Decoding Dimensions is a long-term program in algebraic geometry and number theory at IPM’s School of Mathematics. Now in its fourth year, it brings together students at different stages of their studies for a connected path of learning, from foundational material accessible with a background in basic algebra to advanced topics and preparation for research.
Our courses are designed as parts of a broader mathematical education rather than as isolated subjects. We explore the ideas, tools, and connections needed to develop a working understanding of algebraic geometry and number theory, including topics not regularly offered in university courses. Through independent reading, problem-solving, and discussion, participants develop both the mathematical background and the habits of inquiry needed for research.
Building a lasting mathematical community is equally important. Over the past three years, students from different universities in the country have come together through the program. More experienced participants, including those now pursuing PhDs, help newer students, while participants who have continued their studies abroad remain involved and contribute talks. These exchanges connect people across universities and stages of study, making learning from one another a central part of the program.
Current Courses — Fall 2026
- Algebraic Geometry and Number Theory Essentials
- Algebraic Geometry II
- Algebraic Number Theory
- An Invitation to Arithmetic
- Homotopical Algebra I
- Stable Homotopy Theory
Decoding Dimensions — Fall 2026 course poster (PDF)
How the courses work
The choice and pace of topics are guided by the program’s longer-term mathematical aims. Courses are connected, and a subject need not be completed within a single semester. We allow time to revisit prerequisites, explore connections, and work through difficult ideas, with the emphasis on developing understanding rather than simply covering a syllabus.
Participants read assigned material and work on exercises before class, bringing questions, remarks, and examples to the meetings. They are encouraged to consult other sources, compare approaches, and take increasing responsibility for their own learning. The meetings provide an opportunity to discuss what they have understood, examine difficulties, and make connections with other parts of mathematics.
This way of working develops skills that matter beyond any particular course: reading unfamiliar mathematics, formulating useful questions, finding and testing examples, and explaining ideas clearly. Regular preparation matters, but participants are not expected to have resolved every difficulty before a meeting. Working through those difficulties together is part of the learning.
Taking part
Interested in joining the program? A background in basic algebra is enough to begin with our foundational courses, while more advanced courses require additional preparation. Please consult the individual course pages or the instructors for details about the material and expected background.
For questions about participation or advice on a suitable starting point, you are welcome to write to agnt@ipm.ir.
News and weekly assignments
Weekly reading assignments, exercises, schedule changes, and program announcements are posted on the News page. Please check it regularly alongside the individual course pages.
Winter schools
In each of the past three years, the program has held a winter school in algebraic geometry and number theory, complementing the regular courses with focused study of selected topics.
Past schools have covered algebraic K-theory, Brauer groups, Galois representations, simplicial methods in algebraic geometry and number theory, curves and surfaces, symmetric domains, Tannakian formalism, and Artin–Schreier–Witt and Kummer theory. These schools are also an opportunity for participants at different stages of their studies to meet, exchange ideas, and learn together.
We intend to hold a fourth winter school during the 2026–2027 academic year. Further details will be announced on the News page.
Previous courses
Courses offered in previous years include the following:
- Algebraic Geometry and Number Theory Essentials (Mohammad Hadi Hedayatzadeh)
- Algebraic Geometry I (Mohammad Hadi Hedayatzadeh)
- Algebraic Geometry II (Mohammad Hadi Hedayatzadeh)
- Algebraic Geometry III (Mohammad Hadi Hedayatzadeh)
- Algebraic Number Theory (Shayan Gholami)
- Category Theory and Homological Algebra (Ali Partofard)
- Class Field Theory, Homological Approach (Ehsan Shahoseini)
- Class Field Theory, via Lubin-Tate Groups (Ehsan Shahoseini)
- Commutative Algebra, a geometric approach (Ali Partofard)
- Commutative Algebra I (Mohammad Hadi Hedayatzadeh)
- Commutative Algebra II (Shayan Gholami)
- Elliptic Curves and Modular Forms (Ali Partofard)
- Galois Theory (Ali Akbar Hosseini)
- Group Schemes (Ali Partofard)
- Homological Algebra (Mohammad Amin Turang)
- Six-Functor Formalism (Ali Partofard)
Program background
For background on the program’s aims and original structure, see the original program description (2023) and the original program poster (2023). Current course information is provided above and on the individual course pages.
