IMPSC

International Math & Physics Summer Camp (IMPSC)

Greetings by the Head Director

Thank you for your interest in the International Math & Physics Summer Camp (IMPSC). I am Dr. Shubhrangshu Dasgupta, a Professor of Physics at the Indian Institutes of Technology (IIT) and the Head Director of both IMC and IMPSC. It is our pleasure to offer this opportunity for high school students in collaboration with the esteemed professors at IIT.

We are organizing a summer camp for students who show exceptional promise in mathematics and physics. Over three weeks of intensive physics and mathematics classes, we aim to provide substantial support to those planning to pursue degrees in science and engineering.

Thank you.

International Math & Physics Summer Camp (IMPSC) 2025

IMPSC was established by Professor S. Dasgupta, a Physics professor at the Indian Institute of Technology (IIT) Ropar campus. It is an online summer camp designed to provide high school students with intensive education in college-level Physics and Math, which are typically not accessible in school, over a three-week period.

Classes are conducted by professors and PhDs from the Indian Institute of Technology (IIT), and daily assignments are checked, with students required to make presentations.

How to Apply

1. Send an email to impsc@imc-impea.orgwith the following information to receive detailed application instructions and a problem set:
• Student Name:
• School:
• Nationality:
• Birth date:
• Grade:
• Email of a Recommending Teacher or Mentor (preferably a school math teacher):

Application Period:January 6, 2025 – April 25, 2025

※We received more applications than we expected. We understand everyone spent good amount of time and effort on the application, we will review each application carefully. In order to do so, we decided to extend the deadline to May 12th 2025.

Eligibility : Students in grades 9–12 worldwide who are capable of communicating in English and were born before August 2010.

2. If you send us your information by email, we will send you the problem set and the essay question. The problem set consists of SET 1 and SET 2. SET 1 contains 9 questions, and we recommend solving at least 40% of them. SET 2 includes 4 questions, and you only need to choose and answer one. You may choose to submit either SET 1 or SET 2 — only one set needs to be submitted.

3. Problem-solving answers, the application form, and the essay should be compressed into a zip file with the file name “Application Number_Student Name” and sent to impsc@imc-impea.org. Please ensure the email subject is also “Application Number_Student Name.”

4. We will send an email directly to your teacher or mentor to request a recommendation letter about you.

5.Admissions results will be announced on a rolling basis and will be individually notified via email.

IMPSC Schedule

• Session 1:June 16, 2025 – July 5, 2025 (Topics: Topology, Physics I)

• Session 2:July 7, 2025 – July 26, 2025 (Topics: Linear Algebra, Ergodic Theory, Physics II)

※It may be extended by a week depending on the situation.

Information on Camp Participation Fee and Payment Method

1. The participation fee for the camp is $2,450.

2. Instructions on how to make the payment will be sent to your personal email.

To participate in IMPSC, the following documents are required

1. A recommendation letter from a teacher or mentor (preferably from a math or physics teacher at your school).

2. Problem-solving (Choose either Set 1 or Set 2, complete the answers, and send the file in doc, tex, or pdf format).

3. Application form and Essay.

4. Certificates and any math or science-related papers.
The application deadline is April 25, 2025, and we will review applications on a rolling basis, notifying each applicant of their acceptance status individually. The registration period for accepted students will begin on May

5. ※We received more applications than we expected. We understand everyone spent good amount of time and effort on the application, we will review each application carefully. In order to do so, we decided to extend the deadline to May 12th 2025.
This is the second year of the IMPSC summer camp, and it is a highly competitive program. Moreover, it is a program that attracts exceptional students from around the world in the fields of math and physics.

※ Important Notice:
Please be aware that using ChatGPT for solving problems or writing essays will be detected. Additionally, you must solve the problems independently and should not seek help from others. If any such violations are found, your entire application will be disqualified.

For further details, please contact impsc@imc-impea.org.

Address: Department of Physics Super Academic Block IIT Ropar, Rupnagar, Punjab, India, PIN 140301 (India)

Application

Program A Curriculum

Session 1

Period: June 16, 2025 – July 5, 2025

Math : Topology

Topology and Data Analysis

Many would have heard about the famous math meme “DonutCoffee cup”. If you have a donut made out of clay, then you can reform it into the shape of a coffee cup, by ‘smoothly’ stretching it, pressing it, and bending it, without ‘ripping’ the body of clay into two. Then what else shapes can we make by playing with a donut-shaped clay? What exactly is the rule of this ‘smooth’ manipulation of clay? In this course, we learn mathematical language that can describe and analyze such mysteries of geometric objects. The mathematical word for ‘smooth’ is ‘continuous’. First we define continuity over a metric space, a space where we can tell the distance whenever two points on the space are given. This is called definition of continuity, the first building block of all modern analysis. However, this is not satisfying enough for the world of geometry. Consider a stormy sea. Given two points on its surface, can you think of a way to calculate the distance? Probably not. But still, we need to understand the nature of stormy sea; how the waves act. In this course, we define Topological spaces, spaces where distance is not given but still something can be said about its geometry. We shall extend the definition of continuity to Topological spaces, define important invariants (homotopy and homology) that describe and classify Topological spaces, and see how these invariants can actually help us understand the real world, by their applications to data analysis.

[Textbooks]

Topology, Munkres
Algebraic Topology, Allen Hatcher
Topological Data Analysis with Applications, Carlsson and Vejdemo-Johansson

[References/Further Reading Materials]

To be updated.

[Approximate Course Schedule]

Part I. Introduction to Topology: basic concepts and important invariants.
Day 1	Continuous functions, argument
Day 2	Open and closed sets (in metric spaces)
Day 3	Topological spaces (basis for a topology, examples of topological spaces, product topology and subspace topology)
Day 4	Quotient spaces and quotient topology
Day 5	Classification of Surfaces; labelling scheme, cutting and pasting
Day 6	Basics of Group theory, Topological troup
Day 7	Homotopy of paths, Fundamental group
Day 8	Covering spaces and Fundamental group of the circle
Day 9	Topological proof of Funamental proof of algebra,overview on further topics on Homotopy.
Part II. Applications of Topology: Homology and Topological Data Analysis.
Day 10	Operations on spaces and Homotopy Equivalence
Day 11	Complexes, Simplicial and Singular homology
Day 12	Introuction to Homological Algebra
Day 13	Computation of Homology, Mayer-vietoris Sequences
Day 14	Introduction to clustering. Different methods of clustering; connectivity-base, centroid-based, distribution-based, and density-based
Day 15	Persistence homology and Persistence landscape
Day 16	Statistics with Persistence landscapes; stability, law of large numbers, and central limit theorem
Day 17	Application of Topological Data Analysis I. Prediction of Financial Crisis via TDA, prediction of Chaotic behavior
Day 18	Application of Topological Data Analysis II. Application to Brain network, TDA on Coherence and TDA on Correlation

Physics I

Classical Mechanics

	Topics
Day 1	Projectile motion
Day 2	Potential energy and stability
Day 3	Simple harmonic oscillators
Day 4	Damped harmonic oscillators
Day 5	Forced damped harmonic oscillators
Day 6	Resonance
Day 7	Central force
Day 8	Equation of orbits
Day 9	Three dimensional rotation
Day 10	Inertia tensor
Day 11	Inertia tensor in a different coordinate system
Day 12	Equation of motions, symmetric top, and stability
Day 13	Calculus of variations
Day 14	Euler's equation and Lagrangian dynamics
Day 15	Examples of Lagrangian dynamics
Day 16	Hamiltonian equations
Day 17	Coupled oscillators
Day 18	General coupled oscillators

Program B Curriculum

Session 2

Period: July 7, 2025 – July 26, 2025

Math : Linear Algebra & Ergodic Theory

Number Theory, Geometry, and Matrices

In this course, we shall learn how certain Number Theoretic problems can be transformed into beautiful Geometric problems, and how these beautiful Geometric problems can be solved by analyzing certain simple-looking matices. Especially, our main goal is to understand (the hyperbolic plane) as a quotient of (the group of matrices with determinant ) and see how rich dynamical structures of can be understood as Linear Algebra problems.For the first half of the course, we shall learn the basics of Linear Algebra and a few useful matrix decompositions. For the later half of the course, we shall head towards the mystery of and . First we learn the basics of Ergodic Theory (ergodicity, recurrence, and mixing). Then we shall move on to continued fractions. We shall first prove the ergodicity in a purely algebraic way. Then we shall learn that the continued fraction map can be understood in the language of geodesic flows on , and discover a beautiful geometric proof, using Linear Algebra. If time allows, we shall have one day at the end to survey modern results in this direction, such as Littlewood’s conjecture and Oppenheim conjecture.