CO Course Descriptions

Combinatorics and Optimization (2009-2010)

A broad introduction to the field of optimization, discussing applications, and solution techniques. Mathematical models for real life applications; algorithms: Simplex, Cutting Plane, and Branch & Bound; linear programming duality. [Offered: F,W]

Linear programming and the Simplex algorithm. Duality. Introduction to continuous optimization in real n-space for large n. Nonlinear Optimization and Lagrange multipliers. Modelling and applications. [Offered: S]

An applications-oriented course that illustrates how various mathematical models and methods of optimization can be used to solve problems arising in business, industry and science. [Offered: W,S]

The algebra of formal power series. The combinatorics of the ordinary and exponential generating series. Lagrange's Implicit Function Theorem, applications to the enumeration of permutations, functions, trees and graphs. Integer partitions, geometric methods, enumerating linear transformations. Introduction to the pattern algebra, applications to the enumeration of strings. Lattice paths, Wiener-Hopf factorization. Enumeration under symmetries. [Offered: F]

A first course in error-correcting codes. Linear block codes, Hamming-Golay codes and multiple error-correcting BCH codes are studied. Various encoding and decoding schemes are considered. [Offered: W]

An introduction to some of the key parts of graph theory: connectivity, planarity and matchings. Connectivity: Menger's Theorem, 3-connected graphs and contractible edges, Kuratowski's Theorem, uniqueness of planar embeddings. Planarity, cycle and co-cycle spaces: peripheral cycles and the cycle space of a 3-connected graph. Matchings: Review of Konig's Theorem, Tutte's Theorem. [Offered: F,S]

A first course in optimization, emphasizing optimization of linear functions subject to linear constraints (linear programming). Problem formulation. Duality theory. The simplex method. Sensitivity analysis.

Review of linear programming. Shortest path problems. The max-flow min-cut theorem and applications. Minimum cost flow problems. Network simplex and primal-dual algorithms. Applications to problems of transportation, distribution, job assignments and critical-path planning. [Offered: F,W,S]

A first course in computational optimization. Linear optimization, the simplex method, implementation issues, duality theory. Introduction to computational discrete and continuous optimization. [Offered: F,S]

Formulations of combinatorial optimization problems, greedy algorithms, dynamic programming, branch-and-bound, cutting plane algorithms, decomposition techniques in integer programming, approximation algorithms. [Offered: F]

Linear optimization: feasibility theorems, duality, the simplex algorithm. Discrete optimization: integer linear programming, cutting planes, network flows. Continuous optimization: local and global optima, feasible directions, convexity, necessary optimality conditions.

A course on the fundamentals of nonlinear optimization, including both the mathematical and the computational aspects. Necessary and sufficient optimality conditions for unconstrained and constrained problems. Convexity and its applications. Computational techniques and their analysis.

An applications-oriented course that illustrates how various mathematical models and methods of optimization can be used to solve problems arising in business, industry and science. [Offered: F,W]

Applications of basic optimization models and techniques for decision making in financial markets. Quadratic optimization subject to linear equality constraints. Derivation of efficient portfolios and the Markowitz efficient frontier. The Capital Market Line. Practical portfolio optimization as a quadratic programming problem. A solution algorithm for quadratic programming problems. [Offered: W]

A course in problem solving. 100 problems are studied. Problems are taken mainly from the elementary parts of algebra, geometry, number theory, combinatorics and probability.

The algebra of Laurent series and Lagrange's Implicit Function Theorem, enumerative theory of planar embeddings (maps). The ring of symmetric functions: Schur functions, orthogonal bases, inner product, Young tableaux and plane partitions. Non-intersecting paths, sieve methods, partially ordered sets and Mobius inversion, strings with forbidden substrings, the Cartier-Foata commutation monoid. Introduction to the group algebra of the symmetric group, enumerative applications of sl(2). [Offered: F]

Pairwise orthogonal latin squares. Transversal designs and finite planes. Balanced incomplete block designs, group divisible designs and pairwise balanced designs. Symmetric designs and Hadamard matrices. Recursive constructions. Wilson's fundamental construction.

An undergraduate seminar in combinatorics. The primary objective is to study current work in specific areas of combinatorics. Course content may vary from term to term.

An in-depth study of one or two topics in graph theory. Course content may vary from term to term. Topics may include planar graphs, extremal graph theory, directed graphs, enumeration, algebraic graph theory, probabilistic graph theory, connectivity, graph embedding, colouring problems.

An in-depth look at the following major topics in graph theory; other topics may also be included: Colouring: Brooks', Vizing's and Grotzsch's Theorems, list colouring. Eigenvalues of adjacency matrices: Moore graphs. Directed graphs: tournaments, kernels, disjoint branchings, Lucchesi-Younger Theorem. [Offered: F]

An introduction to the methods of and some interesting current topics in algebraic graph theory. Topics covered will include vertex-transitive graphs, eigenvalue methods, strongly regular graphs and may include graph homomorphisms, Laplacians or knot and link invariants.

Characterizations of optimal solutions and efficient algorithms for optimization problems over discrete structures. Topics include network flows, optimal matchings, T-joins and postman tours, matroid optimization. [Offered: F]

Formulation of problems as integer linear programs. Solution by branch-and-bound and cutting plane algorithms. Introduction to the theory of valid inequalities and polyhedral combinatorics.

Network design under constraints on cost, capacity, distance and reliability. Approximation algorithms. The set covering problem. Tree solutions: spanning trees, Steiner trees, Gomory-Hu trees, optimum communication spanning trees. Connectivity, survivability and reliability. Network design with concentrators: the terminal layout problem. Location problems on networks.

An overview of practical optimization problems that can be posed as scheduling problems. Characterizations of optimal schedules. Simple and efficient combinatorial algorithms for easy problems. A brief overview of computational complexity, definition of P, NP, NP-Complete and NP-hard. Integer programming formulations, the Traveling Salesman Problem, heuristics, dynamic programming and branch-and-bound approaches. Polynomial-time approximation algorithms. [Offered: S]

A broad introduction to game theory and its applications to the modeling of competition and cooperation in business, economics and society. Two-person games in strategic form and Nash equilibria. Extensive form games, including multi-stage games. Coalition games and the core. Bayesian games, mechanism design and auctions.

An undergraduate seminar in optimization. The primary objective is to study recent work in specific areas of optimization. Course content may vary from term to term.

An introduction to the modern theory of convex programming, its extensions and applications. Structure of convex sets, separation and support, set-valued analysis, subgradient calculus for convex functions, Fenchel conjugacy and duality. Lagrange multipliers, minimax theory. Algorithms for nondifferentiable optimization. Lipschitz functions, tangent cones and generalized derivatives, introductory non-smooth analysis and optimization.

Theory and practical algorithms for nonlinear continuous optimization. Fundamentals of unconstrained optimization: conjugate gradient methods and Newton-type methods. Nonlinear least squares problems. Fundamentals of constrained optimization: optimality conditions, quadratic programming, penalty and barrier methods, interior-point methods, sequential quadratic programming. [Offered: W]

Optimization over convex sets described as the intersection of the set of symmetric, positive semidefinite matrices with affine spaces. Formulations of problems from combinatorial optimization, graph theory, number theory, probability and statistics, engineering design, and control theory. Theoretical and practical consequences of these formulations. Duality theory and algorithms.

An in-depth examination of the origins of mathematics, beginning with examples of Babylonian mathematics. Topics may include Pythagorean triples, solution of equations, estimation of pi, duplication of the cube, trisection of an angle, the Fibonacci sequence, the origins of calculus.

Basics of computational complexity; basics of quantum information; quantum phenomena; quantum circuits and universality; relationship between quantum and classical complexity classes; simple quantum algorithms; quantum Fourier transform; Shor factoring algorithm; Grover search algorithm; physical realization of quantum computation; error-correction and fault-tolerance; quantum key distribution.

An in-depth study of public-key cryptography. Number-theoretic problems: prime generation, integer factorization, discrete logarithms. Public-key encryption, digital signatures, key establishment, secret sharing. Proofs of security. [Offered: F]

A broad introduction to cryptography, highlighting the major developments of the past twenty years. Symmetric ciphers, hash functions and data integrity, public-key encryption and digital signatures, key establishment, key management. Applications to Internet security, computer security, communications security, and electronic commerce. [Offered: W]

[Offered: F,W,S]