Algebra and Combinatorics
Algebra refers to the use and manipulation of symbols, often with each representing some mathematical entity such as a quantity (think integer or real number), a set with special structure (think group, ring, topological space, or vector bundle) or an element of such a set, or a relation (think function, partial order, or homomorphism). Manipulation of symbols usually follows specified rules that allow for operations such as addition, multiplication, composition, or action of one object upon another. In representation theory, for example, groups act on vector spaces; and in commutative algebra, elements of rings are viewed as functions on spaces.
Functions are representations of relations between sets, and in particular are useful for representing the changing states of a system: the velocity of a projectile, the frequencies present in a sound signal, the color of a pixel in a digital image, or the prices of a portfolio of stocks. The mathematical field of analysis seeks to formulate methods to analyze quantitatively the change exhibited by the outputs of functions with respect to their inputs, as a way of distilling important information about the underlying systems---such as the way stock prices change over time.
In recent decades, an explosive synergy between biology and mathematics has greatly enriched and extended both fields. Indeed, given its ability to reveal otherwise invisible worlds in all kinds of biological systems, mathematics has been called the "new microscope in biology." In turn, biology has stimulated the creation of new realms of mathematics. Duke's Mathematics Department has a large group of mathematicians who work on specific biological and medical applications and as well as the development of new applied mathematical techniques and new mathematical theorems inspired by biological applications. The range of applications is large and the mathematical techniques diverse, including dynamical systems, partial differential equations, stochastic dynamics, fluid dynamics, geometry, topology, and algebra.
Computational Mathematics involves mathematical research in areas of science and engineering where computing plays a central and essential role. Topics include for example developing accurate and efficient numerical methods for solving physical or biological models, analysis of numerical approximations to differential and integral equations, developing computational tools to better understand data and structure, etc. Computational mathematics is a field closely connected with a variety of other mathematical branches, as for often times a better mathematical understanding of the problem leads to innovative numerical techniques.
Geometry:Researchers at Duke use geometric methods to study:
the geometry and arithmetic of algebraic varieties;
the geometry of singularities;
general relativity and gravitational lensing
exterior differential systems;
the geometry of PDE and conservation laws;
geometric analysis and Lie groups;
control theory and Finsler geometry;
symplectic and contact geometry
Mathematical physics seeks to apply rigorous mathematical ideas to problems in physics, or problems inspired by physics. As such, it is a remarkably broad subject. Mathematics and Physics are traditionally very closely linked subjects. Indeed historical figures such as Newton and Gauss are difficult to classify as purely physicists or mathematicians. Traditionally mathematical physics has been quite closely associated to ideas in calculus, particularly those of differential equations. In recent years however, in part due to the rise of superstring theory, there has been a great enlargement of branches of mathematics which can now be categorized as part of mathematical physics. It is often joked that, in additional to unifying all of physics, superstring theory also encompasses all of mathematics!
Number theory is the study of the integers (i.e. whole numbers) and related objects. Topics studied by number theorists include the problem of determining the distribution of prime numbers within the integers and the structure and number of solutions of systems of polynomial equations with integer coefficients. Many problems in number theory, while simple to state, have proofs that involve apparently unrelated areas of mathematics. A beautiful illustration is given by the use of complex analysis to prove the “Prime Number Theorem,” which gives an asymptotic formula for the distribution of prime numbers. Yet other problems currently studied in number theory call upon deep methods from harmonic analysis.
PDE & Dynamical Systems
Partial differential equations (PDEs) are one of the most fundamental tools for describing continuum phenomena in the sciences and engineering. Early work on PDEs, in the 1700s, was motivated by problems in fluid mechanics, wave motion, and electromagnetism. Since that time, the range of applications of PDEs has expanded rapidly. For example, PDEs are used in mathematical models of weather and climate, in medical imaging technologies, in the design of new composite materials, in models of elementary particle interaction and of the formation of galaxies, in models of cancerous tumor growth or of blood flow in the heart, in simulating semiconductor devices, in models of bacterial colonies, in models of financial markets and asset price bubbles, in describing the flocking behavior of birds and fish. PDEs also have played an important part in the development of other branches of mathematics, including harmonic analysis, differential geometry, probability, optimization and control theory. The phenomena described by PDEs are as complex as the world around us; the mathematical techniques needed to study PDEs are very diverse.
Mathematical research in physical modeling focuses on the formulation and analysis of mathematical representations of problems motivated by other branches of science and engineering. In addition to generating novel problems with new computational and analytical challenges, constructing accurate models for complex systems may uncover the need for fundamental extensions to the governing equations.
Probability and Stochastic process is the study of randomness. It is at once a theoretical and abstract subject and one which is highly applied. Probability is both an increasingly core subject in mathematics and has long been an indispensable tool in applied modeling. Probability has been central in a number of recent Fields Medals. Probability is a required tool in topics ranging from modern population modeling to Bayesian statistics to phase transitions.
Signals, Images, and Data
A large number of signals is collected from a wide variety of physical and biological phenomena, and in a variety of forms, ranging from acoustics, radar, camera images, hyper-spectral images, movies, and many others. These signals are the ones that our own human sensors are used to measuring, and our own brains have evolved to efficiently interpret. Huge data sets of this type are collected daily, from consumer cameras to music studio recordings to satellite images. The large collections of signals need to be analyzed, in order to to order them and to extract useful information. For example, technology companies of all sizes develop and use tools to automatically categorize and recognize objects and faces in images, or in art works; in digital pathology one would like to be able to automatically categorize cell types, or, in neurobiology, cell activity; in applications to agriculture the condition of the crops is monitored by aerial hyper-spectral imaging; in the study of simulations of biological molecules one would like to understand how large molecules move in the high-dimensional space of possible configurations. There are of course many data types that are very different from signals that we are used to hearing and seeing, and require different analysis tools. Examples may include the activity in a social network in time, or a set of transactions in an economic network, or the interactions between gene and proteins through time in a biological network.
Topology is the study of shapes and spaces. What happens if one allows geometric objects to be stretched or squeezed but not broken? In fact there’s quite a bit of structure in what remains, which is the principal subject of study in topology.
The modern field of topology draws from a diverse collection of core areas of mathematics. Much of basic topology is most profitably described in the language of algebra – groups, rings, modules, and exact sequences. But topology has close connections with many other fields, including analysis (analytical constructions such as differential forms play a crucial role in topology), differential geometry and partial differential equations (through the modern subject of gauge theory), algebraic geometry (the topology of algebraic varieties), combinatorics (knot theory), and theoretical physics (general relativity and the shape of the universe, string theory). In addition, topology can strikingly be applied to study the structure of large data sets.