- 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:
- :Â symmetric functions, graph theory
- : Algebraic graph theory
- ±õ²¹²ÔÌý³Ò´Ç³Ü±ô»å±ð²Ô: Combinatorial enumeration
- : Algebraic-enumerative combinatorics, moduli spaces of curves, intersection theory, Integrable Hierarchies
- : Combinatorial optimization, entropy optimization, log-concave polynomials
- : Algebraic combinatorics, symmetric functions, statistical mechanics
- : Analytic combinatorics, computer algebra
- :Â Schubert calculus, symmetric function theory, dynamics
- : Algebraic combinatorics, algebraic geometry
- Bruce Richmond: Asymptotic enumeration
- : Enumerative combinatorics, statistical mechanics, matroid theory
- : Combinatorics in quantum field theory
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 about continuous optimization.
Members:
- : Convex analysis, monotone operator theory, projection methods and splitting techniques
- Approximation / Hardness of Approximation, Polynomial Maximization over Convex Sets, Functional Analysis, Asymptotic Convex Geometry, Spectral theory, Sum of Squares
- : Entropy optimization and log-concave polynomials
- : Mathematical optimization and Mathematics of Operations Research
- : Continuous optimization, numerical analysis
- : 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:
- : Number theory, elliptic curves, isogeny-based cryptography
- : Curve-based cryptography, protocols, provable security
- : Quantum key distribution, quantum-safe cryptography
- : Applied cryptography, internet security
-
: Quantum-safe 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 tree-growing procedures to constructions of Hilbert bases of integer lattices. Read more about discrete optimization.
Members:
- Approximation / Hardness of Approximation, Polynomial Maximization over Convex Sets, Functional Analysis, Asymptotic Convex Geometry, Spectral theory, Sum of Squares
- : Network optimization, Parallel & Sequential Graph Algorithms
- : Integer programming, combinatorial optimization
- : Combinatorial optimization, polyhedral combinatorics, matroids, matchings, and generalizations
- : Mixed-integer programming, Computational Optimization, operations research
- : Combinatorial optimization
- : Approximation algorithms, combinatorial optimization, algorithmic game theory
- :Â Combinatorial optimization, entropy optimization, log-concave polynomials
- : Matroid theory
- :Â Mathematical optimization, algorithmic game theory, operations research
- : Combinatorial optimization, approximation algorithms, algorithmic game theory, stochastic optimization
- : 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 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 about graph theory.
Members:
- : Random graph theory
- : Graph minors
- : Algebraic graph theory
- : Signed graphs
- ±Ê±ð²Ô²Ô²âÌý±á²¹³æ±ð±ô±ô: Extremal combinatorics, graph theory
- : Extremal combinatorics
- : Graph colouring, topological and structural graph theory, probabilistic and extremal combinatorics
- : Graph theory & Topology of Surfaces
- : 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:
- : Quantum Information Processing
- : Quantum computing, Algorithms and Cryptography
- : Quantum Computation and Quantum Information, theoretical computer science
- :Theory of quantum computing, quantum algorithms, quantum complexity theoryÂ