## Algebraic Combinatorics

As a simple example, to solve an enumeration problem one often encodes combinatorial data into an algebra of formal power series by means of a generating function. Algebraic manipulations with these power series then provide a systematic way to solve the original counting problem. Methods from complex analysis can be used to obtain asymptotic solutions even when exact answers are intractable. Read more.

Members:

C. Godsil : Algebraic Graph Theory

I.P. Goulden : Combinatorial Enumeration

D.M. Jackson : Algebraic-Enumerative Combinatorics, Moduli Spaces of Curves, Intersection Theory, Integrable Hierarchies

E. Katz: Combinatorial Algebraic Geometry

K. Purbhoo : Algebraic Combinatorics, Algebraic Geometry

L.B. Richmond : Asymptotic Enumeration

D. Wagner : Enumerative Combinatorics, Statistical Mechanics, Matroid Theory

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## Continuous Optimization

Continuous optimization is the core mathematical science for real-world problems ranging from design of biomolecules to management of investment portfolios. Continuous optimization means finding the minimum or maximum value of a function of one or many real variables, subject to constraints. The constraints usually take the form of equations or inequalities. Continuous optimization has been the subject of study by mathematicians since Newton, Lagrange and Bernoulli. Read more.

Members:

M.J. Best : Portfolio Optimization, Quadratic Programming

T. Coleman : Large-Scale Computational Optimization

L. Tunçel : Mathematical Optimization and Mathematics of Operations Research

S. Vavasis : Continuous Optimization, Numerical Analysis

H. Wolkowicz : Applications of Optimization and Matrix Theory to Algorithmic Development

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## Cryptography

Caesar and probably beyond. In the classical case, some algorithm, depending on a secret piece of information called the key, which had to be exchanged in secret in advance of communication, was used to scramble and descramble messages. (Note that, in a properly designed system, the secrecy should rely only on the key. It should be assumed that the algorithm is known to the opponent.) Read more.

Members:

D. Jao : Number Theory, Elliptic Curves

A.J. Menezes : Curve-Based Cryptography, Protocols, Provable Security

E. Teske-Wilson : Public-Key Cryptography, Computational Number Theory

Michele Mosca : Post-Quantum Cryptography, Quantum Key Distribution

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## Discrete Optimization

Much of combinatorial optimization is motivated by very simple and natural problems such as routing problems in networks, packing and covering problems in graph theory, scheduling problems, and sorting problems. But the methodology of the subject encompasses a variety of techniques ranging from elementary tree-growing procedures to constructions of Hilbert bases of integer lattices. Read more.

Members:

J. Cheriyan : Network Optimization, Parallel & Sequential Graph Algorithms

B. Cook : Integer Programming, Combinatorial Optimization

W.H. Cunningham : Combinatorial Optimization, Polyhedral Combinatorics, Matroids, Matchings, and Generalizations

R. Fukasawa : Mixed-integer programming, Computational Optimization, Operations Research

J. Geelen : Matroid Theory, Matchings

B. Guenin : Combinatorial Optimization

J. Koenemann : Approximation Algorithms, Combinatorial Optimization, Algorithmic Game Theory

L. Sanita: Combinatorial Optimization, Network Design

C. Swamy : Combinatorial Optimization, Approximation Algorithms, Algorithmic Game Theory, Stochastic Optimization

L. Tunçel : Mathematical Optimization and Mathematics of Operations Research

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## Graph Theory

A graph consists of a set of elements together with a binary relation defined on the set. Graphs can be represented by diagrams in which the elements are shown as points and the binary relation as lines joining pairs of points. It is this representation which gives graph theory its name and much of its appeal. However, the true importance of graphs is that, as basic mathematical structures, they arise in diverse contexts, both theoretical and applied. The concept of a graph was known already to Euler in the early eighteenth century, but it was the notorious Four-Colour Problem, posed by F. Guthrie in the mid-nineteenth century, that spurred the development of this simple concept into a flourishing theory. Read more.

Members:

J. Gao : Random Graph Theory

J. Geelen : Graph Minors

C. Godsil : Algebraic Graph Theory

B. Guenin : Signed Graphs

P.E. Haxell : Extremal Combinatorics, Graph Theory

L. Postle : Graph Colouring, Topological and Structural Graph Theory

B. Richter : Graph Theory & Topology of Surfaces

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## Quantum Computing

Today's computers and other information processing devices manipulate information using what is known as the "classical" approximation to the laws of physics. However, we have known for some time that there is a more accurate description of the laws - the one provided by quantum mechanics. The study of how the laws of quantum mechanics affect computing, cryptography, and similar information processing tasksis known as quantum information processing. Read more.

Members:

A. Childs : Quantum Computation, Quantum Information Processing

D.W. Leung : Quantum Information Processing

M. Mosca : Quantum Computing, Algorithms and Cryptography

A. Nayak : Quantum Computation and Quantum Information, Theoretical Computer Science

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