These modules build on the ideas in MATH20201: Algebraic Structures 1, introducing new kinds of algebraic objects, studying the structure of groups and rings more deeply and exploring their applications across many parts of pure and applied mathematics.
The rigorous approach to calculus introduced in, for example, MATH10242: Sequences & Series and MATH20101: Real & Complex Analysis is the just beginning of a long story that involves deep, far-reaching generalisations of such basic ideas as continuity and distance and—as the list of electives shows—has applications across much of modern mathematics.
The core courses listed below build on such modules as MATH10222: Calculus & Applications and MATH20401: PDE's & Vector Calculus to describe the mechanics of fluids and such deformable solids as rubber and steel, while the electives combine the development of sophisticated analytical tools with applications that range across the natural world.
These modules apply ideas from probability to model the pricing of financial instruments. Fundamental early results in this area, proved in the 1970's, lead to the development of a galaxy of new products that are now so numerous and valuable that, for example, the volume of trade in financial derivatives dwarfs that of the underlying commodities and stocks on which they depend.
Geometry is among the most ancient parts of mathematics and its basic objects results—curves and surfaces, their properties and their many modern generalisations—touch on almost every part of mathematics and its applications. The modules listed below reflect this richness and would be enjoyable for students with a solid grounding in both rigorous calculus and linear algebra.
Logic is the study of arguments: what they are and what it means to say that they are sound. As such it's central to Mathematics and the School has a strong tradition in the area. Manchester's first great logician, Alan Turing, is famous for his cryptographic work in the second World War, but mathematicians also know his visionary work on logic and the foundations of computation. The modules listed below provide a solid grounding in formal logic.
Numerical analysis is the study of algorithms and approximations for solving the problems of continuous mathematics, that is, problems involving real or complex numbers. For such problems, which include many key applications in science, engineering and finance, exact, closed-form solutions often do not exist or are not readily obtainable, so numerical analysts develop and analyse algorithms that compute numerical approximations. The best work in the subject blends essentially pure-mathematical work on foundations with a solid grounding in the practicalities of practical computation and this is reflected in the lists of modules below, which include both programming courses and such theoretical modules as Matrix Analysis.
Probability is the part of mathematics that enables us to reason about chance and uncertainty. Its theorems and methods are central to problems across the whole of science, engineering and economics and—in addition to the many beautiful, powerful methods and results of its own—has illuminated many other parts of modern mathematics. The modules listed below build on MATH10141: Probability 1 and MATH20701: Probability 2.
Statistics is the science of data and its interpretation and so combines rigorous analysis (in the sense of the Analysis theme above) with the development of methods and tools suited to specific, real-world problems. Thus, as in the Numerical Analysis theme, the electives include both practical, computational modules as well as conventional lecture courses such as Time Series and Generalised Linear Models, that introduce standard approaches to statistical modelling.