Desirable qualities for applicants:
- Vocation for the study of science
- Capacity for scientific research
- Capacity for analysis and synthesis
- Skills and interest in theoretical work technological development, and problem solving
- Interest in teaching
Information for Bachelor of Mathematics applicants.
By Sergio Arratia Dávila.
What is Mathematics? History and perspectives
It’s difficult to define what mathematics is in a simple context. Many people don't consider it a science because it doesn’t follow all the steps of the scientific method; however, the scientific method would be inconceivable without mathematics. Bertrand Russell defined mathematics as "the subject in which we never know what are we talking about, nor whether what we are saying is true" This definition has no modesty and is in fact an expression full of boastful pride. Mathematicians assert that their work can be applied to the universe and our world because it was designed to be applied to all possible worlds and universes that may exist within the context of logic. Mathematics can be defined as the universal language.
In the beginning, the first two branches of mathematics were geometry and arithmetic which were used to solve practical problems like counting, meditation, and construction. This knowledge was quickly taken into a more scientific context, very likely to be used in gastronomy by Babylonians and Egyptians. Afterwards, the Greeks gave formality and logical abstraction to geometry and the mathematical language became universal. The advancement of human knowledge was parallel to the advancement of mathematics. The Arabs invented algebra. Later, the discovery of calculus accelerated scientific development since it made possible to have a convincing model of the laws that explain the universe in which we live. From that moment on, the development of mathematics has been greatly accelerated. Mathematics of amazing sophistication and abstraction has been created, for which there have barely been found any real applications. Due to its universal nature, we can say that mathematics is at the core of all sciences and knowledge.
Human civilization nowadays is be inconceivable without the laws of physics, technology, and the intellectual technics developed as a result of mathematical research. Mathematics can be linked to philosophy, economics, technology, space travel, industrial processes, social phenomena, computing, music, games of chance, as well as nuclear physics, relativity, quantum physics, etc. Currently, most scientific research is unacceptable if it’s not validated by mathematical models. Even in areas of knowledge such as biology and medicine, the use of mathematical models has become widespread; the progress in genetics is an example of the use of models based on mathematical developments.
Student's profile for the Bachelor of Mathematics
The mathematics program offers one of the largest and most interesting challenges that a person might face throughout their educational development. The study of mathematics does not only involve learning a series of abstract and obscure topics, as many believe. It also involves the development of a logical and rational way of thinking. It is a mental training "exercise" that fosters in the persons that practice it, a great development of inductive and deductive skills. A mathematician is a person who constantly practices a discipline of thought that allows him/her to solve a wide range of issues that covers essentially all branches of knowledge. Any professional that has the task of solving problems that require the use of complex quantitative conceptual models must turn to a mathematician to guide his work.
The world's most famous institutes of research and technological development inevitably have a team of mathematicians among their leading researchers.
The mathematician can grow professionally in different areas ranging from teaching, industry, economics, and business, to pure and applied research, whether developing abstract theories or creating socioeconomic, physical, technological, or industrial models.
Areas of development for the mathematician
Math students can choose between two important fields of specialization: Pure Mathematics and Applied Mathematics
The main objective of pure mathematics is to extend the frontiers of mathematical knowledge; there’s no interest in its practical application. Pure mathematicians consider their work an art and judge its value by the elegance of its logic and reason. They are able to develop both infinite-dimensional or zero-dimensional geometries, as well as algebraic structures as simple or as complicated as they want. They can create theories for continuous deformation of spaces and transform a piece of paper into a bottle which bottom coincides with its mouth According to mathematicians, mathematics itself is not a subject to be proved, as they are "independent of the human mind itself" (Kurt Gödel). It was not in vain that the great physicist Albert Einstein was influenced by developments in non-Euclidean geometries by great pure mathematicians such as Riemann, Minkowski and Poincare and others when he formulated his theory of general relativity.
Nowadays, pure mathematics includes the following main areas of development.
Set Theory. It studies the forms of schematization of all the structures of objects, mainly numerical ones, to create robust mathematical theories. It has enabled the development of a new kind of arithmetic for the treatment of the infinity, which is called either Cantorian or transfinite arithmetic.
Symbolic Mathematical Logic. It reduces mathematical reasoning to a notation with mathematical symbols of universal logical nature. One of the most important results of this discipline is Gödel’s theorem, which shows that any useful branch of mathematics cannot be built on a consistent set of axioms without facing unsolvable problems within the axiomatic framework itself.
Non-Euclidean geometry. When Riemann began the study of a new kind of geometry that departed from the conventional Euclidean flat universe we're used to, he started what is known in mathematics as a non-planar geometry. The development of this kind of geometry has reached enormous levels of abstraction and has found applicability in describing not only the universe we live in, but also of possible universes outside our own, which existence still remain a part of abstract speculation in the minds of mathematicians and astronomers. It’s worth mentioning that the mathematic model for Einstein's theory of general relativity would not have been possible without the development of these geometries.
Algebraic Topology. While topology is a special form of geometry that is in charge of the study of how the surfaces can be distorted, folded, or stretched on multidimensional spaces to acquire certain forms, abstract algebra deals with the determination of abstract properties of algebraic structures, in particular of transformation groups. Algebraic topology is the answer to an attempt to unify the two branches mentioned before.
Functional Analysis. There have been huge advances since Leibniz and Newton started the formal study of the functions of real numbers and the subsequent development of differential and integral calculus. Functional analysis comes to unite the results about the properties of defined functions not only for real numbers but for a huge range of numeric and algebraic structures. Results that have been obtained in other areas of mathematics, like the complex variable theory, real analysis, topology, and the measure theory, reach their culmination in the modern functional analysis.
The Applied Mathematics
Today, more than ever, mathematics has extended its dominance on a wide range of applications, practically on every field of the human development. It’s common to formulate real physic problems in mathematical terms. In industry, the use of applied mathematical tools to optimize the process of production and to control and improve the quality of products is increasing through the use of statistic models. Mathematical simulation has made possible the achievement of space travel and the development of the aeronautic and automotive industry. The prediction of the behavior of economics variables, with a degree of uncertainty that can be delimitated, is possible thanks to statistic models of series of time and the application of mathematic theory of stochastic processes. The decision theory based on the theory of games formulated by the mathematician von Neumann, has enabled safer decision making in business. Electoral results can be anticipated through studies of opinion based on probabilistic sampling. The optimum operation of the modern communication networks and internet can be made through the applications of optimization theories of networks and topology.
Applied mathematics encompasses, among others, the next fields:
Probability and Statistics. The study of the laws that govern chance, such as uncertainty and randomness. Pierre-Simon de Laplace, French mathematician of the XVIII century wrote :An intelligence which at given instant knew all the forces acting in nature and the position of every object in the universe - if endowed with a brain sufficiently vast to make all necessary calculations - could describe with a single formula the motions of the largest astronomical bodies and those of the smallest atoms. To such an intelligence, nothing would be uncertain; the future, like the past, would be an open book. Nowadays it is known that is not possible to reach the knowledge that Laplace dreamed of. It has been proved that the subatomic particles act on a random way, that apparently can't be predicted, and the universe is so large that scientists are convinced that they will never understand with complete certainty all the forces that act in it. While the behavior of the individual particles can't be predicted, when they're collectively constituting everything that surrounds us, it is possible to predict their behavior through a given accuracy and known error possibility, that can be validated through the use of probability. It’s not weird then, that most models used to describe phenomena and processes in disciplines as diverse as the socioeconomic, nature sciences, in technology and industry, are founded on the probability and statistic theories. Practically all mathematic branches are used for the development of the probability and statistic theories. In probability and statistics we may include: the game and decision theory, stochastic processes and time series, the lineal model theories, regression, experiment design, and more.
Operations research. The need to use limited resources on the best possible way dates back to the origins of mankind. As civilization developed and technology appeared, the need to have more sophisticated models to optimize processes increased. Once, conventional calculus and simple methods were enough to solve most problems, but when industrialization brought mass production and with it the increase of size and diversity of problems to solve, more sophisticated techniques had to be created. It was during World War II (hence the name of Investigation of Military Operations) that a revolution began, and is still undergoing, in the development of mathematic techniques of algorithmic-numeric character for the solution of this kind of problems. With the invention and evolution of the computer, it’s now possible to solve complex problems of optimization, logistics, or simulate multiples stages for a process on periods of time that only a few past decades might have been inconceivable. The most popular techniques used in operation investigation are classic optimization, linear and nonlinear programming, optimum control theory, simulation, heuristics techniques, and the network theories, among others. Likewise, the theory of algorithms, numerical analysis, information and computing are an essential part of the techniques on the investigation of operations.
Lately, a new specialty has emerged to make mathematic comprehension easier. Educational Mathematics, which offers a focus, from the point of view of knowledge theories, for the application of didactic techniques in the teaching of mathematics.
As can be noted, the future of mathematics is now wider and more promising than ever before.