Decoding Dimensions

Title: Decoding Dimensions: a Long-term program on Algebraic Geometry and Number Theory
Description: This long-term program hosted at IPM’s School of Mathematics delves into the intriguing realms of Algebraic Geometry and Number Theory. For a description of the program, click here.
The courses for the Spring semester 2024: Algebraic Geometry, Advanced Commutative Algebra, Category Theory and Homological Algebra (for more details, please visit here).
The courses for the Fall semester 2023: Galois Theory, Commutative Algebra, Algebraic Number Theory, Elliptic Curves and Modular Forms.
Poster: here

Introductory Seminars

Speaker: Ali Rajaei, Tarbiat Modarres University
Title: Decoding Class Field Theory...
Abstract: Class Field Theory is a profound and significant branch of algebraic number theory, primarily concerned with the study of abelian extensions of local and global fields by utilizing intrinsic objects from the ground field. Through the examination of complex interrelations and principles such as reciprocity laws, conductors, and Lfunctions, this theory illuminates the elegant structures governing global fields. In the forthcoming lecture, our focus will be on number fields, aiming to elucidate the core concepts and methodologies with a keen emphasis on motivational examples and historical context. It is hard to do justice to the field in one lecture, so expect a lot of bias and injustice!!
Date: Thursday, March 7, 2024
Time: 14:00-15:30 (Tehran local time)
Location: Lecture Hall 1 at IPM School of Mathematics
Poster: here

Speaker: Rahim Zaare-Nahandi, University of Tehran
Title: Decoding Birational Geometry...
Abstract: Birational Geometry is a unique aspect of algebraic geometry that distinguishes it from other geometries. It was extensively utilized by the Italian school in the birational classification of algebraic curves and surfaces during the early 20th century. This culminated in the work of Zariski and his collaborators, including Hironaka, on the resolution of singularities in the 1940s, a pursuit that is still ongoing. The birational classification of higher dimensional varieties, as demonstrated by scholars like Mori in the 1980s, and the subsequent development of the Minimal Model Program, have underscored the significance of birational geometry. The purpose of this presentation is to introduce some initial concepts in birational geometry. After recalling some fundamental concepts in algebraic geometry, we will provide several examples of birational equivalence and rationality, and briefly discuss results on the resolution of singularities. We will also explore the birational classification of varieties, including the birational classification of surfaces via the Castelnuovo Contraction Theorem. Finally, we will provide a brief overview of the ”Minimal Model Program.”
Date: Wednesday, January 3, 2024
Time: 15:30-17:00 (Tehran local time)
Location: Lecture Hall 1 at IPM School of Mathematics
Poster: here

Speaker: Amir Jafari, Sharif University of Technologoly
Title: Decoding Elliptic Curves...
Abstract: Elliptic curves lie at the intersection of analysis, geometry, algebra, and arithmetic. They have a rich and profound history in mathematics going back to antiquity. From the analytic point of view, elliptic curves are Riemann surfaces of genus one and are the natural domain of the definition of elliptic functions (a generalization of trigonometric functions with two independent periods). From the algebraic point of view, elliptic curves are abelian varieties of dimension one. From the arithmetic point of view, elliptic curves can be defined over global fields, local fields, or finite fields that give rise to Galois representations. The arithmetic of elliptic curves is linked to deep questions in diophantine equations that go back to the investigations of Fermat in the seventeenth century. The solution to Fermat's notorious last theorem by Wiles and Taylor was through elliptic curves and modular forms. This talk is an elementary introduction to this fascinating part of mathematics. We will define elliptic curves and discuss some of their analytical, geometrical, algebraical, and arithmetical properties.
Date: Thursday, November 23, 2023
Time: 13:30-15:00 (Tehran local time)
Location: Lecture Hall 1 at IPM School of Mathematics
Poster: here

Speaker: Jafar Shaffaf, Shahid Beheshti University
Title: Decoding Modular Forms...
Abstract: Modular forms are a fascinating and rich area of mathematics that have connections to many other fields, such as number theory, algebraic geometry, representation theory, and physics. They were first introduced by Carl Friedrich Gauss in the 19th century, who studied the properties of modular functions, which are special functions, defined on the upper-half plane, that are invariant under a group of transformations called modular transformations. Modular forms are generalizations of modular functions that also have a certain growth condition at infinity. One of the main motivations for studying modular forms is that they encode deep arithmetic information about various objects, such as elliptic curves, modular curves, Galois representations, and L-functions. In this seminar, we will give an overview of the history and development of modular forms, and explain some of their basic definitions and examples. We will also discuss some of the main applications and open problems in the field.
Date: Thursday, October 26, 2023
Time: 13:30-15:00 (Tehran local time)
Location: Lecture Hall 1 at IPM School of Mathematics
Poster: here

Winter School

The School of Mathematics of the Institute for Research in Fundamental Sciences (IPM) is organizing a one-week-long winter school on Algebraic Geometry and Number Theory. This is part of the Decoding Dimensions long-term program organized at IPM. The winter school will cover three main topics: Algebraic K-Theory, Brauer Groups, Profinite Groups.
For more details, please visit here.

When: February 1-7, 2024
Where: IPM-Isfahan, Isfahan, Iran
Poster: here