Analysis
In the broadest view, mathematical analysis comprises those subjects of mathematics which are based on the notion of limits. If you have studied calculus, then you have seen many of the fundamental concepts of analysis, including convergence, continuity, differentiation and integration. Indeed, analysis is often described as "calculus and what comes after."
This catchphrase has historical and descriptive value; analysis developed out of nineteenth-century efforts to understand and justify the methods of calculus, and to expand the scope of its applications. By the mid-19th century it had reached the modern formulation (in terms of limits) recognizable to today's students. A sesquicentenary later, analysis has become one of the largest branches of mathematics, and is an essential tool for many others, such as partial differential equations, dynamical systems, numerical and computational methods and mathematical physics. Such scientific fields as signal and image processing, quantum physics, optics, fluid dynamics and mathematical biology also rely heavily on the methods of analysis.
One of the most dramatic innovations in analysis has been to change the focus from the detailed study of a single function or infinite series to the simultaneous consideration of collections of these objects, each thought of as a "point" in a "function space." The desire to understand properties of function spaces and the maps between them has led to the introduction of techniques from such diverse areas as algebra, topology, combinatorics and geometry. This interplay among different branches of mathematics and other sciences makes the study of modern analysis especially fascinating, fundamentally important and deeply rewarding.
Some crucial examples of these branches of mathematics, each of which also represents an area of expertise of one or more ASU analysis faculty, are: operator algebras, which comprises the mathematics that grew out of quantum mechanics; harmonic analysis, where functions solving a differential equation are resolved into orthogonal components (as are vectors in three dimensions); representation theory, where a general group of symmetries is used to decompose the space of functions; and generalized special functions, in which the descriptions of important classical families of functions are "twisted" to produce variations with remarkable properties.
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