Schedule

Pieces From Ulam's Book "Adventures of a mathematician"  (Abdelghani Zeghib):

TBA

Incompressible Ideal Fluids: Introduction to the vortex patch problem and time periodic solutions(Haroune Houamed) : 

  The Euler equations stand for the mathematical model that characterizes the evolution of ideal fluids, in general. In my talk, I will focus on the homogeneous, two dimensional, incompressible version of these equations.  

First, I will introduce the set of equations and give a short overview of classical and recent results. Then, in the heart of the session, I will concentrate on a specific type of solution, the so-called vortex patch solution. I will try to explain the general idea behind the construction of these solutions as well as the interesting subsequent questions, related to specific geometries involved in the problem. If the time allows, I will further consider particular cases of the vortex patches by shedding light on the so-called time-periodic solutions.  

The talk is not expected to be too technical and all the requirements will be introduced throughout the presentation.

Cardinality of sets: some infinities are larger than others  (Sara Bentrifa): 

  We are interested in comparing sets based on their cardinality, where the main point is to see how there are numerous different kinds of infinity, and some infinities are bigger than others. We present Cantor's diagonal argument to prove that R is uncountable, and Cantor's theorem to prove that the power set of a set A has strictly greater cardinality than A. Finally we discuss the Continuum hypothesis asserting that no set has a cardinality between that of N and its power set.

Exploration of Constructible and Algebraic numbers through the famous Greek problems (Malek Hanounah) : 

  The three classical Greek problems - squaring the circle, doubling the cube, and trisecting an angle - have fascinated mathematicians for more than 2000 years. While these problems were originally approached using only a ruler and compass, their solutions eventually involved the use of algebraic techniques. In this talk, we will explore the relationship between geometry and algebra through the lens of these problems, focusing on the concept of constructible numbers, which can be constructed using only a ruler and compass, and algebraic numbers, which are solutions to polynomial equations with rational coefficients. We will see how the solutions to the Greek problems can be expressed in terms of these types of numbers.

Abel-Ruffini Theorem (Mohamed Aliouane):

  The Abel-Ruffini theorem theorem says that in general roots of polynomials of degree bigger or equal than 5 cannot be expressed by +,-,/ or (n-th ) square root.

In Galois theory language it means that such a polynomial is not solvable by radical. In the talk I will start by historical introduction and solution of polynomial equations of degree 2,3 and 4 then I will give an idea about Galois theory and how it solves the problem and if time allows i will give an idea about how to construct a polynomial which is non solvable by radical of any given degree

The Cauchy Rigidity Theorem (Abderrahim Mesbah):

   Cauchy's rigidity theorem is a fundamental theorem of rigidity theory. It states that if two convex polyhedra have isometric faces, glued in the same combinatorics, then the polyhedra are of the same modulo translation and rotation. Another objective of the talk is to learn more about the boundary geometry of convex polyhedra.

Knot Theory (Younes Benyahia):

   Knot theory is a very lively mathematical field, it was born in the second half of the 19th century and continues to have countless open problems. In this talk, we will mention some history, and we will discuss the fundamental question of classifying all knots, we will also talk about various applications in math and in other sciences.

Dynamical Systems Via Examples (Hamza Ounesli):

   The purpose of the talk is to give a general introduction to

dynamical systems and show how the topic is connected with different

fields of mathematics (Geometry, topology, probability theory..)... all

this via studying some simple examples of dynamical systems in dimension

one and higher.