Correlation between advancement in Mathematics with advancement in Computer Science
Mathematics and Computer Science are two very closely knitted subjects, more so now in this modern time when computing is undergoing an explosive exponential growth. New developments in Pure Mathematics, particularly in the fields of Algebra, Analysis, Functional Analysis, Geometry and Topology, have triggered an explosive growth in Applied Mathematics and Computer Science causing more and more applications of computing to arise in traditional fields such as Biomedical Science, Medicine, Social Science and even the Humanities and Arts. For example, new developments in the fields of Differential Geometry and Topology has triggered advancement in the field of Computer Graphics and 3D Image Rendering which has now found important applications in the Filming industry as now we are seeing more 3D animation films being released. In Medicine, images are now being produced from CT, X-Ray, MRI and EKG scans with very high resolution due to improvement in image processing algorithms which are based on Wavelets and Harmonic Analysis in Mathematics resulting in more accurate and timely diagnosis and treatment of diseases such as Lymphoma, Breast Cancer and Trigeminal Neuralgia is now possible. Also, numerical algorithms are now becoming increasingly popular because of their power to perform very accurate and precise computations quickly enabling Theoretical Physicists and Cosmologists to learn more about matter, space and the universe, and Engineers to design and build machines and structures with great durability and efficiency.
How does Mathematics form the basis of modern computing?
In these modern times, everything has to do with models. Traditionally, theories were solely based on empirical evidence and procedures were invented by trial and error which sometimes resulted in difficult consequences. In fact, there were some phenomena that could not have been tested by experimental means and thus was abandoned for years. The use of Mathematics as a tool for solving real world problems is called Mathematical Modelling and forms a very important branch of Mathematics today. Modern Medical Technology, Economic Theory as well as Computing and Engineering technologies are based on the ability to represent information from real world problems mathematically and then to apply analytical methods to analyze and solve the problems. Functional programming languages such as Haskell and Scheme implement a technique called Recursion which allows for various data structures such as stacks and queues and search and sort algorithms to work. It involves defining a function that takes previous states of itself as argument in order to product the next state. Recursion has also found applications in other fields of computing such as Software Engineering, Database Systems, Operating Systems, Networking and Digital Systems. Asymptotic Analysis of Algorithms has its basis in Mathematical Analysis, Polynomial Analysis and Proof Theory. Automata Theory, Theory of Computation and Decidability, and Cryptography has its basis in Graph Theory and Networks, Abstract Algebra and Elementary Number Theory.
Necessity of Mathematics in Computer Science Education
Mathematics education is absolutely vital to studying Computer Science today. Depending on the specialization that is chosen, certain courses in Mathematics are required, however courses in Mathematical Modelling is foundation regardless of the field is chosen. As Computing Technology advances, so will the need for more in-depth knowledge of Mathematics and techniques of Mathematical Modelling as Science and Engineering are now at a stage where information and processes are represented by analytical models that required advanced mathematical knowledge to solve them. Common topics important for modelling are: Ordinary and Partial Differential Equations, Numerical Analysis and Scientific Computing, Stochastic Analysis and Multivariate Statistics, Operations Research and Optimization Theory, and Mathematical Programming. The ability to integrate the theoretical aspects of Mathematics and Computer Science with the practical/skill based aspects will be an invaluable attribute for Analytical and Computational Scientists as our technological days unfold.