## 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