern do you see?

Thread C — Propagation Counting:
For the path Pₙ with S = {1}, there is exactly one propagation sequence: (1, 2, 3, …, n). For the 2×2 grid with an appropriate S, count all valid propagation sequences. Can you find a formula for the number of propagation sequences for the n×n grid?

1:45–2:00 — Break

2:00–4:00 — Mini-Presentations

Each student presents one result from Week 2 (5–8 minutes, whiteboard):

A proof that Z(Kₙ) = n-1
The game analysis on a specific graph
A worked example of a propagation sequence and its 2D structure
The linear algebra connection explained in their own words

End of Week 2: collect new sticky-note questions for Week 3.

Week 3 — Sequences and Geometry

Week 3 Goals

Understand sequences, subsequences, and the concept of monotonicity.
Prove the Erdős–Szekeres theorem using the pigeonhole principle.
Understand what PMI sequences are and how they relate to zero forcing propagation.
Understand the centerpoint theorem: why it is needed, what it says, and a sketch of why it holds.
Know at least one genuinely open problem well enough to explain it to a non-expert.

Day 9 (Week 3, Monday) — Sequences and Erdős–Szekeres

Theme: Order in disorder.

0:00–0:30 — Warm-up: A Sequence Puzzle

(Answer: 4²+1 = 17.)

Let students explore small cases first: n=5 (length 5 → monotone subseq of length 3), n=9 (length 9+1 = 10 → length 4), etc.

0:30–1:45 — Main Session: Sequences and the Theorem

Distribute Handout: sequences_pmi.md.

Definitions (20 min):

Sequence, subsequence
Increasing / decreasing / monotone subsequence
Longest increasing subsequence (LIS)
Work puzzles A and B from the handout together

Patience sorting (20 min):

Describe the patience sorting algorithm informally — read numbers left to right, build “piles” where each new number goes on the leftmost pile with a smaller top. If none, start a new pile. The number of piles at the end equals the LIS length.

Trace through example: (3, 1, 4, 1, 5, 9, 2, 6):

Card	Action	Piles (tops)
3	New pile	[3]
1	New pile	[3][1] — No: place on leftmost pile with top > 1. Top of pile 1 is 3 > 1.

(Refine: place each card on the leftmost pile whose top card is ≥ the card. Start new pile only if all tops are smaller.)

Trace carefully through the full example. Count piles. Verify it equals LIS length.

The Erdős–Szekeres Theorem (45 min):

State the theorem. Do two worked examples illustrating the bound is tight. Then give the full pigeonhole proof (see Handout sequences_pmi.md Section 2 for the complete proof — walk through it on the board step by step).

Emphasize the structure of the proof: assign a label to each element, observe that labels from a bounded set, apply pigeonhole, derive a contradiction.

1:45–2:00 — Break

2:00–3:00 — Problem Session

Problem Set 3A:

Find the LIS of (1, 5, 2, 4, 3, 8, 7, 6). Use patience sorting to verify.
Prove the Erdős–Szekeres bound is tight: construct a sequence of exactly n² distinct numbers with no increasing or decreasing subsequence of length n+1. (This is Problem H8 from the handout.)
Let σ be a permutation of {1, …, n}. The LIS of σ and the number of piles in patience sorting are equal. Prove that the LIS is the length of the longest chain in the partial order where i ≤ j iff position(i) < position(j) and value(i) < value(j).
(Dilworth’s theorem) For any finite partially ordered set (P, ≤): the minimum number of chains needed to cover P equals the maximum size of an antichain. Verify this for the Erdős–Szekeres poset.
(Challenge) The RSK correspondence is a bijection between permutations of {1,…,n} and pairs (P,Q) of standard Young tableaux of the same shape. Under RSK, the LIS of the permutation equals the length of the first row of P. Can you find (from a reference) what the longest decreasing subsequence corresponds to?

3:00–4:00 — Towards 2D: Point Sets in the Plane

Introduce 2D sequences of points. Define a “2D increasing subsequence” (both coordinates increase). Prove or conjecture a 2D Erdős–Szekeres theorem.

Work through Puzzle E from the handout. Introduce the connection to zero forcing propagation sequences for grid graphs (preview of Day 10).

Homework: H8 and H9 from sequences_pmi.md.

Day 10 (Week 3, Tuesday) — PMI Sequences and Controllability

Theme: The heart of the research.

0:00–0:30 — Warm-up

Return to the grid propagation sequences from Week 2, Day 7. Ask: looking at these sequences as 2D point sequences, what monotonicity property do they satisfy?

Let students observe and conjecture before the formal definition.

0:30–1:45 — Main Session: PMI Sequences

(The coordinator presents the formal definition of PMI sequences as given in the professor’s research notes. The following is a framework into which that definition should be inserted.)

Step 1: Review the 1D picture (15 min)

For path graphs, propagation sequences are exactly increasing sequences (in the natural labelling). The Erdős–Szekeres theorem gives bounds on their structure. The zero forcing number of a path is 1 — corresponding to the unique starting point of an increasing sequence.

Step 2: The 2D picture (25 min)

For grid graphs:

Vertices are labeled by (row, column) pairs.
A propagation sequence is a sequence of these 2D labels.
The condition “blue vertex with exactly one white neighbor forces it” translates into a constraint on the order of the 2D labels.

Step 3: PMI — Formal Definition (30 min)

[INSERT DEFINITION FROM PROFESSOR’S RESEARCH HERE]

Present the definition carefully with:

The formal statement
A concrete example of a PMI sequence
A concrete example of a sequence that fails to be PMI and why
The theorem connecting PMI to zero forcing propagation

Step 4: Main theorem statement (15 min)

State the theorem connecting PMI sequences to network controllability:

[THEOREM FROM PROFESSOR’S RESEARCH — e.g., “S is a zero forcing set whose propagation sequence is a PMI sequence if and only if…”]

1:45–2:00 — Break

2:00–3:00 — Exploration Problems on PMI

Students work through Problems E1, E2, E3 from the sequences handout, now interpreted in the PMI framework.

Additional explorations:

For the 2×2 grid, list every valid propagation sequence starting from the minimum zero forcing set. Which are PMI?
For the 2×3 grid, is every zero forcing propagation sequence a PMI sequence? Or are there forcing sequences that are not PMI?
Conjecture: is the minimum Z(G) always achieved by a zero forcing set whose propagation sequence is PMI? Test your conjecture on small grids and trees.

3:00–4:00 — Open Problems and Research Directions

Present 2–3 open problems from the professor’s research that the students can now understand:

Open Problem 1: [OPEN PROBLEM FROM PROFESSOR’S RESEARCH]

Open Problem 2: [OPEN PROBLEM FROM PROFESSOR’S RESEARCH]

Open Problem 3 (standard, safe):

Students discuss which open problem they find most compelling and why.

Homework: H10 from sequences_pmi.md; attempt to formulate one new conjecture related to PMI sequences.

Day 11 (Week 3, Wednesday) — The Centerpoint Theorem

Theme: A completely separate gem of discrete geometry.

0:00–0:30 — Warm-up: The Mars Data Problem

Present the problem cold — no hints.

Take answers. Likely responses:

“The average” → challenge: what if one GPS is broken and reports (10⁶, 10⁶)?
“The median of both coordinates” → great idea — but does it work?

Write the challenge on the board: the coordinatewise median can be badly non-central. This motivates the session.

0:30–1:45 — Main Session: Centerpoint Theorem

Distribute Handout: centerpoint.md.

Act 1 — The Failure of the Average (15 min)

Work through the outlier counterexample from the handout. Emphasize the concept of breakdown point: the average’s breakdown point is 0 (one bad point ruins it), the median’s is 50%.

Act 2 — The Median in 1D (15 min)

The median of 1D data: every halfline through the median contains at least n/2 points. Proof: if the median were not a centerpoint, more than half the data would be on one side — contradicting the definition of median.

Act 3 — The Naïve 2D Median Fails (25 min)

Work through the explicit counterexample from the handout (5 points: (3,0),(4,0),(5,0),(0,3),(0,4)).

Coordinatewise median = (3,0). Draw this on the board. Find the halfplane through (3,0) that contains only 1 point (the point itself). Conclude: (3,0) has Tukey depth only 1/5 < 1/3.

Ask students: where should the “real” centerpoint be? Have them guess by looking at the picture.

Act 4 — The Centerpoint Definition and Theorem (30 min)

Define Tukey depth and centerpoint formally. State the Centerpoint Theorem. Verify the theorem on the 5-point example by identifying a valid centerpoint.

Compare 1D median (guarantee n/2) vs. 2D centerpoint (guarantee n/3): the “cost” of an extra dimension.

State the general d-dimensional version (guarantee n/(d+1)) and the Helly connection.

Act 5 — Helly’s Theorem (15 min)

State Helly’s theorem. Prove the 1D special case (intervals on a line). Sketch why it implies the centerpoint theorem (intersect all halfplanes containing at least n/3 of the data — if no common point, Helly’s theorem would be violated).

1:45–2:00 — Break

2:00–3:00 — Problem Session

Problem Set 3B:

Find a centerpoint for the set {(0,0),(4,0),(2,4)} (triangle). Verify your answer satisfies the n/3 condition.
Prove: the set of all centerpoints of a finite point set is convex. (Hint: if c₁ and c₂ are both centerpoints, show any convex combination is also a centerpoint by considering halfplanes and using linearity.)
Prove Helly’s theorem in 1D: if n intervals on the real line are pairwise intersecting, they all share a common point.
Show that the centerpoint bound n/3 is tight in 2D: construct a set of n points such that no point (not even from outside the set) has every halfplane through it containing more than n/3 of the points. (Hint: use a triangle arrangement.)
(Challenge) The Radon partition theorem says: any set of d+2 points in ℝᵈ can be partitioned into two sets whose convex hulls intersect. Use Radon’s theorem (you may cite it without proving it) to give a short proof of Helly’s theorem in the plane.

3:00–4:00 — Applications and Open Problems

Discuss applications of the centerpoint:

Robust statistics: Centerpoint as a high-dimensional analogue of the median. Used in depth-based data analysis.
Computational geometry: Finding centerpoints efficiently. Best known: O(n log n) in 2D. Can it be done in O(n)?
Voting theory: The “median voter theorem” is the 1D centerpoint theorem. Plott’s condition is the 2D analogue — and it almost never holds! This is why 2D voting is typically indeterminate.
Machine learning: Outlier-robust classifiers use depth-based notions.

Open Problems (all genuinely open):

Algorithmic Centerpoint: Find a centerpoint of n points in ℝ² in O(n) time.
Colorful Centerpoint: If each point is colored one of 3 colors with n/3 of each color, must there be a “rainbow” centerpoint? (Every halfplane through it has points of all 3 colors.)
First Selection Lemma (proved, but state it): For n points in ℝ², some point of ℝ² lies inside at least cn³ triangles formed by the data points, where c > 1/27 is a universal constant. The best known c is approximately 1/27 — achieving exactly 1/27 is open.

Homework: H11 and H12 from centerpoint.md; prepare one slide (one piece of paper / one whiteboard section) for tomorrow’s final presentation.

Day 12 (Week 3, Thursday) — Final Presentations and Open Problems

0:00–0:30 — Warm-up: Open Problem Speed Round

Write 5 open problems on the board (one from each topic area). Students have 15 minutes to write a one-sentence description of each problem accessible to a 14-year-old. Share and compare.

0:30–1:45 — Final Student Presentations

Each student gives a 10–12 minute presentation. They may choose any topic from the three weeks and should:

State one definition clearly.
Give one example they worked out themselves.
State one theorem and give a sketch of the proof (or at least explain why the result is not obvious).
State one open problem related to the topic and explain why it is hard.

The coordinator and other students ask questions. Aim for mathematical depth over polish.

Suggested topics if students are unsure:

Zero forcing on trees (characterize the zero forcing sets of a given tree)
The Erdős–Szekeres theorem: statement, proof, and tight example
Centerpoint theorem: the Mars motivation, why coordinatewise median fails, the theorem
The zero forcing game and game zero forcing number
PMI sequences and their connection to controllability

1:45–2:00 — Break

2:00–3:30 — Open Problems Discussion

A round-table discussion of genuinely open problems from all three weeks. Students vote on which problem they find most compelling.

Format: each student nominates one open problem. The coordinator (or the student) explains the problem in full. The group discusses:

Why is it hard?
What approaches might work?
What small cases can we check?
What would a solution even look like?

This is explicitly not a problem-solving session — the goal is to understand what open research looks like.

3:30–4:00 — Closing Ceremony

Each student writes on an index card: one thing I now understand that I didn’t three weeks ago and one question I still have.
Coordinator reads them out (anonymously if preferred).
Discuss next steps: recommended reading, research summer programs, competition mathematics, university mathematics.

Resources to share:

Resource	For	Link/Note
Diestel, Graph Theory	Week 1–2 deepening	Free at diestel-graph-theory.com
Alon & Spencer, The Probabilistic Method	Combinatorics (advanced)	Classic textbook
Art of Problem Solving	More puzzles	artofproblemsolving.com
Graph Theory (Coursera/MIT OCW)	Structured video learning	Free online
Combinatorics: Topics, Techniques, Algorithms (Cameron)	Accessible graduate-level	Cambridge University Press

Coordinator’s Reference Guide

Pacing Notes

The schedule is intentionally generous — each main session has room to run long or short without derailing the week. Rough flexibility:

If a group is strong: push harder on proofs, add more challenge problems, introduce directed graphs and digraph zero forcing.
If a group is struggling: spend more time on worked examples, reduce the number of problems in each set, cut the game analysis from Day 7 and use that time for more practice.
Never sacrifice the closing presentations — they are the most educationally valuable session of each week.

Common Sticking Points by Topic

Graphs: Students often confuse “path” (no repeated vertices) with “walk” (repetition allowed). Be explicit about this distinction early. Also: “isomorphic” is conceptually hard — use physical analogies (same network, different labelling = same city map rotated and renamed).

Zero Forcing: The forcing rule seems simple but students regularly misapply it — they try to force a vertex that has more than one white neighbor. Slow down, trace step by step, and have students narrate each forcing step aloud before writing it down.

Erdős–Szekeres: The proof is elegant but the pigeonhole step is subtle (assigning (eᵢ, dᵢ) pairs). Work through it at least twice: once quickly, then very slowly with students narrating each line. The “tight example” (construction achieving exactly n²) is the best way to consolidate understanding.

Centerpoint: The coordinatewise median counterexample is the crux of the whole session — do not rush past it. Make sure students can compute a halfplane through the coordinatewise median that contains very few points. Then the gap between “what we have” and “what we need” is clear.

Proof-Writing Guidance

Most A-level students have not written mathematical proofs. Introduce the following scaffold explicitly on Day 2 and reinforce throughout:

Proof structure:
"We want to show: [restate the claim]"
"Assume [set up notation/hypotheses]"
"Notice that [key observation]"
"Therefore [logical consequence]"
"This proves [restate what was proved] □"

Encourage students to circle “key steps” in their proofs — the one line that carries the argument. Everything else is set-up and clean-up.

Differentiating for Strong Students

Ask “can you generalize this to directed graphs?”
Push on tight examples: not just proving the theorem but constructing extremal cases.
Introduce the Zero Forcing Conjecture (maximum nullity conjecture): for any graph G, Z(G) + M(G) = n where M(G) is the maximum nullity of a matrix from the “family” associated with G. This is open.
Introduce the connection to minimum rank problems (symmetric matrices with prescribed zero pattern).

Differentiating for Students Who Find It Hard

Stay with n = 3,4,5 vertex examples on every topic.
For zero forcing: replace proof-writing with careful tracing of examples. Can they consistently apply the rule without error? That is the first milestone.
For Erdős–Szekeres: focus on the statement and the tight example, skip the proof on first pass.
For centerpoint: the counterexample to coordinatewise median is accessible even without the full theorem.

Week 4 — Communicating Research

Week 4 Goals

Understand the conventions and purpose of a scientific talk, poster, and short written summary.
Learn to identify the core narrative of their research experience and shape it for a non-expert audience.
Design clear, minimal slides or a poster with effective mathematical figures.
Practice delivery: pacing, signposting, handling questions, managing nerves.
Produce a complete, rehearsed presentation ready for an external audience.

The Week 4 Deliverable

By the end of Thursday (Day 16), each student will have a 12–15 minute presentation — either a slide talk or a poster — ready to deliver to an audience that includes people outside the lab (other students, parents, teachers, university staff). The exact format of the follow-up event should be confirmed with the professor before Week 4 begins and communicated to students on Day 13 morning.

Day 13 (Week 4, Monday) — What Is a Research Presentation?

Theme: Audiences, formats, and the art of mathematical storytelling.

0:00–0:30 — Warm-up: Watch and Critique

Show a short mathematical video — ideally a 5–7 minute excerpt from a well-made talk (e.g., a 3Blue1Brown video on a topic students already know, or a recorded conference lightning talk). Then ask:

What did the speaker do well?
At what point did you get lost? Why?
What was the single central idea of the talk?

Key observation: Even experts lose audiences by assuming too much background, moving too fast, or presenting too many results. A great talk does one thing well.

0:30–1:45 — Main Session: Formats and Structure

Distribute Handout: presentation_guide.md.

Part 1 — Formats (25 min):

Format	Length	Setting	Strengths
Seminar talk (slides)	45–60 min	University room	Deep, linear narrative
Conference talk	20–25 min	Conference hall	Broad audience, sell the result
Lightning talk	5 min	Any	One idea, maximum impact
Poster	Displayed + 3–5 min pitches	Exhibition space	Conversational, own pace
Blog post / write-up	Reading time 5–10 min	Online	Permanent, searchable

For this program, students will prepare either a 12–15 minute slide talk or a poster with a 5-minute pitch, depending on the follow-up event format. Both require the same underlying work: knowing your story.

Part 2 — Anatomy of a Mathematical Talk (45 min):

Every good mathematics talk, regardless of length, has the same four acts:

Act 1 — The Hook (10–15% of talk time)

Open with a concrete question, puzzle, or surprising fact — not a definition.
The audience must care about the problem before they are given the tools to solve it.
Bad opening: “Let G = (V,E) be a graph. We define the zero forcing number Z(G) as…”
Good opening: “How many accounts would you need to control on Twitter to eventually make your opinion dominate the entire network?”

Act 2 — The Setup (20–25%)

Introduce only the definitions and prior work the audience needs for this talk.
Every definition must be accompanied by an example. No naked definitions.
If you need to define “graph”, do it with a picture, not with symbols.

Act 3 — The Result (40–50%)

Your main theorem, discovery, or exploration.
State it clearly, then give intuition before the proof.
If there is a proof, every step should feel inevitable.
If there is no proof (an exploration or conjecture), present evidence: examples, patterns, failed attempts.

Act 4 — The So What (15–20%)

Why does this matter? What does it connect to?
What is still open? What would you do next?
End with a single clear sentence the audience will remember.

Exercise (10 min): Students individually sketch a four-act outline for their own topic, using just bullet points. No sentences yet.

1:45–2:00 — Break

2:00–3:00 — The Elevator Pitch

The hardest communication skill is compression. An elevator pitch is a 60–90 second explanation of your work that a curious non-expert can follow.

Structure of a good elevator pitch:

One sentence: the problem. (“We study how few controllers you need to steer a network.”)
One sentence: why it’s hard. (“The answer depends on the graph structure in a subtle way.”)
One sentence: the key idea. (“We discovered that the minimum number equals the size of a special coloring game.”)
One sentence: one cool consequence. (“This means you can tell if a Twitter network is controllable just by playing a coloring puzzle on it.”)

Workshop:

Each student writes their own elevator pitch for their chosen topic (they pick from the week 1–3 material). Then they deliver it to the group. Everyone gives one piece of feedback:

Was the problem clear?
Was there too much jargon?
Did you understand why it mattered?

Repeat with refinements. Target: a pitch each student could give to a family member at dinner.

3:00–4:00 — Choosing a Topic and Angle

Each student finalises which topic from Weeks 1–3 they will present, and which angle they will take.

Possible topics and angles:

Topic	Possible Angle
Zero forcing	The game / physical intuition first; theorem as punchline
Erdős–Szekeres	Puzzle-first (“can you find 17 numbers with no long monotone run?”); theorem as resolution
Centerpoint	Mars robots story; show coordinatewise median failure dramatically
Graph controllability	Social media influence; zero forcing as the mathematical model
PMI sequences	Propagation sequences on grids; connection to 1D case as the through-line

Discussion: Which angle is most accessible? Which is most technically satisfying? These may be different — the choice depends on the audience.

By end of Day 13: Every student has:

A chosen topic
A chosen angle
A four-act outline (bullet points)
An elevator pitch they can deliver from memory

Homework: Read Sections 1–3 of the presentation_guide.md handout. Expand the four-act outline into one full sentence per slide/section. Bring it to Day 14.

Day 14 (Week 4, Tuesday) — Structuring the Narrative

Theme: From outline to script. Choosing what to cut.

0:00–0:30 — Warm-up: The One-Slide Rule

Challenge: explain zero forcing (or your chosen topic) using exactly ONE slide — one image or diagram, no equations, a title, and at most 20 words of text.

Students sketch this on paper. Share. Discuss: what did you have to cut? What survived? The thing that survived is your core message.

0:30–1:45 — Main Session: The Narrative Arc in Practice

Part 1 — Peer Outline Review (30 min):

Students exchange their four-act outlines (from homework). Each student reads their partner’s outline and answers:

Is the hook a question or puzzle, or is it a definition? (If a definition: rewrite it as a question.)
Does Act 2 (Setup) contain anything that isn’t needed for Act 3 (Result)?
Is there a “so what” moment — a sentence that answers “why should I care?”
What is the single most confusing part?

Written feedback exchanged; 10 minutes to discuss.

Part 2 — What to Cut (30 min):

The most common mistake in student presentations: trying to include everything. A 12-minute talk has room for approximately:

1 hook example
3–4 definitions
1 main theorem
1 worked example
1 open problem
1 closing sentence

Everything else goes in the “appendix” — slides prepared but not shown unless asked about in Q&A.

Exercise: Each student lists every concept they want to include. Then rank them: (A) essential, (B) helpful, (C) nice to have. Cut all C’s from the main talk. Move B’s to backup slides.

Part 3 — Signposting (25 min):

Signposting is the art of telling the audience where they are and where they are going. Examples:

“I’ll start with the question that motivated this, then give you the key definition, and finish with the main result.”
“This is the tricky step — take a moment to check you’re comfortable with it before we move on.”
“We’ve just seen the definition. Now here’s why it turns out to be the right one.”
“I’ll skip the proof details here, but the intuition is…”

Good signposting means the audience is never lost. They may not follow every detail, but they always know where they are in the talk.

Students annotate their outlines: add a signpost sentence at the start of each act.

1:45–2:00 — Break

2:00–3:00 — Figures and Mathematical Diagrams

The most powerful tool in a mathematics presentation is a well-drawn figure. A good diagram:

Makes an abstract definition concrete
Replaces a paragraph of text
Gives the audience something to look at while the speaker talks

Workshop — Drawing for an Audience:

Each student picks the single most important concept from their talk and draws the figure they would show to explain it. Requirements:

Must work on a whiteboard or as a simple slide diagram
Must be labeled (no unlabeled graphs)
Must have a one-sentence caption
Must work in black and white (if the venue projector is grey-scale)

Students present their figure to the group and explain it in 2 minutes. Feedback: is the figure self-contained? Does the label help or clutter?

Common figures for this program:

A small graph with colored (blue/white) vertices for zero forcing
A sequence written out with the increasing subsequence highlighted
A 2D scatter plot showing the coordinatewise median counterexample
A grid with a zero forcing propagation traced step-by-step

3:00–4:00 — First Draft of Slides or Poster Layout

Students begin building their slides or poster layout (on paper first, or directly in their chosen software).

For slide talks — design principles (10 min briefing):

One idea per slide. If you have two ideas, use two slides.
Title of each slide = the takeaway sentence, not a topic label. (“Zero forcing spreads like a chain reaction” not “Zero Forcing Definition.”)
No slide should be read aloud verbatim — the slide shows the picture; you provide the commentary.
Font size ≥ 24pt for body text; ≥ 36pt for titles.
Aim for 1 slide per minute as a rough guide (12-15 slides for a 12-15 min talk).

For posters — layout principles (10 min briefing):

Three columns: Problem / Method / Results.
Every section needs a figure. A poster with no figures will not draw people in.
The title should be readable from 3 metres away.
Aim for 40% white space — crowded posters are unread posters.

Students sketch their full slide deck or poster layout by end of session (rough thumbnails are fine — no need for polished slides yet).

Homework: Complete a first draft of all slides or the poster. Bring it in digital or printed form to Day 15. Read Section 4 of presentation_guide.md (delivery).

Day 15 (Week 4, Wednesday) — Delivery and Rehearsal

Theme: From script to performance.

0:00–0:30 — Warm-up: Two-Minute Talks

Each student delivers a completely unscripted two-minute talk on the topic of their choice — but it must be a topic from Week 1, 2, or 3 that is not their presentation topic. No preparation allowed.

Purpose: speaking about mathematics without notes forces you to use plain language and identify what you actually understand vs. what you are memorising. The stumbles here are instructive, not embarrassing.

0:30–1:45 — Main Session: Delivery Skills

Part 1 — Pacing and Timing (20 min):

Record (or have a volunteer time) a student giving a 3-minute version of their talk. Play it back (or just discuss timing). Common issues:

Too fast: nerves compress the talk; audience can’t process definitions at speed.
Too slow on setup, too rushed at the result: the interesting part gets the least time.
No pauses: pauses are not dead air — they are thinking time for the audience.

Exercise: read one definition from their talk aloud, then pause for 5 full seconds. It feels much longer than it is. This is fine.

Part 2 — Eye Contact and the Room (15 min):

When speaking, the instinct is to look at your slides (or your notes). Instead:

Look at the audience for 80% of the time.
Look at the slide only briefly to orient yourself, then return to the audience.
When writing on a whiteboard, pause, write, then turn back and speak.

Practice: each student picks one paragraph from their talk and delivers it while maintaining eye contact with the group (no notes, no looking at the board).

Part 3 — Handling Questions (30 min):

The Q&A is often what audiences remember most. Rules:

Repeat the question before answering (others may not have heard it; it buys you thinking time).
“That’s a great question” is fine once. Not every time.
“I don’t know” is a perfectly good answer — but follow it with “my guess would be…” or “that’s something worth investigating.”
“That’s slightly outside what I covered, but briefly…” lets you engage without derailing.
If the question is aggressive or confusing: “Can you say a bit more about what you’re asking?” is always legitimate.

Mock Q&A drill: Coordinator (and students) fire rapid questions at each student about their topic. Students practice the above techniques. Questions range from genuine curiosity (“Why is the bound 1/3 and not 1/4?”) to deliberately awkward (“How is this useful in the real world?”).

Part 4 — Managing Nerves (15 min):

Honest discussion: everyone gets nervous presenting. Strategies:

Preparation reduces nerves more than anything else. Know your first sentence and last sentence by heart. The middle will follow.
Start with the thing you know best — the hook or the main example.
If you lose your place: pause, look at your slide title, say “So, the key point here is…” and you’re back.
The audience wants you to succeed. They are not looking for errors.

Share the “known but nervous” feeling: professional mathematicians get nervous before talks. This is normal.

1:45–2:00 — Break

2:00–4:00 — Full Rehearsal (First Run)

Each student gives their complete 12–15 minute presentation in full, with slides or poster, to the group. Coordinator times the talk.

After each talk, structured feedback (5 minutes):

The group uses this rubric:

Criterion	Question to answer
Hook	Was the opening question/puzzle clear and compelling?
Setup	Were definitions accompanied by examples?
Core result	Was the main point stated clearly and early enough?
Figures	Did the visuals help or clutter?
Signposting	Did you always know where you were in the talk?
Pacing	Too fast, too slow, or right?
One-thing-to-fix	Name exactly one specific improvement for tomorrow.

Each presenter hears: two things that worked, one specific thing to fix. Not more. The “one thing to fix” is the only homework.

Day 16 (Week 4, Thursday) — Final Presentations

Theme: Ready for the outside world.

0:00–0:45 — Morning Polish

Students have 45 minutes of quiet time to implement the one fix from Day 15 feedback and make any final adjustments. Coordinator is available for questions but does not give new feedback — changes at this stage should be targeted and small.

Checklist to run through before presenting:

First sentence memorised
Last sentence memorised
All figures labeled
Slides legible from the back of the room (or poster readable from 2 metres)
Backup slides ready for expected Q&A questions
Timing confirmed (practice the talk in your head once)

0:45–2:15 — Final Presentations

Each student delivers their polished 12–15 minute talk to a full audience. If the follow-up event allows it, invite one or two external guests (a professor from a neighboring group, a PhD student, or a teacher from the students’ school) to form a small audience. This gives the talks stakes and provides feedback from a fresh perspective.

Format:

Talk (12–15 min)
Q&A (5 min): the group and any guests ask questions. Coordinator ensures every presenter gets at least two substantive questions.
30-second applause, then next speaker.

Coordinator does not give feedback during this session — this is the real presentation, not a rehearsal.

2:15–2:30 — Break

2:30–3:30 — Reflection and Looking Forward

Individual reflection (10 min, written):

Each student answers on paper:

What was the hardest part of this program — content, thinking, or communicating?
What is one thing you understand now that you couldn’t have explained three weeks ago?
If you spent another week on this material, what would you investigate?
What would you tell a friend about what research mathematics is actually like?

These are kept by the students — they are not collected or assessed. But they make excellent material for personal statements, scholarship applications, or university interviews.

Group debrief (20 min):

Open discussion:

What is the relationship between understanding something and being able to explain it?
What was the most surprising thing you learned — about mathematics or about yourself?
What does “research” mean, now that you have done a version of it?

Resources for continuing (10 min):

Resource	Purpose
Diestel, Graph Theory (free online)	Rigorous graph theory text
Alon & Spencer, The Probabilistic Method	Advanced combinatorics
The Art of Problem Solving (artofproblemsolving.com)	Competitions and enrichment puzzles
Mathematical Writing by Knuth, Larrabee, and Roberts	Short guide to writing and presenting maths
YouTube: 3Blue1Brown	Visual mathematics — study the presentation style
YouTube: ICM 2022 talks	What professional mathematical talks look like
UK Mathematics Trust (ukmt.org.uk)	Senior Mathematical Challenge and Olympiad preparation
Undergraduate research programs (e.g., Nuffield, Sutton Trust)	Next step for research experience

3:30–4:00 — Closing

Certificate of completion (if prepared). Photos. Informal conversation.

Coordinator’s closing message: “You have spent four weeks working on problems that professional mathematicians find difficult. You asked questions, got stuck, found examples, wrote proofs, and explained hard ideas to other people. That is what mathematics research looks like. You are already doing it.”

Complete Materials Checklist

Print one copy per student on Day 1:

handouts/survey.md
handouts/networks_intro.md

Print on Day 5 (Week 2, Monday):

handouts/zero_forcing.md

Print on Day 9 (Week 3, Monday):

handouts/sequences_pmi.md

Print on Day 11 (Week 3, Wednesday):

handouts/centerpoint.md

Print on Day 13 (Week 4, Monday):

handouts/presentation_guide.md

Supplies (for the full four weeks):

Whiteboard markers in at least 3 colors (blue and white are thematic for zero forcing)
Graph paper, 5+ sheets per student per session
Sticky notes (for open questions board)
Index cards (for Day 12 closing exercise)
Colored dot stickers (for physically playing zero forcing on large drawn graphs)
Large paper or poster boards for the grid game on Day 7
A4 paper for sketching slide layouts and poster thumbnails (Week 4)
Access to a projector or TV for slide rehearsals on Days 15–16
Optional: printed A3 poster template for students choosing the poster format

Quick Reference: All Key Definitions

Concept	Definition
Graph G = (V,E)	V = vertices, E = edges (pairs of vertices)
Degree d(v)	Number of edges at v
Handshaking Lemma	Σ d(v) = 2\|E\|
Tree	Connected acyclic graph
Bipartite	2-colorable with no monochromatic edges
χ(G)	Chromatic number: min colors for proper coloring
Euler’s formula	V - E + F = 2 (connected planar graph)
Color change rule	Blue vertex with exactly 1 white neighbor forces it blue
Zero forcing set S	S such that propagation colors all vertices blue
Z(G)	Minimum size of a zero forcing set
Z(Pₙ)	1
Z(Cₙ)	2
Z(Kₙ)	n-1
Z(K₁,ₙ)	n-1
Z(Pₘ × Pₙ)	min(m,n) (conjectured in Week 2; prove in Week 3)
Propagation sequence	Ordering of V recording zero forcing order
LIS	Longest increasing subsequence
Erdős–Szekeres	Sequence of mn+1 → increasing subseq of length m+1 or decreasing of length n+1
PMI sequence	[Definition from professor’s research]
Tukey depth	Fraction of data in worst-case halfplane through a point
Centerpoint	Point of maximum Tukey depth; depth ≥ 1/(d+1) always
Helly’s theorem	d+1 pairwise-intersecting convex sets in ℝᵈ → common intersection