Course Descriptions

CREDIT HOURS: 3

This course will provide students with an introduction to advanced topics in partial differential equations in a variety of settings. Topics may include: reaction diffusion systems, pattern formation, numerical methods, applications to physical sciences, variational methods, Sobolev Theory.

CROSS-LISTING: MATH 4230.03

CREDIT HOURS: 3

Most mathematical models of physical systems cannot be solved exactly. Often such systems have a naturally occurring small parameter which may be exploited using asymptotic analysis techniques. In this course, we will study a variety of physical systems which illustrate many of the common approaches used in asymptotic analysis. Focus will be on applications to ordinary and partial differential equations.

CROSS-LISTING: MATH 4250.03

CREDIT HOURS: 3

Various graph algorithms will be presented and analyzed. Specifically we will treat the algorithms for the problems: minimum spanning tree, shortest path, maximal flow, minimum cost flow, maximum matching. For each problem, various algorithms will be presented and compared. The link with Linear Programming, especially LP-Duality, will receive special attention.

PREREQUISITES: Some knowledge of linear programming and the theory of algorithms is recommended.

CROSS-LISTING: MATH 4320.03

CREDIT HOURS: 3

This course is intended for math and computer science students. Items to be selected from the following topics: graphs and matrices, graphs and groups, network analysis, extremal graph theory, enumeration problems, and algebraic methods in graph theory.

CROSS-LISTING: MATH 4330.03, CSCI 4115.03

CREDIT HOURS: 3

This course will cover current research in combinatorics. Selected topics may include: graph polynomials, simplicial complexes, partial orders, enumeration problems and algebraic methods in combinatorics.

FORMAT: Lecture

FORMAT COMMENTS: Lecture

LECTURE HOURS PER WEEK: 3

LAB HOURS PER WEEK: 0

TUTORIAL HOURS PER WEEK: 0

PREREQUISITES: Familiarity with basic graph theory (materials covered in Math 3330/CSCI 3110) is recommended.

EXCLUSIONS: MATH 4331

CREDIT HOURS: 3

This course will cover basics of probability and stochastic processes, and then focus on areas where probability and combinatorics interact. Topics include: probabilistic method, stochastic graph models for complex networks, probabilistic algorithms. Probabilistic techniques include: expectation and concentration of random variables, stochastic processes, conditional expectation, Markov chains, martingales, branching processes.

CROSS-LISTING: MATH 4340.03

CREDIT HOURS: 3

This course introduces a common framework for combinatorial structures (graphs, diagraphs, hypergraphs, posets, preorders, lattices, finite topologies, simplicial complexes), with an emphasis on how to model these structures with other fields of mathematics, such as matrix theory and linear algebra, commutative algebra, topology, analysis, probability and logic.

CROSS-LISTING: MATH 4360.03

CREDIT HOURS: 3

A self-contained introduction to cosmology will be given and no prior knowledge of differential geometry or general relativity will be assumed (although some knowledge of elementary differential equations will be useful). A cosmological model is a model of the universe, as a whole, on the largest scales; the emphasis of the course will be on the modelling aspects of cosmology.

CROSS-LISTING: MATH 4410.03, PHYC 4660.03/5660.03

CREDIT HOURS: 3

This course will cover the basic elements of Lp-spaces, convolution, interpolation, maximal functions, Fourier analysis of functions, and the theory of generalized functions, or distributions. Further topics may include L2-Sobolev spaces, boundary values of harmonic functions, spherical harmonics, singular integral operators, or multipliers.

EXCLUSIONS: MATH 4500.03

CREDIT HOURS: 3

This course is a self-contained introduction to manifold theory. Topics include: elements of surface theory, the tangent space, vector fields, differential forms and more general tensors, the Lie derivative, connections, Riemannian geometry, applications in mechanics and general relativity.

CROSS-LISTING: MATH 4530.03

CREDIT HOURS: 3

This course is an introduction to the methods of modern applied analysis. Topics include: Fourier series, tensor calculus, and the calculus of variations. The course is suitable for advanced undergraduates and the graduate students specializing in applied mathematics, relativity, differential geometry, and differential equations.

EXCLUSIONS: MATH 4540.03

CREDIT HOURS: 3

A review of differential geometry will be given followed by an introduction to the general theory of relativity. Various topics will be discussed, including: linearized theory and gravitational radiation, spherically symmetric metrics and the Schwarzchild solution, gravitational collapse, black holes, and cosmology.

CROSS-LISTING: MATH 4650.03, PHYC 4650.03/5650.03

CREDIT HOURS: 3

This is a course on formal languages and computational models. Topics covered include finite automata, pushdown automata, Turing machines, undecidability and recursive and recursively enumerable functions. Some applications to computer science are also discussed such as compiler design and text processing.

CROSS-LISTING: MATH 4660.03, CSCI 4112.03

CREDIT HOURS: 3

This course covers topics of current interest in logic and/or the foundations of computation. Suitable topics include: formal logic, soundness and completeness, GĂ¶del's incompleteness theorem, formal set theory, the Zermelo-Fraenkel axioms, non-standard models, independence of axioms, lambda calculus and foundations of functional programming languages, proof theory, semantics.

CROSS-LISTING: MATH 4680.03

CREDIT HOURS: 3

This course looks at two-player games of strategy where there are no chance devices and both players have perfect information. The surprising mathematical structure underlying these games will be introduced along with the evaluation scheme and its application to specific games in the classes of hot, all-small and impartial games.

CROSS-LISTING: MATH 4900.03

CREDIT HOURS: 0

Weekly seminars focused on topics in Category Theory.

NOTES: Students are expected to register in this course each term, receiving a grade of IP until all course requirements are completed.

CREDIT HOURS: 0

Weekly seminars focused on topics in Combinatorics and Game Theory.

NOTES: Students are expected to register in this course each term, receiving a grade of IP until all course requirements are completed.

CREDIT HOURS: 0

Weekly seminars focused on topics in Relativity.

NOTES: Students are expected to register in this course each term, receiving a grade of IP until all course requirements are completed.

CREDIT HOURS: 0

Weekly seminars focused on topics in Analysis.

NOTES: Students are expected to register in this course each term, receiving a grade of IP until all course requirements are completed.

CREDIT HOURS: 0

Weekly seminars focused on topics in Applied Math.

NOTES: Students are expected to register in this course each term, receiving a grade of IP until all course requirements are completed.

**MATH 5230 Partial Differential Equations**CREDIT HOURS: 3

This course will provide students with an introduction to advanced topics in partial differential equations in a variety of settings. Topics may include: reaction diffusion systems, pattern formation, numerical methods, applications to physical sciences, variational methods, Sobolev Theory.

CROSS-LISTING: MATH 4230.03

**MATH 5250 Asymptotic Analysis**CREDIT HOURS: 3

Most mathematical models of physical systems cannot be solved exactly. Often such systems have a naturally occurring small parameter which may be exploited using asymptotic analysis techniques. In this course, we will study a variety of physical systems which illustrate many of the common approaches used in asymptotic analysis. Focus will be on applications to ordinary and partial differential equations.

CROSS-LISTING: MATH 4250.03

**MATH 5320 Combinatorial Optimization**CREDIT HOURS: 3

Various graph algorithms will be presented and analyzed. Specifically we will treat the algorithms for the problems: minimum spanning tree, shortest path, maximal flow, minimum cost flow, maximum matching. For each problem, various algorithms will be presented and compared. The link with Linear Programming, especially LP-Duality, will receive special attention.

PREREQUISITES: Some knowledge of linear programming and the theory of algorithms is recommended.

CROSS-LISTING: MATH 4320.03

**MATH 5330 Topics in Graph Theory**CREDIT HOURS: 3

This course is intended for math and computer science students. Items to be selected from the following topics: graphs and matrices, graphs and groups, network analysis, extremal graph theory, enumeration problems, and algebraic methods in graph theory.

CROSS-LISTING: MATH 4330.03, CSCI 4115.03

**MATH 5331 TOPICS IN COMBINATORICS**CREDIT HOURS: 3

This course will cover current research in combinatorics. Selected topics may include: graph polynomials, simplicial complexes, partial orders, enumeration problems and algebraic methods in combinatorics.

FORMAT: Lecture

FORMAT COMMENTS: Lecture

LECTURE HOURS PER WEEK: 3

LAB HOURS PER WEEK: 0

TUTORIAL HOURS PER WEEK: 0

PREREQUISITES: Familiarity with basic graph theory (materials covered in Math 3330/CSCI 3110) is recommended.

EXCLUSIONS: MATH 4331

**MATH 5340 Discrete Random Structures**CREDIT HOURS: 3

This course will cover basics of probability and stochastic processes, and then focus on areas where probability and combinatorics interact. Topics include: probabilistic method, stochastic graph models for complex networks, probabilistic algorithms. Probabilistic techniques include: expectation and concentration of random variables, stochastic processes, conditional expectation, Markov chains, martingales, branching processes.

CROSS-LISTING: MATH 4340.03

**MATH 5360 Combinatorial Modelling**CREDIT HOURS: 3

This course introduces a common framework for combinatorial structures (graphs, diagraphs, hypergraphs, posets, preorders, lattices, finite topologies, simplicial complexes), with an emphasis on how to model these structures with other fields of mathematics, such as matrix theory and linear algebra, commutative algebra, topology, analysis, probability and logic.

CROSS-LISTING: MATH 4360.03

**MATH 5410 Cosmology**CREDIT HOURS: 3

A self-contained introduction to cosmology will be given and no prior knowledge of differential geometry or general relativity will be assumed (although some knowledge of elementary differential equations will be useful). A cosmological model is a model of the universe, as a whole, on the largest scales; the emphasis of the course will be on the modelling aspects of cosmology.

CROSS-LISTING: MATH 4410.03, PHYC 4660.03/5660.03

**MATH 5500 Introduction to Harmonic Analysis**CREDIT HOURS: 3

This course will cover the basic elements of Lp-spaces, convolution, interpolation, maximal functions, Fourier analysis of functions, and the theory of generalized functions, or distributions. Further topics may include L2-Sobolev spaces, boundary values of harmonic functions, spherical harmonics, singular integral operators, or multipliers.

EXCLUSIONS: MATH 4500.03

**MATH 5530 Differential Geometry**CREDIT HOURS: 3

This course is a self-contained introduction to manifold theory. Topics include: elements of surface theory, the tangent space, vector fields, differential forms and more general tensors, the Lie derivative, connections, Riemannian geometry, applications in mechanics and general relativity.

CROSS-LISTING: MATH 4530.03

**MATH 5540 Applied Analysis**CREDIT HOURS: 3

This course is an introduction to the methods of modern applied analysis. Topics include: Fourier series, tensor calculus, and the calculus of variations. The course is suitable for advanced undergraduates and the graduate students specializing in applied mathematics, relativity, differential geometry, and differential equations.

EXCLUSIONS: MATH 4540.03

**MATH 5650 General Relativity**CREDIT HOURS: 3

A review of differential geometry will be given followed by an introduction to the general theory of relativity. Various topics will be discussed, including: linearized theory and gravitational radiation, spherically symmetric metrics and the Schwarzchild solution, gravitational collapse, black holes, and cosmology.

CROSS-LISTING: MATH 4650.03, PHYC 4650.03/5650.03

**MATH 5660 Theory of Computation**CREDIT HOURS: 3

This is a course on formal languages and computational models. Topics covered include finite automata, pushdown automata, Turing machines, undecidability and recursive and recursively enumerable functions. Some applications to computer science are also discussed such as compiler design and text processing.

CROSS-LISTING: MATH 4660.03, CSCI 4112.03

**MATH 5680 Topics in Logic and Computation**CREDIT HOURS: 3

This course covers topics of current interest in logic and/or the foundations of computation. Suitable topics include: formal logic, soundness and completeness, GĂ¶del's incompleteness theorem, formal set theory, the Zermelo-Fraenkel axioms, non-standard models, independence of axioms, lambda calculus and foundations of functional programming languages, proof theory, semantics.

CROSS-LISTING: MATH 4680.03

**MATH 5900 Combinatorial Game Theory**CREDIT HOURS: 3

This course looks at two-player games of strategy where there are no chance devices and both players have perfect information. The surprising mathematical structure underlying these games will be introduced along with the evaluation scheme and its application to specific games in the classes of hot, all-small and impartial games.

CROSS-LISTING: MATH 4900.03

**MATH 7030 Category Theory Seminar**CREDIT HOURS: 0

Weekly seminars focused on topics in Category Theory.

NOTES: Students are expected to register in this course each term, receiving a grade of IP until all course requirements are completed.

**MATH 7050 Combinatorics and Game Theory Seminar**CREDIT HOURS: 0

Weekly seminars focused on topics in Combinatorics and Game Theory.

NOTES: Students are expected to register in this course each term, receiving a grade of IP until all course requirements are completed.

**MATH 7060 Relativity Seminar**CREDIT HOURS: 0

Weekly seminars focused on topics in Relativity.

NOTES: Students are expected to register in this course each term, receiving a grade of IP until all course requirements are completed.

**MATH 7190 Analysis & AMP Seminar**CREDIT HOURS: 0

Weekly seminars focused on topics in Analysis.

NOTES: Students are expected to register in this course each term, receiving a grade of IP until all course requirements are completed.

**MATH 7400 Applied Math Seminar**CREDIT HOURS: 0

Weekly seminars focused on topics in Applied Math.

NOTES: Students are expected to register in this course each term, receiving a grade of IP until all course requirements are completed.