Nurturing Future Mathematicians

There are two strands, one for students aged 10/11, and one for age 12/13. Students worldwide are eligible to apply. With fewer than 10 students per class, the program ensures personalized attention, fostering a learning environment conducive to young minds. 

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Some foundational courses best taught by mathematicians are listed below. In any instruction
season at Mathnuts, we offer two to four of these, depending on the maturity of the students
admitted.

I. Set Theory and Beyond.
Students will go beyond basic ideas of countable and uncountable sets to study topics such as
the Cantor-Schroeder-Bernstein Theorem, consequences of the axiom of choice, and others.
These topics prepare students to study structures that appear throughout mathematics such as
graphs and posets, groups and rings, topological and metric spaces, etc.

II. Geometric Transformations
For over two thousand years Euclidean geometry remained the synthetic geometry of Euclid’s
Elements; lengths and angles were not measured but compared. Then, Rene Descartes (1596-
1650) introduced the method of doing geometry using algebra and numbers – analytic
geometry. Later, Felix Klein (1849-1925) introduced a more general method (transformations),
still involving algebra, that was applicable to other geometries as well. The fundamental
geometrical notion of congruence has to do with moving (transforming) a figure to coincide with
another. The course studies geometric transformations that preserve distance in the Euclidean
plane. These transformations will be seen to describe ALL the symmetry patterns in the plane
(and 3-D which we don't cover).
The course begins with an introduction to Analytic geometry which is also useful for Calculus. 

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III Affine and Projective geometries
The goal of this course is to develop a clear grasp of the various geometries and their
relationships to one another. For this reason, we study projective geometry for it entails a unified
view and closely relates the different geometries, including Euclidean geometry and non-
Euclidean geometries, as special cases. And the geometry most closely related to projective
geometry is affine geometry; studying affine geometry enables greater understanding of
projective geometry; so we introduce the former first.
A way to see the various geometries, without needing prerequisites in advanced algebra, is the
axiomatic approach which we use in the course.

IV Non-Euclidean geometry
Although Non-Euclidean geometry means geometries other than Euclidean
geometry, the reference in mathematics is to two particular geometries – Elliptic geometry and
Hyperbolic geometry. The impression of school students and the public in general is that
geometry consists of Euclidean geometry, whereas Euclidean geometry is only one of many
geometries. However, there are two geometries which are related to Euclidean geometry and
whose study gives a greater understanding of what Euclidean geometry is really about.

There are three geometries called constant curvature geometries where a ‘plane’ of the
geometry has a curvature that does not change. In Euclidean geometry, the plane is flat in the
sense that its curvature is zero. In contrast, a hyperbolic plane has negative curvature like that
of a horse saddle and an elliptic plane has positive curvature like a sphere. While the emphasis
in this course is these two geometries, we begin with those concepts of Euclidean geometry in
which it differs -- from the other two constant-curvature geometries -- and the study leads us to
Elliptic and Hyperbolic geometry.
This course is ideal for building in the student an appreciation of the idea of proof in
mathematics and to improve the student’s ability to construct proofs.

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