 Algebraic combinatorics
 Continuous optimization
 Cryptography
 Discrete optimization
 Graph theory
 Quantum computing
Algebraic combinatorics
Algebraic combinatorics is the mathematical area concerned with the relationships between discrete and algebraic objects. Combinatorial objects give rich and detailed insight into algebraic problems in representation theory and beyond, and algebraic tools give new understanding of discrete structures, for example in systematic tools for enumeration of combinatorial objects using generating functions. These and other aspects of algebraic combinatorics are unified by the centrality of beautiful, explicit combinatorial objects and also find applications in statistical mechanics and quantum field theory. Read more about algebraic combinatorics.
Members:
 Logan Crew: symmetric functions, graph theory
 Chris Godsil: Algebraic graph theory
 Ian Goulden: Combinatorial enumeration
 David Jackson: Algebraicenumerative combinatorics, moduli spaces of curves, intersection theory, Integrable Hierarchies
 Jonathan Leake: Combinatorial optimization, entropy optimization, logconcave polynomials
 Olya Mandelshtam: Algebraic combinatorics, symmetric functions, statistical mechanics
 Stephen Melczer: Analytic combinatorics, computer algebra
 Oliver Pechenik: Schubert calculus, symmetric function theory, dynamics
 Kevin Purbhoo: Algebraic combinatorics, algebraic geometry
 Bruce Richmond: Asymptotic enumeration
 David Wagner: Enumerative combinatorics, statistical mechanics, matroid theory
 Karen Yeats: Combinatorics in quantum field theory
Continuous optimization
Continuous optimization is the core mathematical science for realworld 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 about continuous optimization.
Members:
 Walaa Moursi: Convex analysis, monotone operator theory, projection methods and splitting techniques
 Vijay Bhattiprolu: Approximation / Hardness of Approximation, Polynomial Maximization over Convex Sets, Functional Analysis, Asymptotic Convex Geometry, Spectral theory, Sum of Squares
 Jonathan Leake: Entropy optimization and logconcave polynomials
 Levent Tunçel: Mathematical optimization and Mathematics of Operations Research
 Steve Vavasis: Continuous optimization, numerical analysis
 Henry Wolkowicz: Applications of optimization and matrix theory to algorithmic development
Cryptography
In classical cryptography, 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 about cryptography.
Members:
 David Jao: Number theory, elliptic curves, isogenybased cryptography
 Alfred Menezes: Curvebased cryptography, protocols, provable security
 Michele Mosca: Quantum key distribution, quantumsafe cryptography
 Douglas Stebila: Applied cryptography, internet security

Samuel Erik Jaques: Quantumsafe cryptography, quantum cryptanalysis
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 treegrowing procedures to constructions of Hilbert bases of integer lattices. Read more about discrete optimization.
Members:
 Vijay Bhattirolu: Approximation / Hardness of Approximation, Polynomial Maximization over Convex Sets, Functional Analysis, Asymptotic Convex Geometry, Spectral theory, Sum of Squares
 Joseph Cheriyan: Network optimization, Parallel & Sequential Graph Algorithms
 Bill Cook: Integer programming, combinatorial optimization
 Bill Cunningham: Combinatorial optimization, polyhedral combinatorics, matroids, matchings, and generalizations
 Ricardo Fukasawa: Mixedinteger programming, Computational Optimization, operations research
 Bertrand Guenin: Combinatorial optimization
 Jochen Koenemann: Approximation algorithms, combinatorial optimization, algorithmic game theory
 Jonathan Leake: Combinatorial optimization, entropy optimization, logconcave polynomials
 Peter Nelson: Matroid theory
 Kanstantsin Pashkovich: Mathematical optimization, algorithmic game theory, operations research
 Chaitanya Swamy: Combinatorial optimization, approximation algorithms, algorithmic game theory, stochastic optimization
 Levent Tunçel: Mathematical optimization and Mathematics of Operations Research
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 FourColour Problem, posed by F. Guthrie in the midnineteenth century, that spurred the development of this simple concept into a flourishing theory. Read more about graph theory.
Members:
 Jane Gao: Random graph theory
 Jim Geelen: Graph minors
 Chris Godsil: Algebraic graph theory
 Bertrand Guenin: Signed graphs
 Penny Haxell: Extremal combinatorics, graph theory
 Peter Nelson: Extremal combinatorics
 Luke Postle: Graph colouring, topological and structural graph theory, probabilistic and extremal combinatorics
 Bruce Richter: Graph theory & Topology of Surfaces
 Sophie Spirkl: Induced subgraphs, structural graph theory
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 about quantum computing.
Members:
 Debbie Leung: Quantum Information Processing
 Michele Mosca: Quantum computing, Algorithms and Cryptography
 Ashwin Nayak: Quantum Computation and Quantum Information, theoretical computer science
 David Gosset:Theory of quantum computing, quantum algorithms, quantum complexity theory