38 undergraduate courses in the 2026–27 Undergraduate Calendar, and 46 graduate courses in the Fall 2026 Graduate Calendar.
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.
A broad introduction to the field of optimization, discussing applications, and solution techniques. Mathematical models for real life applications; algorithms; aspects of computational complexity; geometry; linear programming duality, focusing on the development of algorithms.
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.
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.
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.
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.
An introduction to some of the key topics in graph theory: connectivity, planarity, and matchings. Connectivity: Menger's theorem, 3-connected graphs. Planarity: Kuratowski's theorem, uniqueness of planar embeddings. Matchings: Review of Konig's theorem, Tutte's theorem.
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.
Formulations of combinatorial optimization problems, greedy algorithms, dynamic programming, branch-and-bound, cutting plane algorithms, decomposition techniques in integer programming, approximation algorithms.
Modelling problems in data science and machine learning as optimization problems, and their analysis. Solution methods, including gradient descent and its extensions, and their consequences in modelling decisions. Convexity and Lagrangian duality. Applications to clustering and classification problems. Analysis and implementation of computational methods.
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.
Computational optimization methodologies underlying portfolio problems in finance. Computational linear algebra, determining derivatives, quadratic, and nonlinear optimization. The efficient frontier problem. Applications of optimization in finance such as volatility surface determination and global minimization for value-at-risk.
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 Lagrange Implicit Function Theorem, the MacMahon Master Theorem. Enumeration of planar triangulations. The transfer matrix method. Sieve methods, inclusion/exclusion, Mobius inversion. Polya enumeration, Enumeration of trees. Basic hypergeometric series, q-analogues, Rogers-Ramanujan identities. Asymptotic methods.
The ring of symmetric functions, standard bases, the Hall inner product. Young tableaux. The Robinson-Schensted-Knuth correspondence, the hook-length formula, the Jacobi-Trudi formula, the Pieri rule, the Littlewood-Richardson rule. Representation theory of the symmetric groups. Enumeration of plane partitions. Enumeration of maps on surfaces. Other topics.
Basics of information theory; Shannon entropy, KL divergence, and mutual information; basic properties of entropic quantities; chain rule, Pinsker's Inequality, Data Processing Inequality; compression; Channel Coding Theorem; error-correction; applications to combinatorics, optimization, cryptography, and computer science.
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.
Colouring: Brooks' Theorem and Vizing's Theorem. Flows: integer and group-valued flows, the flow polynomial, the 6-flow theorem. Extremal graph theory; Ramsey's theorem, Turan's theorem, Mader's theorem on graphs with no n-clique-minor. Probabilistic methods: Lower bounds for Ramsey numbers, graphs with large girth and chromatic number.
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.
This is an introductory course on matroid theory, with particular emphasis on graphic matroids and on topics that are applicable to graph theory. The topics include matroid intersection and partition, graphic matroids, matroid connectivity, regular matroids, and representable matroids. Applications include matching, disjoint paths, disjoint spanning trees, the 8-flow theorem for graphs, planarity testing, and recognizing totally unimodular matrices.
The probabilistic method is a collection of proof techniques to show the existence of a combinatorial structure by showing a random element in an appropriate probability space has the desired structure with a positive probability. The course will introduce the fundamental techniques and give numerous applications in a wide variety of areas, including graph theory, combinatorics, number theory, geometry, optimization, randomized algorithms, and information theory. In particular, normally the following topics will be discussed: method of the first moment, Chebyshev’s inequality, alterations, the Lovasz local lemma, correlation inequalities, exponential concentration inequalities, and applications.
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.
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.
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 travelling salesman problem, heuristics, dynamic programming, and branch-and-bound approaches. Polynomial-time approximation algorithms.
A broad introduction to game theory and its applications to the modelling of competition and co-operation 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, subgradient calculus for convex functions, Fenchel conjugacy and duality, Lagrange multipliers. Ellipsoid method for convex optimization.
Numerical algorithms for nonlinear optimization. Newton, variable-metric, quasi-Newton and conjugate gradient methods. Obtaining derivatives. Convexity. Trust region methods. Constrained optimization including optimality conditions, sequential quadratic programming, interior point, and active set strategies.
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, elliptic curve cryptography, post-quantum cryptography. Proofs of security.
The primary objective is to study current work in specific areas of quantum information. Course content may vary from term to term.
A broad introduction to modern cryptography, highlighting the tools and techniques used to secure internet and messaging applications. Symmetric-key encryption, hash functions, message authentication, authenticated encryption, public-key encryption and digital signatures, key establishment, key management.
Reading course as announced by the department.
From the Fall 2026 Graduate Calendar.
Elementary counting, bijections. Sets, weights, decompositions. The ordinary generating series, sum, product, composition and differentiation lemmas; formal use in combinatorial identities. The algebra of formal power series. Strings on a finite alphabet. Unique factorization of strings. Pattern and noncommutative methods. The Maximal Decomposition Theorem. Rational and algebraic generating series, linear recurrence equations. Integer partitions. Functional equation. Ferrers graph, geometric decompositions, Euler's Pentagonal Number Theorem. Vector spaces over GF(q), linear transformations, finite forms of partition theorems, Gaussian coefficients. Inversions in permutations. The exponential generating series, rule of sum and product, composition and differentiation lemmas. Disjoint cycles and permutations, functions, set partitions, graphs. Lagrange's Implicit Function Theorem. Lattice paths, Wiener-Hopf Factorization. Lattice polygons, polyominos, trees and functions. Formal use in combinatorial identities. Enumeration under group action, Orbit-Stabilizer Theorem, Burnside's Lemma, Polya's Theorem. Necklaces, abstract trees and trees embedded in the plane.
Linear optimization: Farkas' Lemma, duality, Simplex method, geometry of polyhedra. Combinatorial optimization: integrality of polyhedra, total unimodularity, flow problems, weighted bipartite matching. Continuous optimization: convex sets, Separation Theorem, convex functions, analytic characterizations of convexity, Karush-Kuhn-Tucker Theorem.
The algebra of formal Laurent series. Multivariate ordinary generating functions and exponential generating functions. The Lagrange Implicit Function Theorem, the MacMahon Master Theorem. Enumeration of planar triangulations. The Transfer Matrix method. Sieve methods, Inclusion/Exclusion, Möbius inversion. Pólya Enumeration, Enumeration of Trees. Basic hypergeometric series, q-analogues, Rogers-Ramanujan identities. Asymptotic methods.
The ring of symmetric functions, standard bases, the Hall inner product. Young tableaux. The Robinson-Schensted-Knuth correspondence, the hook-length formula, the Jacobi-Trudi formula, the Pieri rule, the Littlewood-Richardson rule. Representation theory of the symmetric groups. Enumeration of plane partitions. Enumeration of maps on surfaces. Other topics.
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.
Consult the schedule of classes for specific topics being offered.
Consult the schedule of classes for specific topics being offered.
Colouring: Brooks' Theorem and Vizing's Theorem. Flows: integer and group-valued flows, the flow polynomial, the 6-flow theorem. Extremal graph theory; Ramsey's theorem, Turan's theorem, Mader's theorem on graphs with no n-clique-minor. Probabilistic methods: Lower bounds for Ramsey numbers, graphs with large girth and chromatic number.
Automorphisms. Cayley graphs and their properties. Arc and distance transitive graphs. Generalised polygons. homomorphisms and covers. Adjacency and incidence matrices. Eigenvectors of graphs. Quotients. Interlacing. Strongly regular graphs. Line graphs and graphs with least eigenvalue -2. Expanders. Shannon capacity.
This is an introductory course on matroid theory, with particular emphasis on graphic matroids and on topics that are applicable to graph theory. The topics include: matroid intersection and partition, graphic matroids, matroid connectivity, regular matroids, and representable matroids. Applications include: matching, disjoint spanning trees, the 8-flow theorem for graphs, planarity testing, and recognizing totally unimodular matrices.
The probabilistic method is a collection of proof techniques to show the existence of a combinatorial structure by showing a random element in an appropriate probability space has the desired structure with a positive probability. The course introduces the fundamental techniques and gives numerous applications in a wide variety of areas, including graph theory, combinatorics, number theory, geometry, optimization, randomised algorithms, and information theory. In particular, normally the following topics are discussed: method of the first moment; Chebyshev’s inequality; alterations; the Lovasz local lemma; correlation inequalities; exponential concentration inequalities; applications.
Characterizations of optimalsolutions and efficient algorithms for optimization problems over discrete structures. Topics include network flows, optimal matchings, T-joins and postman tours, matroid optimization.
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.
An introduction to the modern theory of convex programming, its extensions and applications. Structure of convex sets, separation and support, subgradient calculus for convex functions, Fenchel conjugacy and duality, Lagrange multipliers. Ellipsoid method for convex optimization.
A course on theory and solution algorithms for the minimization of a convex quadratic function subject to linear constraints. Karush-Kuhn-Tucker conditions, duality theory. Active set solution algorithms and their parametric extensions. Quadratic programmes as linear complementarity problems. Applications in portfolio optimization and structural analysis.
Numerical algorithms for nonlinear optimization. Newton, variable-metric, quasi-Newton and conjugate gradient methods. Obtaining derivatives. Convexity. Trust region methods. Constrained optimization including optimality conditions, sequential quadratic programming, interior point and active set strategies.
Optimization over convex sets described as the intersections 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.
Techniques for formulating data science models as optimization problems. Algorithms for solving data science problems with an emphasis on scalability, efficiency and parallelizability including gradient-descent based algorithms, derivative-free algorithms, and randomized algorithms. Theoretical analyses of algorithms and approaches for recognizing the most suitable algorithm for solving a particular problem.
Review of basics of quantum information and computational complexity; Simple quantum algorithms; Quantum Fourier transform and Shor factoring algorithm: Amplitude amplification, Grover search algorithm and its optimality; Completely positive trace-preserving maps and Kraus representation; Non-locality and communication complexity; Physical realizations of quantum computation: requirements and examples; Quantum error-correction, including CSS codes, and elements of fault-tolerant computation; Quantum cryptography; Security proofs of quantum key distribution protocols; Quantum proof systems. Familiarity with theoretical computer science or quantum mechanics will also be an asset, though most students will not be familiar with both.
An in-depth study of public-key cryptography, including: number-theoretic problems - prime generation, integer factorization, discrete logarithms; public-key encryption; digital signatures; key establishment; elliptic curve cryptography; post-quantum cryptography; and security proofs.
A broad introduction to modern cryptography, highlighting the tools and techniques used to secure internet and messaging applications. Symmetric-key encryption, hash functions, message authentication, authenticated encryption, public-key encryption and digital signatures, key establishment, key management.
Consult the schedule of classes for specific topics being offered.
Methods of obtaining asymptotic estimates for sums arising in enumeration. Application to Bell numbers, the distribution of balls into cells, and random graphs. Methods for obtaining asymptotic estimates for the coefficients of generating functions. Application to random planar graph problems and to unimodality problems.
Consult the schedule of classes for specific topics being offered.
The probabilitistic method proves the existence of a combinatorial structure by showing a random element in an appropriate probability space has the desired properties with positive probability. The course will introduce the basic techniques and give applications in graph theory, combinatorics, discrete optimization, theoretical CS. In particular, some of the following will be considered: linearity of expectation; alterations; the second moment method; bounding of large deviations; the local lemma; correlation inequalities; martingales and tight concentration; discrepancy; derandomization.
Consult the schedule of classes for specific topics being offered.
An introduction to the concepts of directed graphs, problems about directed circuits and directed cuts, and minimax equalities relating these and other graphical objects.
A general study of algorithms for the solutions of problems in graph theory. Included are algorithms for graph isomorphism, planarity, connectedness, chromatic number, spanning trees, circuits and many others.
Consult the schedule of classes for specific topics being offered.
Consult the schedule of classes for specific topics being offered.
Consult the schedule of classes for specific topics being offered.
The course studies some of the successful paradigms for designing approximation algorithms and for proving approximation guaranteees, such as the greedy method, formulating and solving linear programming relaxations, the primal-dual method, randomized rounding, semidefinite programming relaxations, approximation of metrics. Also, the PCP theorem, and its application to the hardness of approximating specific problems, e.g., set covering and shortest lattice vector.
Consult the schedule of classes for specific topics being offered.
Mathematical Programming
Consult the schedule of classes for specific topics being offered.
Mathematical Operations Research
Basic optimization: quadratic minimization subject to leanear equality constraints. Effecient portfolios: the efficient frontier, the capital market line, Sharpe ratios and threshold returns. Practical portfolio optimization: short sales restrictions target portfolios, transactions costs. Quadratic programming theory. Special purpose quadratic programming algorithms for portfolio optimization: today's large investment firms expect to solve problems with at least 1000 assets, transactions costs and various side constraints in just a few minutes of computation time. This requires very specialized QP algorithms. An overview of such algorithms will be presented with computational results from commercial problems. The efficient frontier, the capital market line, Sharpe ratios and threshold returns in practice.
Consult the schedule of classes for specific topics being offered.
Consult the schedule of classes for specific topics being offered.
Fundamental problems of elementary and algebraic number theory from an algorithmic and computational complexity point of view with emphasis placed on analysis of algorithms. Topics include basic arithmetic algorithms; computation over finite fields; primality testing; algorithms for integer factorization; algorithms in number fields.
Consult the schedule of classes for specific topics being offered.
Seminar in Combinatorics
Seminar in Graph Theory
Seminar in Discrete Optimization
Seminar in Continuous Optimization
Seminar in Operations Research