University of Calgary

# Pure Math

Submitted by admin on Fri, 05/06/2011 - 5:08pm.

The creation, development, and exploration of the form and knowledge of Mathematics itself, with elegance, aesthetic balance, and connections with other areas of Mathematics as the sought after, delineating features.

Pure Mathematics (PMAT)  is concerned with the development and exploration of structures and techniques that have proven invaluable in illuminating questions that arise in a wide range of problems in mathematics and sciences. The focus is on general methods, powerful ideas that are widely applicable, and theories that capture an elegance of design revealing a deeper richness and unity within mathematics.

At the undergraduate level, the student begins with the basics of algebra, calculus and discrete math, then progresses towards a selection of courses in the theory of numbers, cryptography, topology and geometry, differential equations, analysis, set theory and logic.

Number theory includes both deep results about our familiar numbers 1,2,3,4,... and their atomic constituents the primes 2,3,5,7,11,..., as well as results about richer collections of numbers obtained by extending the natural numbers through algebraic means. Number theory is intimately connected with cryptography, the science of passing secrets through encoded messages. The Department offers a concentration in cryptography as part of its undergraduate program in pure math.

Topology and geometry involve the basic shapes of mathematics: from rigid objects such as lines, planes, circles and spheres, to bendable creations like loops, knots, and surfaces in space. Patterns, relations, transformations, and invariants for these objects are all of mathematical interest. Tools from algebra and analysis are both developed through topological and geometrical methods, as well as applied to study these creations.

Differential equations model the physical world, from the propagation of seismic waves through the earth under our feet, to the transmission of heat through the expanding universe. Mathematicians are concerned with the general properties of differential equations, the nature of their solutions, their use in making accurate models of the real world, and efficient methods to compute answers from these equations.

Analysis encompasses a wide range of techniques to study the properties of functions on a space. It is calculus taken to a higher level. Real and complex analysis examine functions on the real line and on the complex plane, while differential geometry examines functions on smooth spaces called manifolds. Analysis gives us a way to organize the infinite dimensions which arise in so many aspects of mathematical theories.

Set theory and logic form the foundations of modern mathematics, assuring the truth of deep results based only on clear, precise axioms and the methods of deductive reasoning for their creation. Set theory concerns itself with collections of elements, and rules for decomposing and recombining sets in well-defined ways. It goes hand-in-hand with logic, the basis of our lemma-theorem-proof approach to establishing mathematical truths.

Career options
A degree in Pure Mathematics is excellent preparation for graduate work in mathematics, and in technical disciplines that make heavy use of mathematics, such as engineering, physics, and computer science.