Advanced Summer Research Programme in Mathematics for Older Teens at Cambridge

Course Description: Advanced Summer Research Programme in Mathematics for Older Teens at Cambridge

Mathematics Advanced Research Programme is an advanced and research-focused intensive programme which welcomes individuals with strong academic background to extend your research interest. It offers an opportunity to strengthen and showcase your ability with the guidance from the best scholars. The programme comes with a wide array of advantages:

• Experience the pinnacle of intellectual discourse with the advanced lectures and talks, featuring esteemed Cambridge Professors*.​

• Be engaged in hands-on learning through workshops and supervision led by current Cambridge scholars*.​

• Discover the latest and most important research on specific subjects. Learn about the most up-to-date discoveries and developments in a chosen field by renowned professors.​

• Compose a customised research proposal by applying the knowledge from the tremendous resources of the University Library.

• Opportunity to be invited to submit a research paper and included in the Conference Proceeding (subject to performance and quality).​

• ​​Study and live like a Cambridge student with other like-minded individuals from around the world in the traditional and world-leading academic atmosphere in Cambridge and the UK.

Student Intake:

This programme represents a higher tier of admission standards compared to the other research programmes, as it is limited to a selected group of 20-25 students.

Certification & Rewards

This provider takes immense pride in celebrating the achievements of students who complete our programme. As a recognition of the dedication and hard work, here are the rewards and benefits that participants will receive:

• Recommendation Letter from Cambridge Professor for the top 50%

For those who excel and rank in the top 50% of the programme will receive a recommendation letter from the Cambridge Professor who delivers the academic lectures in the programme. This letter will not only acknowledge outstanding performance but also open doors to future academic and professional opportunities.

• Programme Recommendation Letter from Programme Director at Cambridge

A reference letter from our Programme Director at Cambridge will highlight students’ participation and accomplishments during their time in the programme, serving as a valuable addition to the academic portfolio.

• Certificate of Attendance

Students will receive an official Certificate of Attendance, co-issued by a Cambridge College and This Provider, to recognise their commitment to academic excellence.

• Programme Report Card

A report card to reflect a comprehensive evaluation of students' learning progress achieved during the programme with individual feedback.

• Reviewed Research Proposal

Students’ research proposals will undergo a thorough review, providing valuable feedback and insights from our experienced faculty. Completing a full research paper based on the research proposal is available through our optional continuation programme, if interested.

• UCAS Points

Completion of the Advanced Research Programme will award students 8 UCAS points, providing a meaningful boost to their university applications.

• University Application Guidance

We offer students professional advice and guidance from Cambridge professors and existing Cambridge students to help them prepare for university applications, especially Cambridge, Oxford and other top universities.

Academic Content

Leading Professor:

Prof. Maciej Dunajski

• Professor of Mathematical Physics, DAMTP, University of Cambridge.

• Fellow & Graduate Tutor, Clare College, Cambridge.

This course offers an accessible and inspiring introduction to  geometry, designed for high school students with a basic understanding of shapes and the Pythagorean theorem. No calculus is required.

The series explores how our understanding of space and form has evolved—from  Euclidean geometry to the  discoveries of curved and non-Euclidean spaces that describe our physical universe.

Through six interconnected lectures, students will:

• Discover what geometry is and why it lies at the heart of mathematics.

• Learn the foundations of Euclidean geometry and explore its logical beauty.

• Journey beyond flat space into non-Euclidean and curved geometries that underpin modern physics and cosmology.

• Learn about projective geometry and its  links to art, design, and perspective.

• Understand how geometry helps describe the shape of the physical world around us.

By the end of the course, students will appreciate geometry not only as a branch of mathematics but as a powerful way of understanding the world — from simple triangles to the structure of the cosmos.

Lectures:

Lecture 1 – What is Geometry?

It will introduce the language and scope of geometry: points, lines, planes, transformations, and invariants. Historical milestones from Euclid to the modern axiomatic approach will be outlined, together with how proofs, constructions, and coordinate systems underpin all later topics.

Lecture 2 – Euclidean Geometry

It will revisit classical plane and solid geometry, focusing on congruence, similarity, circles, triangles, and polygons. Vector and coordinate methods will be used alongside synthetic proofs to show how distances, angles, and rigid motions define Euclidean space.

Lecture 3 – Non-Euclidean Geometry

It will explore geometries that relax Euclid’s parallel postulate, including hyperbolic and spherical models. Key ideas such as geodesics, angle sums of triangles, and area–curvature relationships will be discussed, with visual models to build intuition.

Lecture 4 – Geometry of Curved Spaces

It will develop tools for understanding curvature in two and three dimensions: metrics, geodesics, and Gaussian curvature. Connections to general relativity will be highlighted to show how geometry describes gravity and the shape of spacetime.

Lecture 5 – Projective Geometry

It will introduce points at infinity, homogeneous coordinates, and projective transformations. Central topics include cross-ratio, duality, and perspective, with examples from art, computer vision, and camera models to illustrate why incidence, not distance, is fundamental here.

Lecture 6 – Geometry of the Physical World

It will examine how geometry informs real phenomena: symmetries and groups in physics, crystallographic patterns, coordinate frames and rotations, and geometric reasoning in navigation, robotics, and graphics. The lecture ties abstract concepts to measurement, modelling, and modern applications.

Small Group Supervisions:

Supervision is a unique feature of the Cambridge education system. It provides students with the opportunity to explore the subjects more deeply by discussing the work process and receiving regular feedback from their supervisors, including feedback on the preparation for the final projects at the end of the programme.

Research Skill Sessions:

Research Planning & Topic Selection

• Gain understanding about the research and related details

• Select a topic for the research presentation with the support of the supervisor

Presentation Structure & Literature Review

• Familiarise presentation structure

• Acquire knowledge on literature review

• Select and refine research methodology and data collection strategies

Data Analysis and Interpretation

• Know the difference between analysis and interpreting data

• Analyse and interpret collected data

Academic Skill Workshops:

Team Building

• Learn effective communication strategies to ensure clear, open, and respectful dialogue within the team.

• Develop skills for working collaboratively, including active listening, sharing responsibilities, and leveraging diverse strengths.

• Understand techniques for resolving conflicts constructively and maintaining a positive team dynamic.

Academic Writing

• Learn to structure your writing with clear sections and logical flow.

• Master techniques for building strong, evidence-based arguments.

• Understand how to write clearly and formally, and improve through revision and editing.

Presentation skills

• Learn how to structure your presentation with a clear introduction, main points, and conclusion to ensure a logical flow and coherent message.

• Master the use of voice modulation, body language, and eye contact to engage your audience and convey confidence.

• Understand how to effectively use visual aids, such as slides and handouts, to enhance your presentation and reinforce your key points.

Master Talks:

This provider invites a number of world class scholars and academics from various disciplines to give seminars on their specialities to promote academic interests. The speakers include world famous researchers, policymakers, industrial leaders, societies, associations, professional bodies and individuals.

Instructional Model

Mathematics Advanced Research Programme goes beyond simply teaching research techniques; as it fosters an unyielding enthusiasm for exploration. Our small, exclusive classes are where serious discussions thrive, and ideas are shared among the best and brightest. This tailored approach allows students to focus their efforts and delve deeper into a specific area of study, fostering a more immersive and fulfilling educational experience.

Professor & Supervisor Total Contact Time 45+ hours 

Research Lecture by Cambridge Professor 12 hours

Subject Revision by Cambridge Scholar 6 hours

Project and Proposal Supervision by Cambridge Scholar 8 hours

Research & Academic Workshop by Cambridge Tutor 8 hours

Proposal Writing Session by Cambridge Supervisor 8 hours

Master Talk by Profound Scholars 2 hours

Subject Presentation 3 hours

Cultural & Social Activities 12 hours

London / Oxford Excursion 2 days

Evaluation

The evaluation for our Advanced Research Programme ensures a comprehensive and fair assessment. Experienced lecturers and dedicated supervisors evaluate participants, allowing them to showcase their learning and enhance their presentation skills and research proposal:

• 15-minute group presentations for research evaluation

• Participants must submit a minimum 2000-word research proposal as a programme requirement

• Assessments by Cambridge Professor and dedicated supervisors

• Emphasis on critical findings, methodologies, and effective communication

• Valuable feedback and constructive criticism provided

• Encourages academic growth and excellence in the programme

Admission Requirements

This programme is suitable for

• The High School students who possess excellent academic ability and want to extend research interest with the guidance from the leading scholars at the University of Cambridge

• Age 16 to 18

Accommodation & Dining *�️

To deliver a comprehensive and enriching experience in Cambridge, participants are accommodated in different colleges within the University of Cambridge. Full board dining is provided throughout the programme, with meals served in the college dining halls (apart from weekend trips).

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Living and dining in these iconic settings not only offers a taste of traditional Cambridge life but also enhances the participants' research and academic learning. The immersive environment fosters a deep connection with the university's rich academic heritage, making their time at Cambridge both intellectually and culturally enriching.

Cambridge Colleges

Some of the collaborating Colleges are:

• Clare Hall, University of Cambridge

Clare Hall was founded in 1966 on the initiative of Clare College. It was envisaged as a college with three main classes of members: university teaching officers whose principal focus was research; visiting academics who would stay in Cambridge for between six months and a year; and graduate students. The President, Fellows, Visiting Fellows and graduate students together make up an integrated academic society that is not constrained by hierarchy and is consequently a culturally rich and intellectually diverse environment for advanced study.

• Sidney Sussex College, University of Cambridge

Sidney Sussex College was founded on St. Valentine's Day in 1596 by the legacy of Lady Frances Sidney, Countess of Sussex. The College is in the historic heart of Cambridge, and its beautiful buildings and gardens are home to a dynamic and diverse academic community. Our students, Fellows and staff come from all backgrounds and all over the world, drawn by the University of Cambridge's world-class reputation and Sidney Sussex's commitment to excellence in teaching and research.

• Murray Edwards College, University of Cambridge

Murray Edwards College was founded in 1954 as New Hall, later renamed in 2008 to honor Dame Rosemary Murray, its first President, and benefactor Ros Edwards. The college is known for its modernist architecture, featuring the iconic Dome, and a large collection of contemporary art. Academically, it offers a wide range of disciplines with a strong emphasis on supporting women in STEM fields. The college structure includes dedicated tutorial systems, research opportunities, and a library that caters to various academic needs. Murray Edwards fosters a collaborative environment, encouraging intellectual growth, leadership, and social responsibility, aiming to empower women to excel in their respective fields.

What Is Included:

• Accommodation for 13 nights in a College of the University of Cambridge*.

• Daily breakfast, lunch and dinners (except for weekend days), with a memorable distinctive formal dining experience in the Cambridge tradition on the day of the graduation ceremony.

• All academic activities, such as lectures, supervisions, revisions/seminars, master talks with world-class speakers, final group presentations, and access to teaching venues.

• A rich social and cultural schedule, including excursions to London and Oxford.

• Programme awards as outlined in the programme details.

Application Procedure:

1. Gather Necessary Documents

2. Fill and Submit Application Form

3. Receive Conditional Offer

4. Attend Interview (if applicable)

5. Pay Deposit

6. Receive Offer Letter and Invoice

7. Pay Programme Fee (deducted deposit)

8. Receive Programme Invitation Letter and Accommodation Letter (for visa purpose)

9. Receive Pre-arrival Pack 

10. Study Aboard

Airport Transfers

We can arrange a private airport pick-up/drop-off transfer between London Heathrow Airport and Cambridge at a price of £200 per way, per person.