1. Program Introduction

Designed specifically for students, this program treats research as a learning journey rather than a performance. Over 18 months, students build solid foundations, practice explaining proofs at the board, run small computational experiments to develop conjectures, and learn how to organize their ideas into clear written notes.

The outcome we value most is a student who can think and communicate mathematically with confidence.

What Students Will Learn

• Write and present solutions with clear logical structure (not just final answers).
  
  

• Explain proofs aloud and respond to questions in a seminar setting.
  
  

• Use computation responsibly to discover patterns and form conjectures.
  
  

• Read mathematical text actively: extract definitions, identify key lemmas, and reproduce arguments.
  
  

• Produce a well-structured research note that documents the learning path, results, and limitations.

2. Target Students, Class Size, and Selection

2.1 Who Should Apply

The program is intended for:

• Grade 9 students (Semester 2) and Grade 10 students who enjoy mathematics and are curious about research.

• Students who want to explore research-style mathematical thinking and strengthen their academic profile through authentic work.

• Students who are willing to commit to consistent weekly effort over a long time horizon (18 months).

2.2 Class Size and Group Structure

• Each intake admits a maximum of 10 students for an 18-month program cycle.

• Students are organized into small groups of 2–3.

• Each group is supervised directly by a professor and a teaching assistant (TA), with regular guidance and close academic oversight.

2.3 Selection Process

Selection is conducted in two stages to identify students with both ability and long-term commitment.

Stage 1: Application File

Applicants submit:

• Academic transcript(s).

• Two recommendation letters from teachers.

• A personal statement describing: interest in mathematics, learning habits and persistence, future goals and what the student hopes to gain from the program.

Stage 2: Interview (1–1, 20 minutes, Zoom)

• A short individual interview with the program director or advisor.

• The interview assesses communication, seriousness of commitment, and the student’s ability to explain mathematical thinking clearly.

3. Program Structure

3.1 Weekly Schedule

The program runs on a weekly schedule with two 90-minute sessions, and all students are expected to attend both.

• Core Session (90 minutes).
  Focuses on building the foundational mathematics required for the research topics. Content is chosen to be directly useful for students’ projects, so students engage with research problems with genuine understanding rather than shortcuts.

• Reading Seminar (90 minutes).
  A student-led seminar in which groups present assigned reading and proofs, followed by discussion. Students present in 2–3-person groups, with presentation responsibility rotating on a fixed schedule. Mentors probe understanding, ask “why” at key steps, and assign focused follow-up work.

3.2 Topics Offered in the First Year

• Topic 1: Two-lines distinct distances

• Topic 2: Pinned distances for well-distributed or grid-like sets

• Topic 3: Unit distances in restricted models

• Topic 4: Rectangle-free matrices → point–line incidence bounds

• Topic 5: Rich lines: structure versus spread

• Topic 6: Distinct dot products in Cartesian product sets

• Topic 7: Pinned angle sets via slopes

• Topic 8: Small-n extremal classification

3.3 Supporting Courses Aligned with the Topics

Month 1 — Mathematical Reasoning and Proof I

• Logic and quantifiers; turning informal claims into precise statements

• Definitions; if and only if, necessary and sufficient conditions

• Proof methods: direct proof, contrapositive, contradiction

• Counterexamples and debugging

• Reading proofs: tracking hypotheses and conclusions

• LaTeX basics

Month 2 — Proof II and Induction

• Induction: common patterns and traps

• Invariants and monotonicity

• Constructing examples and proving impossibility

• Organizing longer arguments using lemmas and claims

• LaTeX intermediate

Month 3 — Counting I: Fundamental Principles

• Pigeonhole principle

• Binomial coefficients and identities

• Double counting and bijections

• Inclusion–exclusion

Month 4 — Counting II: Recurrences and Generating Functions

• Deriving recurrences from combinatorial decompositions

• Generating functions: definitions and standard examples

• Extremal constructions

• Writing practice: definitions and examples before proofs

Month 5 — Graph Theory I: Core Language and Proof Skills

• Graph definitions, degrees, paths, cycles, trees, connectivity

• Handshaking lemma and average-degree arguments

• Greedy methods and minimal counterexample idea

• Algorithmic intuition: BFS and DFS (conceptual only)

Month 6 — Graph Theory II: Extremal and Structural Ideas

• Bipartite graphs and matchings

• Coloring basics and planar intuition

• Mantel’s theorem and the Turán idea

• Ramsey theory: R(3,3) = 6

Month 7 — Inequalities as Counting Tools

• Cauchy–Schwarz in discrete counting

• Hölder inequality as a multi-tuple viewpoint

• Energy as repetition counting

• Error control and overcounting

Month 8 — Incidence Thinking (Before Szemerédi–Trotter)

• Point–line incidences and incidence graphs

• Rectangle-free 0–1 matrices

• First incidence bounds via Cauchy–Schwarz

• Rich lines: definitions and structural implications

Month 9 — Coordinate Geometry for Combinatorics

• Slopes, collinearity, parametrizing lines

• Circles and intersections without square roots

• Pinned versus unpinned distances, dot products, angles

• Degeneracies and symmetry traps

Month 10 — Dot Products and Direction Sets

• Geometric meaning of dot products in the plane

• Direction sets and angle counting

• Two-case arguments: structure versus spread

• Restricted toy models

Month 11 — Communication and Literature Skills

• How to read a paper efficiently

• Definitions-first writing style

• LaTeX advanced

• Short talks with serious Q&A

• Peer review

Month 12 — Research Readiness and Reproducible Workflow

• Turning curiosity into a solvable question

• Responsible experimentation

• Weekly notebooks and progress diaries

• Planning proof attempts and recording dead ends

• Writing a clear project proposal

3.4 Intensive Research Phase (Months 13–18)

Students transition from coursework-style training to sustained work on one carefully scoped project in a restricted model, aiming for a complete research note and a final presentation.

3.5 Quarterly Progress Review

A structured progress meeting every three months to review each group’s work and set objectives for the next quarter. This helps maintain momentum and ensure genuine understanding.

4. Deliverables

4.1 Primary Deliverable: Research Note

Each student or group produces a written note explaining:

1. The topic and guiding questions

2. The learning path and key definitions and lemmas

3. Computational experiments (if applicable)

4. Conjectures or theorem statements

5. Proofs, partial progress, or clear explanations of obstacles

The note is written as a learning and verification document, not as a journal paper.

4.2 Final Presentation

A final internal seminar where students present their work and answer questions.

4.3 Publication (Optional)

Submission may be considered only if results are correct and genuinely non-trivial, and if the student’s role constitutes a primary contribution. Submission is not guaranteed; education and integrity come first.

Primary Contribution (Definition).
A primary contribution may be a strong idea, careful experimentation, or high-quality execution under guidance. The key requirement is ownership: the student can clearly explain the main question, their work, and the meaning and limits of the result.