University General Course Catalog 2021-2022 
    
    May 20, 2024  
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8. Course Descriptions

Note: Sequencing rules in effect for many Math courses prohibit students from earning credit for a lower numbered Math course after receiving credit for a higher numbered Math course. Sequencing rules are included in the course descriptions of applicable courses.
 

Mathematics
  

  • MATH 422 - Optimal Analysis

    (3 units)
    Analysis of extrema of real-valued functions and functionals with applications. Introduction to calculus of variations and optimal control.

    Grading Basis: Graded
    Units of Lecture: 3
    Student Learning Outcomes
    Upon completion of this course, students will be able to:
    1. derive extremal equations and solve some practically important variational problems in single and multiple independent variables including ones with unknown boundary data and integration limits.
    2. solve Isoperimetric and other constrained variational problems using Lagrange multipliers, and apply direct methods to solve variational problems.
    3. apply Hamiltonian principle to derivation of canonical equations and Hamilton-Jacobi equation, and use maximum principle for solution of some optimal control problems.


    Click here for course scheduling information. | Check course textbook information
  

  • MATH 429 - Topics in Applied Analysis

    (1 to 3 units)
    Variable content chosen from such topics as: integral transforms, approximation of functions, nonlinear mathematics, stability theory, matrix exponentials.

    Maximum units a student may earn: 6

    Grading Basis: Graded
    Student Learning Outcomes
    Upon completion of this course, students will be able to:
    1. demonstrate an advanced level of competency in the area of Applied Analysis.


    Click here for course scheduling information. | Check course textbook information
  

  • MATH 430 - Linear Algebra II

    (3 units)
    Vector spaces; duality, direct sums; linear maps: eigenvalues, eigenvectors, rational and Jordan forms; bilinear maps, quadratic forms; inner product spaces: symmetric, skewsymmetric, orthogonal maps, spectral theorem.

    Prerequisite(s): MATH 330 .

    Grading Basis: Graded
    Units of Lecture: 3
    Offered: Every Fall - Odd Years

    Student Learning Outcomes
    Upon completion of this course, students will be able to:
    1. demonstrate understanding of the vocabulary of Linear Algebra at an advanced level.
    2. demonstrate competence at an advanced level by rigorously proving results from Linear Algebra I.
    3. work with self-adjoint, normal and positive operators on inner product spaces.
    4. work with special decompositions and forms of matrices: e.g. the singular value decomposition, Jordan form.


    Click here for course scheduling information. | Check course textbook information
  

  • MATH 439 - Topics in Algebra

    (1 to 3 units)
    Variable content chosen from such topics as Galois theory, number theory topological groups, combinatorial analysis, theory of graphs.

    Maximum units a student may earn: 6

    Grading Basis: Graded
    Student Learning Outcomes
    Upon completion of this course, students will be able to:
    1. demonstrate an advanced level of competency in the area of Algebra.


    Click here for course scheduling information. | Check course textbook information
  

  • MATH 440 - Topology

    (3 units)
    General topological spaces, continuity, compact and locally compact spaces, connectedness, path connectedness, product and quotient topologies, countability and separation axioms, metric spaces.

    Prerequisite(s): MATH 310 .

    Grading Basis: Graded
    Units of Lecture: 3
    Offered: Every Spring

    Student Learning Outcomes
    Upon completion of this course, students will be able to:
    1. demonstrate an understanding of topological spaces and continuous functions through proofs and examples.?
    2. analyze topological properties such as compactness, connectedness and path-connectedness, and prove their consequences including the Extreme Value and Intermediate Value Theorem from Calculus.?
    3. analyze continuity or discontinuity of functions defined on a quotient space.


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  • MATH 441 - Intro Algebraic Topology

    (3 units)
    Topological spaces, functors, homotopy, the fundamental group, covering spaces, higher homotopy groups, simplicial complexes, homology theories.

    Prerequisite(s): MATH 440 . Corequisite(s): MATH 331 .

    Grading Basis: Graded
    Units of Lecture: 3
    Offered: Every Fall - Even Years

    Student Learning Outcomes
    Upon completion of this course, students will be able to:
    1. demonstrate understanding of covering spaces for a topological space through proofs and examples.
    2. use functorial properties of homotopy groups to convert topological problems to algebraic ones, in order to solve them.
    3. compute or describe the fundamental group of a topological space by applying the Seifert-van Kampen theorem.


    Click here for course scheduling information. | Check course textbook information
  

  • MATH 442 - Differential Geometry

    (3 units)
    Geometry of curves and surfaces in space; Frenet’s formulas; Cartan’s frame fields, Gaussian curvature; intrinsic geometry of surface; congruence of surfaces; the Gauss-Bonnet theorem.

    Prerequisite(s): MATH 295  with a “C” or better; MATH 285  with a “C” or better; MATH 330  with a “C” or better. Recommended corequisite: MATH 310 . 

    Grading Basis: Graded
    Units of Lecture: 3
    Offered: Every Fall - Odd Years

    Student Learning Outcomes
    Upon completion of this course, students will be able to:
    1. analyze the intrinsic geometry of curves in space, such as arclength and curvature.
    2. analyze the intrinsic geometry of surfaces in space, such as surface area, Gaussian curvature.
    3. determine whether two surfaces cannot be related by an isometric bijection, due to different intrinsic geometry.
    4. use the Gauss-Bonnet theorem to relate the geometry of an oriented surface without boundary to its topology.


    Click here for course scheduling information. | Check course textbook information
  

  • MATH 449 - Topics in Geometry and Topology

    (1 to 3 units)
    Variable content chosen from such topics as differential topology, algebraic topology, convexity, topological vector spaces. Mathematical structures of special relativity.

    Maximum units a student may earn: 6

    Grading Basis: Graded
    Student Learning Outcomes
    Upon completion of this course, students will be able to:
    1. demonstrate an advanced level of competency in geometry and topology.


    Click here for course scheduling information. | Check course textbook information
  

  • MATH 455 - Elementary Theory of Numbers I

    (3 units)
    Congruences, primitive roots, arithmetic functions, quadratic reciprocity, distribution of prime numbers, diophantine equations, rational approximations, algebraic numbers.

    Prerequisite(s): MATH 331  with a “C” or better.

    Grading Basis: Graded
    Units of Lecture: 3
    Offered: Every Spring - Odd Years

    Student Learning Outcomes
    Upon completion of this course, students will be able to:
    1. demonstrate knowledge of divisibility, congruences and primitive roots.
    2. solve Diophantine equations using the theory of congruences, diophantine approximation and algebraic numbers.
    3. analyze the distribution of primes and mean values of arithmetic functions.


    Click here for course scheduling information. | Check course textbook information
  

  • MATH 466 - Numerical Methods I

    (3 units)
    Numerical solution of linear systems, including linear programming; iterative solutions of non-linear equations; computation of eigenvalues and eigenvectors, matrix diagonalization. (CS 466 and MATH 466 are cross-listed; credit may be earned in one of the two.)

    Prerequisite(s): MATH 330 .

    Grading Basis: Graded
    Units of Lecture: 3
    Offered: Every Fall

    Student Learning Outcomes
    Upon completion of this course, students will be able to:
    1. implement a numerical method to solve a nonlinear equation: e.g. bisection method, Newton’s method.
    2. solve linear systems using direct and iterative methods.
    3. construct interpolating functions.


    Click here for course scheduling information. | Check course textbook information
  

  • MATH 467 - Numerical Methods II

    (3 units)
    Numerical differentiation and integration; numerical solution of ordinary differential equations, two point boundary value problems; difference methods for partial differential equations. (CS 467 and MATH 467 are cross-listed; credit may be earned in one of the two.)

    Prerequisite(s): MATH 285 .

    Grading Basis: Graded
    Units of Lecture: 3
    Offered: Every Spring

    Student Learning Outcomes
    Upon completion of this course, students will be able to:
    1. use Taylor and Runge-Kutta methods to solve IVP’s for ODE’s.
    2. use the shooting and finite difference methods to solve BVP’s for ODE’s.
    3. use Numerical techniques to solve elliptic, parabolic and hyperbolic PDE’s.


    Click here for course scheduling information. | Check course textbook information
  

  • MATH 475 - Euclidean and Non-Euclidean Geometry

    (3 units)
    Axiom systems, models, independence, consistency; incidence, distance, betweenness, congruence, convexity; inequalities, quadrilaterals, limit triangles, the non-Euclidean geometry of Bolyai-Lobatchevsky.

    Prerequisite(s): MATH 373 .

    Grading Basis: Graded
    Units of Lecture: 3
    Student Learning Outcomes
    Upon completion of this course, students will be able to:
    1. prove properties of lines, angles, and circles from the Euclidean axioms.
    2. prove properties of lines, angles and circles in non-Euclidean geometry.?
    3. distinguish between geometric properties that depend on the Euclidean parallel postulate and those that are independent of any parallel postulate assumptions.
    4. describe the logical consistency of geometries using models.


    Click here for course scheduling information. | Check course textbook information
  

  • MATH 485 - Graph Theory & Combinator

    (3 units)
    Counting rules; generating functions; recurrence relations; inclusion-exclusion; pigeonhole principle; Ramsey theory; fundamental graph theory concepts (connectedness, coloring, planarity); Eulerian/Hamiltonian chains and circuits; matching.

    Recommended Preparation: MATH 330 .

    Grading Basis: Graded
    Units of Lecture: 3
    Offered: Every Fall

    Student Learning Outcomes
    Upon completion of this course, students will be able to:
    1. solve simple and complex counting problems, by using the addition and product rules, recognizing permutation/combinations with and without replacement, rephrasing as occupancy problems, and/or using the inclusion-exclusion principle/derangements.
    2. demonstrate an understanding of the concept of computational complexity for algorithms, including the use of “Big O” notation.
    3. demonstrate an understanding of the main concepts of graph theory, including graphs versus digraphs, connectedness, graph coloring, planarity, as well as the properties of bipartite graphs, complete graphs, and trees.
    4. demonstrate knowledge of some of the great historical problems and results in graph theory/combinatorics, including the four-color problem, the Konigsberg bridge problem, Euler’s formula, the travelling salesman problem, the hatcheck problem, Kuratowski’s theorem, Ramsey’s theorem, and the solution to the Fibonacci recursion.
    5. make simple “discrete” arguments/proofs, such as using mathematical induction or making “combinatorial arguments”.


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  • MATH 486 - Game Theory

    (3 units)
    Extensive form games; Nash, perfect equilibrium; matrix/bimatrix games; minmax theorem; TU/NTU solutions; marriage, college admissions, and housewrapping games; core; Shapley value; power indices.

    Recommended Preparation: MATH 330 .

    Grading Basis: Graded
    Units of Lecture: 3
    Offered: Every Fall

    Student Learning Outcomes
    Upon completion of this course, students will be able to:
    1. model real world problems from the social and biological sciences as cooperative or noncooperative games, and to choose appropriate solution concepts to analyze them.
    2. solve strategic form matrix and bimatrix games for both minimax solution and Nash equilibria, use backward induction for finding perfect Nash equilibria in perfect-information extensive form games, find the TU solution for bimatrix games, run the Deferred Acceptance Procedure and Top Trading Cycle algorithms for ordinal preference games, and solve for the core and Shapley Value for n-player TU games.
    3. demonstrate an understanding of the underlying theory behind the models, including the minimax theorem, Shapley-Bondareva theorem, Nash’s theorem, the Folk Theorem for repeated games, and the relationship between some of these and linear programming.
    4. demonstrate an understanding of some of the subtleties of game theoretic modelling, such as the role and modelling of information, the advantages and disadvantages of modelling using strategic vs extensive form, the difference between cooperative and noncooperative games, and the implications of the “TU-assumption” for cooperative games.


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  • MATH 487 - Deterministic Operations Research

    (3 units) CO13
    Linear programming and duality theory, integer programming, dynamic programming, PERT scheduling, EOQ inventory model, and nonlinear programming. Emphasis on both theory and applications.

    Prerequisite(s): General Education courses (CO1-CO3) completed; at least 3 courses from CO4-CO8 completed; Junior or Senior standing. Recommended Preparation: MATH 330 .

    Grading Basis: Graded
    Units of Lecture: 3
    Offered: Every Spring

    Student Learning Outcomes
    Upon completion of this course, students will be able to:
    1. use methods of deterministic operations research to model real-world situations, and interpret the results to reach sound conclusions.
    2. communicate, in written form, the results of a model in the context of current thought on the situation being modeled.
    3. distinguish between sound and unsound interpretations of model results applied to issues affecting society.
    4. analyze a problem’s societal context and the impact of context on sound interpretation of mathematical models applied to real-world situations.


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  • MATH 488 - Partial Differential Equations

    (3 units)
    Partial differential equations; first order equations, initial and mixed boundary-value problems for the second order Laplace, heat and wave equations; finite difference approximation.

    Prerequisite(s): MATH 285  with a “C” or better; MATH 330  with a “C” or better.

    Grading Basis: Graded
    Units of Lecture: 3
    Offered: Every Spring

    Student Learning Outcomes
    Upon completion of this course, students will be able to:
    1. apply separation of variables to solve a partial differential equation.
    2. solve second order constant coefficient partial differential equations by applying transforms.
    3. apply the method of characteristics to solve first order partial differential equations.


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  • MATH 490 - Internship

    (1 to 6 units)
    Individual study for the purposes of obtaining credit for high math content work related experience.

    Maximum units a student may earn: 6

    Grading Basis: Graded
    Student Learning Outcomes
    Upon completion of this course, students will be able to:
    1. communicate effectively and professionally and work in teams with colleagues and supervisors in a workplace.
    2. translate business questions and issues into mathematical/statistical problems,?and communicate the technical solutions to the business audience.
    3. practice professional behavior and communications’ standards including verbal and written communication, timeliness, and ethics.


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  • MATH 495 - Introduction to Algebraic Combinatorics

    (3 units)
    Walks in Graphs, Posets and Sperner Property, Partitions of integers, Enumeration under Group Action, Young Tableaux, Enumeration problems in Graph Theory: Spanning Trees, Eulerian circuits, Matchings and Path-systems, Vector Spaces in Graphs, Dimension and Polynomial methods in Combinatorics, Algebraic Combinatorics Gems.

    Maximum units a student may earn: 3

    Prerequisite(s): MATH 330. Recommended Preparation: MATH 331; MATH 485.

    Grading Basis: Graded
    Units of Lecture: 3
    Offered: Every Spring - Even Years

    Student Learning Outcomes
    Upon completion of this course, students will be able to:
    1. demonstrate understanding of combinatorial properties (like Sperner property, unimodality) and its implications in various posets.
    2. apply Group-theoretic results like, Burnside’s Lemma and Polya-Redfield counting for enumeration problems involving symmetries.
    3. demonstrate use of determinants in some enumeration problems in Graph Theory and its applications.
    4. apply other linear-algebraic arguments like rank, dimension, orthogonality, polynomials, etc in various combinatorial problems.


    Click here for course scheduling information. | Check course textbook information
  

  • MATH 499 - Independent Study

    (1 to 3 units)
    Individual study conducted under the direction of a faculty member. 

    Maximum units a student may earn: 6

    Grading Basis: Graded
    Student Learning Outcomes
    Upon completion of this course, students will be able to:
    1. demonstrate an advanced level of competency in the subject of study of this course.


    Click here for course scheduling information. | Check course textbook information
  

  • MATH 601 - Set Theory

    (3 units)
    Formalism, inference, axiomatic set theory, unicity, pairs, relations, functions ordinals, recursive definition, maximality, well ordering, choice, regularity, equinumerosity, cardinal arithmetic.

    Grading Basis: Graded
    Units of Lecture: 3
    Student Learning Outcomes
    Upon completion of this course, students will be able to:
    1. discuss ZF and ZFC.
    2. demonstrate understanding of the terminology of set theory.
    3. explain concepts and prove facts about ordinals and cardinals.


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  • MATH 610 - Complex Analysis

    (3 units)
    Complex numbers, analytic and harmonic functions. Cauchy-Riemann equations, complex integration, the Cauchy integral formula, elementary conformal mappings. Laurent series, calculus of residues.

    Grading Basis: Graded
    Units of Lecture: 3
    Offered: Every Fall - Even Years

    Student Learning Outcomes
    Upon completion of this course, students will be able to:
    1. work with holomorphic and harmonic functions.
    2. demonstrate an understanding of the complex logarithm and complex roots.
    3. integrate complex valued functions using residues.


    Click here for course scheduling information. | Check course textbook information
  

  • MATH 611 - Real Analysis

    (3 units)
    Continuity, monotonicity, differentiability; uniform convergence and continuity and differentiability; Stone-Weierstrass Theorem; multivariable functions, linear transformations, differentiation, inverse and implicit functions, Jacobians and change of variable; Lebesgue measure and integration.

    Grading Basis: Graded
    Units of Lecture: 3
    Student Learning Outcomes
    Upon completion of this course, students will be able to:
    1. distinguish between different modes of convergence.
    2. demonstrate understanding of the Stone Weierstrass theorem.
    3. apply the inverse and implicit function theorems.
    4. demonstrate understanding of the basic theorems of Lebesgue integration.


    Click here for course scheduling information. | Check course textbook information
  

  • MATH 612 - Functional Analysis

    (3 units)
    Normed vector spaces, Banach and Hilbert spaces, linear functionals and operators, the Hahn-Banach, closed graph and uniform boundedness theorems with applications, dual spaces, self adjoint operators, compact operators.

    Grading Basis: Graded
    Units of Lecture: 3
    Offered: Every Spring - Even Years

    Student Learning Outcomes
    Upon completion of this course, students will be able to:
    1. demonstrate an understanding of operators and functionals on Banach spaces and Hilbert spaces.
    2. work with the central concepts and theorems of functional analysis: e.g. the Hahn-Banach theorem, uniform boundedness and open mapping theorems.
    3. apply the spectral theory of bounded self-adjoint operators.


    Click here for course scheduling information. | Check course textbook information
  

  • MATH 619 - Topics in Analysis

    (1 to 3 units)
    Variable content chosen from such topics as differential forms, analytic functions, distribution theory, measure and integration, constructive analysis.

    Maximum units a student may earn: 6

    Grading Basis: Graded
    Student Learning Outcomes
    Upon completion of this course, students will be able to:
    1. demonstrate an advanced level of competency in the area of Analysis.


    Click here for course scheduling information. | Check course textbook information
  

  • MATH 620 - Mathematical Modeling

    (3 units)
    Formulation, analysis and critique of methods of mathematical modeling; selected applications in physics, biology, economics, political science and other fields.

    Grading Basis: Graded
    Units of Lecture: 3
    Offered: Every Fall and Spring

    Student Learning Outcomes
    Upon completion of this course, students will be able to:
    1. choose and apply key mathematical and statistical techniques for solving problems in a diverse collection of scientific disciplines.
    2. organize and clean data; critically assess the origin of the data and method of data analysis.
    3. interpret the results of the modeling process to reach sound scientific conclusions within the problem’s economic, scientific, and social context.
    4. propose a project (individually or in a group) and devise strategies and practices to do the research work that will lead, with the support of computational software (e.g. Maple, Mathematica, R, Matlab), to the writing of a technical report using professional typesetting software (e.g., LaTeX).


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  • MATH 621 - Introduction To Applied Dynamical Systems

    (3 units)
    Continuous and discrete dynamical systems; fixed points; limit cycles; stability analysis; bifurcation analysis; numerical solutions; model derivation

    Grading Basis: Graded
    Units of Lecture: 3
    Student Learning Outcomes
    Upon completion of this course, students will be able to:
    1. conduct an equilibrium stability analysis for ODE models and discrete maps.
    2. recognize common equilibrium bifurcations.
    3. apply computational techniques to the of study dynamical systems models.
    4. complete a research project and effectively communicate their findings.


    Click here for course scheduling information. | Check course textbook information
  

  • MATH 622 - Optimal Analysis

    (3 units)
    Analysis of extrema of real-valued functions and functionals with applications. Introduction to calculus of variations and optimal control.

    Grading Basis: Graded
    Units of Lecture: 3
    Student Learning Outcomes
    Upon completion of this course, students will be able to:
    1. derive extremal equations and solve some practically important variational problems in single and multiple independent variables including ones with unknown boundary data and integration limits.
    2. solve Isoperimetric and other constrained variational problems using Lagrange multipliers, and apply direct methods to solve variational problems.
    3. apply Hamiltonian principle to derivation of canonical equations and Hamilton-Jacobi equation, and use maximum principle for solution of some optimal control problems.


    Click here for course scheduling information. | Check course textbook information
  

  • MATH 629 - Topics in Applied Analysis

    (1 to 3 units)
    Variable content chosen from such topics as: integral transforms, approximation of functions, nonlinear mathematics, stability theory, matrix exponentials.

    Maximum units a student may earn: 6

    Grading Basis: Graded
    Student Learning Outcomes
    Upon completion of this course, students will be able to:
    1. demonstrate an advanced level of competency in the area of Applied Analysis.


    Click here for course scheduling information. | Check course textbook information
  

  • MATH 630 - Linear Algebra II

    (3 units)
    Vector spaces; duality, direct sums; linear maps: eigenvalues, eigenvectors, rational and Jordan forms; bilinear maps, quadratic forms; inner product spaces: symmetric, skewsymmetric, orthogonal maps, spectral theorem.

    Grading Basis: Graded
    Units of Lecture: 3
    Offered: Every Fall - Odd Years

    Student Learning Outcomes
    Upon completion of this course, students will be able to:
    1. demonstrate understanding of the vocabulary of Linear Algebra at an advanced level.
    2. demonstrate competence at an advanced level by rigorously proving results from Linear Algebra I.
    3. work with self-adjoint, normal and positive operators on inner product spaces.
    4. work with special decompositions and forms of matrices: e.g. the singular value decomposition, Jordan form.


    Click here for course scheduling information. | Check course textbook information
  

  • MATH 639 - Topics in Algebra

    (1 to 3 units)
    Variable content chosen from such topics as Galois theory, number theory topological groups, combinatorial analysis, theory of graphs.

    Maximum units a student may earn: 3

    Grading Basis: Graded
    Units of Lecture: X
    Student Learning Outcomes
    Upon completion of this course, students will be able to:
    1. demonstrate an advanced level of competency in the area of Algebra.


    Click here for course scheduling information. | Check course textbook information
  

  • MATH 640 - Topology

    (3 units)
    General topological spaces, continuity, compact and locally compact spaces, connectedness, path connectedness, product and quotient topologies, countability and separation axioms, metric spaces.

    Grading Basis: Graded
    Units of Lecture: 3
    Offered: Every Spring

    Student Learning Outcomes
    Upon completion of this course, students will be able to:
    1. demonstrate an understanding of topological spaces and continuous functions through proofs and examples.?
    2. analyze topological properties such as compactness, connectedness and path-connectedness, and prove their consequences including the Extreme Value and Intermediate Value Theorem from Calculus.?
    3. analyze continuity or discontinuity of functions defined on a quotient space.


    Click here for course scheduling information. | Check course textbook information
  

  • MATH 641 - Intro Algebraic Topology

    (3 units)
    Topological spaces, functors, homotopy, the fundamental group, covering spaces, higher homotopy groups, simplicial complexes, homology theories.

    Grading Basis: Graded
    Units of Lecture: 3
    Offered: Every Fall - Even Years

    Student Learning Outcomes
    Upon completion of this course, students will be able to:
    1. demonstrate understanding of covering spaces for a topological space through proofs and examples.
    2. use functorial properties of homotopy groups to convert topological problems to algebraic ones, in order to solve them.
    3. compute or describe the fundamental group of a topological space by applying the Seifert-van Kampen theorem.


    Click here for course scheduling information. | Check course textbook information
  

  • MATH 642 - Differential Geometry

    (3 units)
    Geometry of curves and surfaces in space; Frenet’s formulas; Cartan’s frame fields, Gaussian curvature; intrinsic geometry of surface; congruence of surfaces; the Gauss-Bonnet theorem.

    Grading Basis: Graded
    Units of Lecture: 3
    Offered: Every Fall - Odd Years

    Student Learning Outcomes
    Upon completion of this course, students will be able to:
    1. analyze the intrinsic geometry of curves in space, such as arclength and curvature.
    2. analyze the intrinsic geometry of surfaces in space, such as surface area, Gaussian curvature.
    3. determine whether two surfaces cannot be related by an isometric bijection, due to different intrinsic geometry.
    4. use the Gauss-Bonnet theorem to relate the geometry of an oriented surface without boundary to its topology.


    Click here for course scheduling information. | Check course textbook information
  

  • MATH 643 - Differential Geometry and Relativity I

    (3 units)
    Manifolds, the tangent bundle, differential forms, exterior differentiation, Lie differentiation, Koszul connections, curvature, torsion, Cartan’s structural equations, integration of differential forms.

    Grading Basis: Graded
    Units of Lecture: 3
    Student Learning Outcomes
    Upon completion of this course, students will be able to:
    1. perform local calculations in differential geometry: covariant derivatives, differential forms, curvature and tensor calculations.?
    2. explain the postulates of General Relativity.
    3. derive geodesic equations in a given spacetime and solve them in special cases.?


    Click here for course scheduling information. | Check course textbook information
  

  • MATH 649 - Topics in Geometry and Topology

    (1 to 3 units)
    Variable content chosen from such topics as differential topology, algebraic topology, convexity, topological vector spaces. Mathematical structures of special relativity.

    Maximum units a student may earn: 6

    Grading Basis: Graded
    Student Learning Outcomes
    Upon completion of this course, students will be able to:
    1. demonstrate an advanced level of competency in geometry and topology.


    Click here for course scheduling information. | Check course textbook information
  

  • MATH 655 - Elementary Theory of Numbers I

    (3 units)
    Congruences, primitive roots, arithmetic functions, quadratic reciprocity, distribution of prime numbers, diophantine equations, rational approximations, algebraic numbers.

    Grading Basis: Graded
    Units of Lecture: 3
    Offered: Every Spring - Odd Years

    Student Learning Outcomes
    Upon completion of this course, students will be able to:
    1. demonstrate knowledge of divisibility, congruences and primitive roots.
    2. solve Diophantine equations using the theory of congruences, diophantine approximation and algebraic numbers.
    3. analyze the distribution of primes and mean values of arithmetic functions.
    4. apply the course concepts to research problems.
    5. synthesize number theoretical methods with other mathematical concepts.


    Click here for course scheduling information. | Check course textbook information
  

  • MATH 659 - Special Topics of Interest in Probability

    (1 to 3 units)
    Variable content based on faculty and student interests.

    Maximum units a student may earn: 9

    Grading Basis: Graded
    Student Learning Outcomes
    Upon completion of this course, students will be able to:
    1. demonstrate understanding of the methods discussed in class.
    2. apply methods discussed in class to research questions.
    3. articulate the relationship of methods discussed in class to the broader field of probability and statistics.


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  • MATH 666 - Numerical Methods I

    (3 units)
    Numerical solution of linear systems, including linear programming; iterative solutions of non-linear equations; computation of eigenvalues and eigenvectors, matrix diagonalization. (CS 666 and MATH 666 are cross-listed; credit may be earned in one of the two.)

    Grading Basis: Graded
    Units of Lecture: 3
    Offered: Every Fall

    Student Learning Outcomes
    Upon completion of this course, students will be able to:
    1. implement a numerical method to solve a nonlinear equation: e.g. bisection method, Newton’s method.
    2. solve linear systems using direct and iterative methods.
    3. construct interpolating functions.


    Click here for course scheduling information. | Check course textbook information
  

  • MATH 667 - Numerical Methods II

    (3 units)
    Numerical differentiation and integration; numerical solution of ordinary differential equations, two point boundary value problems; difference methods for partial differential equations. (CS 667 and MATH 667 are cross-listed; credit may be earned in one of the two.)

    Grading Basis: Graded
    Units of Lecture: 3
    Offered: Every Spring

    Student Learning Outcomes
    Upon completion of this course, students will be able to:
    1. use Taylor and Runge-Kutta methods to solve IVP’s for ODE’s.
    2. use the shooting and finite difference methods to solve BVP’s for ODE’s.
    3. use Numerical techniques to solve elliptic, parabolic and hyperbolic PDE’s.


    Click here for course scheduling information. | Check course textbook information
  

  • MATH 675 - Euclidean and Non-Euclidean Geometry

    (3 units)
    Axiom systems, models, independence, consistency; incidence, distance, betweenness, congruence, convexity; inequalities, quadrilaterals, limit triangles, the non-Euclidean geometry of Bolyai-Lobatchevsky.

    Grading Basis: Graded
    Units of Lecture: 3
    Student Learning Outcomes
    Upon completion of this course, students will be able to:
    1. prove properties of lines, angles, and circles from the Euclidean axioms.
    2. prove properties of lines, angles and circles in non-Euclidean geometry.?
    3. distinguish between geometric properties that depend on the Euclidean parallel postulate and those that are independent of any parallel postulate assumptions.
    4. describe the logical consistency of geometries using models.


    Click here for course scheduling information. | Check course textbook information
  

  • MATH 685 - Graph Theory & Combinator

    (3 units)
    Counting rules; generating functions; recurrence relations; inclusion-exclusion; pigeonhole principle; Ramsey theory; fundamental graph theory concepts (connectedness, coloring, planarity); Eulerian/Hamiltonian chains and circuits; matching.

    Grading Basis: Graded
    Units of Lecture: 3
    Offered: Every Fall

    Student Learning Outcomes
    Upon completion of this course, students will be able to:
    1. solve simple and complex counting problems, by using the addition and product rules, recognizing permutation/combinations with and without replacement, rephrasing as occupancy problems, and/or using the inclusion-exclusion principle/derangements.
    2. demonstrate an understanding of the concept of computational complexity for algorithms, including the use of “Big O” notation.
    3. demonstrate an understanding of the main concepts of graph theory, including graphs versus digraphs, connectedness, graph coloring, planarity, as well as the properties of bipartite graphs, complete graphs, and trees.
    4. demonstrate knowledge of some of the great historical problems and results in graph theory/combinatorics, including the four-color problem, the Konigsberg bridge problem, Euler’s formula, the travelling salesman problem, the hatcheck problem, Kuratowski’s theorem, Ramsey’s theorem, and the solution to the Fibonacci recursion.
    5. make simple “discrete” arguments/proofs, such as using mathematical induction or making “combinatorial arguments”.


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  • MATH 686 - Game Theory

    (3 units)
    Extensive form games; Nash, perfect equilibrium; matrix/bimatrix games; minmax theorem; TU/NTU solutions; marriage, college admissions, and housewrapping games; core; Shapley value; power indices.

    Grading Basis: Graded
    Units of Lecture: 3
    Offered: Every Fall

    Student Learning Outcomes
    Upon completion of this course, students will be able to:
    1. model real world problems from the social and biological sciences as cooperative or noncooperative games, and to choose appropriate solution concepts to analyze them.
    2. solve strategic form matrix and bimatrix games for both minimax solution and Nash equilibria, use backward induction for finding perfect Nash equilibria in perfect-information extensive form games, find the TU solution for bimatrix games, run the Deferred Acceptance Procedure and Top Trading Cycle algorithms for ordinal preference games, and solve for the core and Shapley Value for n-player TU games.
    3. demonstrate an understanding of the underlying theory behind the models, including the minimax theorem, Shapley-Bondareva theorem, Nash’s theorem, the Folk Theorem for repeated games, and the relationship between some of these and linear programming.
    4. demonstrate an understanding of some of the subtleties of game theoretic modelling, such as the role and modelling of information, the advantages and disadvantages of modelling using strategic vs extensive form, the difference between cooperative and noncooperative games, and the implications of the “TU-assumption” for cooperative games.


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  • MATH 687 - Deterministic Operations Research

    (3 units)
    Linear programming and duality theory, integer programming, dynamic programming, PERT scheduling, EOQ inventory model, and nonlinear programming. Emphasis on both theory and applications.

    Grading Basis: Graded
    Units of Lecture: 3
    Offered: Every Spring

    Student Learning Outcomes
    Upon completion of this course, students will be able to:
    1. use methods of deterministic operations research to model real-world situations, and interpret the results to reach sound conclusions.
    2. communicate, in written form, the results of a model in the context of current thought on the situation being modeled.
    3. distinguish between sound and unsound interpretations of model results applied to issues affecting society.
    4. analyze a problem’s societal context and the impact of context on sound interpretation of mathematical models applied to real-world situations.


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  • MATH 688 - Partial Differential Equations

    (3 units)
    Partial differential equations; first order equations, initial and mixed boundary-value problems for the second order Laplace, heat and wave equations; finite difference approximation.

    Grading Basis: Graded
    Units of Lecture: 3
    Student Learning Outcomes
    Upon completion of this course, students will be able to:
    1. apply separation of variables to solve a partial differential equation.
    2. solve second order constant coefficient partial differential equations by applying transforms.
    3. apply the method of characteristics to solve first order partial differential equations.


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  • MATH 690 - Internship

    (1 to 6 units)
    Individual study for the purposes of obtaining credit for high math content work related experience. Maximum of 6 credits except under special circumstances.

    Maximum units a student may earn: 6

    Grading Basis: Graded
    Units of Internship/Practicum: X
    Student Learning Outcomes
    Upon completion of this course, students will be able to:
    1. communicate effectively and professionally and work in teams with colleagues and supervisors in a workplace.
    2. translate business questions and issues into mathematical/statistical problems,?and communicate the technical solutions to the business audience.
    3. practice professional behavior and communications’ standards including verbal and written communication, timeliness, and ethics.


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  • MATH 695 - Introduction to Algebraic Combinatorics

    (3 units)
    Walks in Graphs, Posets and Sperner Property, Partitions of integers, Enumeration under Group Action, Young Tableaux, Enumeration problems in Graph Theory: Spanning Trees, Eulerian circuits, Matchings and Path-systems, Vector Spaces in Graphs, Dimension and Polynomial methods in Combinatorics, Algebraic Combinatorics Gems.

    Maximum units a student may earn: 3

    Recommended Preparation: MATH 685.

    Grading Basis: Graded
    Units of Lecture: 3
    Offered: Every Spring - Even Years

    Student Learning Outcomes
    Upon completion of this course, students will be able to:
    1. demonstrate understanding of combinatorial properties (like Sperner property, unimodality) and its implications in various posets.
    2. apply Group-theoretic results like, Burnside’s Lemma and Polya-Redfield counting for enumeration problems involving symmetries.
    3. demonstrate use of determinants in some enumeration problems in Graph Theory and its applications.
    4. apply other linear-algebraic arguments like rank, dimension, orthogonality, polynomials, etc in various combinatorial problems.
    5. use these concepts in their research problems and synthesize combinatorial ideas with other branches of Mathematics.


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  • MATH 699 - Independent Study

    (1 to 3 units)
    Individual study conducted under the direction of a faculty member. Limited to 6 credits except under special circumstances.

    Maximum units a student may earn: 3

    Grading Basis: Graded
    Student Learning Outcomes
    Upon completion of this course, students will be able to:
    1. demonstrate an advanced level of competency in the subject of study of this course.


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  • MATH 701 - Numerical Analysis and Approximation

    (3 units)
    Norms of vectors and matrices, computation of eigenvalues and eigenvectors, matrix transformations, Weierstrass’ approximation theorem, Chebyshev polynomials, best and uniform approximation, splines, approximation in abstract spaces.

    Grading Basis: Graded
    Units of Lecture: 3
    Student Learning Outcomes
    Upon completion of this course, students will be able to:
    1. state and prove basic theorems in numerical linear algebra including Schur’s Theorem, quadratic order of convergence of Newton’s methods and the Gershgorin circle theorem.
    2. demonstrate familiarity common algorithms including the shifted QR algorithm of Francis, the algorithms for computation of matrix norms, Gram Schmidt algorithm, Thomas algorithm and Newton’s divided difference formula.
    3. recall and define terminology including error, condition number, matrix norm, machine epsilon, unitary matrix, Haar subspace, order of convergence positive definite matrix and stability.
    4. write practical computer programs.


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  • MATH 702 - Numerical Analysis and Approximation

    (3 units)
    Norms of vectors & matrices, computation of eigen values and eigen vectors, matrix transformations, weierstrass’ approximation theorem, chebyshev polynomials, best and uniform approximation, splines, approximation in abstract spaces.

    Grading Basis: Graded
    Units of Lecture: 3
    Student Learning Outcomes
    Upon completion of this course, students will be able to:
    1. demonstrate familiarity with the solution of numerical differential equations and understand the classification of PDEs using the terms parabolic, elliptic, and hyperbolic.
    2. study and explain partial differential equations such as Burger’s equation, the non-linear Schroedinger equation and the KdV equation using appropriate numerical algorithms such as Crank-Nicolson, ADI methods, upwind, implicit, explicit, Runge-Kutta, finite differences and psuedo-spectral methods.
    3. demonstrate understanding of concepts such as truncation error, stability, Fourier mode analysis, conserved quantities and dissipation.
    4. state and apply theorems such as the Lax equivalence theorem and the Peano kernel theorem.
    5. demonstrate understanding of algorithmic considerations related to parallel processing and write practical parallel computer programs.


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  • MATH 705 - Applied Functional Analysis

    (3 units)
    This foundational course in the modern methods of applied mathematics and modern computational techniques is geared towards an interdisciplinary audience. Learn to work with finite and infinite-dimensional linear spaces, normed spaces, and inner product spaces including spaces of matrices, polynomials and functions, Banach spaces, Lp-spaces, Hilbert spaces and Sobolev spaces, and linear operators on these spaces.

    Maximum units a student may earn: 3

    Recommended Preparation: MATH 330 .

    Grading Basis: Graded
    Units of Lecture: 3
    Offered: Every Spring - Odd Years

    Student Learning Outcomes
    Upon completion of this course, students will be able to:
    1. work with finite and infinite dimensional linear spaces, normed spaces, and inner product spaces.
    2. perform computations in the spaces of matrices, polynomials and functions.
    3. work with linear operators on Banach spaces, Lp-Spaces and Sobolev spaces.


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  • MATH 713 - Abstract and Real Analysis

    (3 units)
    Lebesgue measure, measurable spaces, integration, convergence theorems, Fubini Theorem, measure–theoretic foundations of probability, topological measures, differentiation. Radon-Nikodym Theorem, Riesz Representation, special topics.

    Grading Basis: Graded
    Units of Lecture: 3
    Offered: Every Fall

    Student Learning Outcomes
    Upon completion of this course, students will be able to:
    1. demonstrate understanding of measure theory.
    2. demonstrate understanding of Lebesgue integration.
    3. demonstrate understanding of the uses of modern theories of measure and integration.


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  • MATH 714 - Abstract and Real Analysis

    (3 units)
    Metric spaces, abstract measures, measurable functions, integration, product measures, Fubini Theorem, topological measures, Haar measure, differentiation, Radon-Nikodym Theorem, linear spaces, Hahn-Banach Theorem, Riez Representation

    Grading Basis: Graded
    Units of Lecture: 3
    Offered: Every Spring - Odd Years

    Student Learning Outcomes
    Upon completion of this course, students will be able to:
    1. demonstrate understanding of advanced topics in measure theory.
    2. demonstrate understanding of advanced topics in modern integration.
    3. demonstrate understanding of the relation between integration and differentiation.


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  • MATH 715 - Complex Function Theory

    (3 units)
    Analytic functions, conformal mappings, Cauchy’s theorem, power series, Laurent series, the Riemann mapping theorem, harmonic functions, subharmonic functions, canonical mappings of multiply connected regions, analytical continuation.

    Grading Basis: Graded
    Units of Lecture: 3
    Offered: Every Spring - Even Years

    Student Learning Outcomes
    Upon completion of this course, students will be able to:
    1. demonstrate knowledge and understanding of the topics listed in the course description and their applications in pure and applied mathematics.


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  • MATH 716 - Complex Function Theory

    (3 units)
    Analytic functions, conformal mappings, Cauchy’s theorem, power series, Laurent series, the Riemann mapping theorem, harmonic functions, subharmonic functions, canonical mappings or multiple connected regions, analytical continuation.

    Grading Basis: Graded
    Units of Lecture: 3
    Student Learning Outcomes
    Upon completion of this course, students will be able to:
    1. demonstrate knowledge and understanding of the topics listed in the course description and their applications in pure and applied mathematics.


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  • MATH 717 - Functional Analysis I

    (3 units)
    The Hahn-Banach Theorem, Open Mapping Theorem and Closed Graph Theorem, the Uniform Boundedness Principle; topological vector spaces, subspaces, extension of linear functionals, convexity and the Krein-Milman theorem.

    Prerequisite(s): MATH 713 .

    Grading Basis: Graded
    Units of Lecture: 3
    Offered: Every Fall - Even Years

    Student Learning Outcomes
    Upon completion of this course, students will be able to:
    1. apply the fundamental results of Hilbert and Banach space theory.
    2. apply the Hahn-Banach theorem, Open Mapping Theorem and Closed Graph theorem.
    3. apply the fundamental theorems to concrete topological vector spaces.


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  • MATH 718 - Functional Analysis II

    (3 units)
    Operators, Banach algebras, Gelfand representation, Gelfand-Naimark Theorem, continuous functional calculus, spectral measures, Borel functional calculus. Applications of the spectral theorem: Fredholm alternative, integral equations.

    Prerequisite(s): MATH 714 ; MATH 717 .

    Grading Basis: Graded
    Units of Lecture: 3
    Offered: Every Spring - Odd Years

    Student Learning Outcomes
    Upon completion of this course, students will be able to:
    1. work with bounded operators on Hilbert spaces and their properties.
    2. apply the concepts of Banach algebras, C*-algebra and of the functional calculus.
    3. demonstrate mastery of spectral theory.


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  • MATH 721 - Nonlinear Dynamics and Chaos I

    (3 units)
    Geometric approach to nonlinear dynamics, Poincare maps, center manifolds, normal forms, averaging method.

    Grading Basis: Graded
    Units of Lecture: 3
    Student Learning Outcomes
    Upon completion of this course, students will be able to:
    1. demonstrate an advanced level of competency in geometric approach to 1 and 2- dimensional flows, stability analysis, Lyapunov stability and Lyapunov function method, classification and analysis of bifurcation of equilibria in 1 and 2-dimentional flows, conservative, reversible and dissipative systems, index theory.
    2. demonstrate an advanced level of competency in application of Poincare-Bendixon theorem, Lienard system, Hopf bifurcations, global bifurcation of cycles, hyperbolic systems.
    3. demonstrate an advanced level of competency in asymptotic methods, weakly coupled nonlinear oscillators, synchronization phenomena in nonlinear systems and real world models.


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  • MATH 722 - Nonlinear Dynamics and Chaos II

    (3 units)
    Local bifurcations of vector fields, bifurcations of maps, global bifurcations and chaos, Milnikov’s method, chaotic attractors, applied chaos.

    Grading Basis: Graded
    Units of Lecture: 3
    Student Learning Outcomes
    Upon completion of this course, students will be able to:
    1. demonstrate an advanced level of competency in center manifold and normal forms theorie, applications of normal forms theory to analysis of local bifurcations in multidimensional systems, normal forms and bifurcations in periodic systems.
    2. demonstrate knowledge of theory and applications of Lorenz’s and Rossler’s systems, chaotic attractors, Lyapunov’s exponents, Eegodicity concept.
    3. demonstrate an advanced level of competency in nonlinear maps and their bifurcations, normal form theory for nonlinear maps, chaotic dynamics of maps.


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  • MATH 731 - Modern Algebra

    (3 units)
    Groups, fields, linear dependence, linear transformations, Galois theory.

    Grading Basis: Graded
    Units of Lecture: 3
    Offered: Every Fall - Odd Years

    Student Learning Outcomes
    Upon completion of this course, students will be able to:
    1. demonstrate an advanced level of competency in the theory of rings (ideals, factorization in commutative rings, localization, rings of polynomials and of power series).
    2. demonstrate an advanced level of competency in the theory of modules (exact sequences, projective and injective modules, modules over principal ideal domains).
    3. demonstrate an advanced level of competency in linear algebra (determinants, rational canonical form, Jordan canonical form, minimal and characteristic polynomials, eigenvalues and eigenvectors).


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  • MATH 732 - Modern Algebra

    (3 units)
    Groups, fields, linear dependence, linear transformations, Galois theory.

    Grading Basis: Graded
    Units of Lecture: 3
    Offered: Every Spring - Even Years

    Student Learning Outcomes
    Upon completion of this course, students will be able to:
    1. demonstrate an advanced level of competency in Galois theory (field extensions, splitting fields, algebraic closure, normality, Galois group of a polynomial, finite fields,separability, cyclotomic extensions, radical extensions).
    2. demonstrate an advanced level of competency in commutative algebra (chain conditions, prime and primary ideals, primary decomposition, Jacobson radical, Hilbert Nullstellensatz).


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  • MATH 735 - Advanced Linear Algebra

    (3 units)
    Linear Algebra and Matrix Theory have been fundamental tools in mathematical and statistical disciplines as well as in science and engineering. This one-semester course gives a mathematically rigorous treatment of Linear Algebra and Matrix Theory.

    Prerequisite: MATH 330  with a “B” or better, or MATH 430 , with a “C” or better.

    Grading Basis: Graded
    Units of Lecture: 3
    Offered: Every Spring

    Student Learning Outcomes
    Upon completion of this course, students will be able to:
    1. demonstrate an advanced level of competency in linear algebra and matrix theory.
    2. demonstrate an advanced level of understanding of various matrix decompositions.
    3. demonstrate an advanced level of competency in their applications in sciences and engineering.


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  • MATH 736 - Numerical Linear Algebra

    (3 units)
    Numerical linear algebra is at the intersection of numerical analysis and linear algebra and has many applications in mathematics, science and engineering. Its purpose is the design and analysis of algorithms for the numerical solution of matrix problems including their stability. Writing proofs and computer algorithms are required.

    Maximum units a student may earn: 3

    Recommended Preparation: MATH 330 with a “B” or better or MATH 466/666 with a “C” or better.

    Grading Basis: Graded
    Units of Lecture: 3
    Offered: Every Spring - Odd Years

    Student Learning Outcomes
    Upon completion of this course, students will be able to:
    1. apply algorithms for solving linear systems of equations and least-squares problems.
    2. compare properties of classical matrix factorizations.
    3. use direct and iterative methods for small and large, dense and sparse matrices.


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  • MATH 737 - Multilinear Algebra

    (3 units)
    Multilinear Algebra is a fundamental subject in mathematics with applications in various areas in mathematics. It studies functions of several variables that are linear in each entry separately, thereby generalizing linear algebra. This one-semester course gives a mathematically rigorous treatment of Multilinear Algebra, which leads to an advanced understanding of the subject. Writing mathematical proofs is required.

    Maximum units a student may earn: 3

    Prerequisite(s): MATH 731  with a “B” or better or MATH 735  with a “B” or better.

    Grading Basis: Graded
    Units of Lecture: 3
    Offered: Every Fall - Odd Years

    Student Learning Outcomes
    Upon completion of this course, students will be able to:
    1. apply finite group representation theory to research problems in mathematics.
    2. describe when multilinear algebra techniques are applicable to research problems in mathematics such as generalized matrix inequalities and combinatorics.
    3. critically assess published results based on multilinear algebra.


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  • MATH 738 - Lie Algebras

    (3 units)
    Lie Algebras are fundamental objects in mathematics as well as in physics. This one-semester course gives a mathematically rigorous treatment of Lie Algebras, which leads to an advanced understanding of the subject. Writing mathematical proofs is required.

    Maximum units a student may earn: 3

    Prerequisite(s): MATH 735  with a “B” or better or MATH 731  with a “B” or better.

    Grading Basis: Graded
    Units of Lecture: 3
    Offered: Every Spring - Even Years

    Student Learning Outcomes
    Upon completion of this course, students will be able to:
    1. apply structural and classification results of Lie algebras to research problems in mathematics.
    2. describe when Lie Algebras are applicable to research problems in mathematics such as matrix algebra, finite reflections groups, Lie groups, and geometry.
    3. critically assess published results based on Lie algebras.


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  • MATH 741 - Topology

    (3 units)
    Topological structures, uniform spaces, metric spaces, compact and locally compact spaces, connectivity, function spaces, topological algebra, elementary homological algebra, singular homology theory, cell complexes, homotopy groups.

    Grading Basis: Graded
    Units of Lecture: 3
    Offered: Every Fall - Even Years

    Student Learning Outcomes
    Upon completion of this course, students will be able to:
    1. compute the homology of a space defined by a simplicial complex.
    2. use homotopy groups or homology groups to demonstrate that spaces are not homeomorphic to one another.
    3. apply homological algebraic arguments to prove results about continuous maps between spaces.


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  • MATH 742 - Topology

    (3 units)
    Topological structures, uniform spaces, metric spaces, compact and locally compact spaces, connectivity, function spaces, topological algebra, elementary homological algebra, singular homology theory, cell complexes, homotopy groups.

    Grading Basis: Graded
    Units of Lecture: 3
    Offered: Every Spring - Odd Years

    Student Learning Outcomes
    Upon completion of this course, students will be able to:
    1. use long exact sequences to compute homology and cohomology groups of topological spaces.
    2. compute homology groups and cohomology groups of a space from the Mayer Vietoris sequence for a decomposition into subspaces.
    3. use the Kunneth formula to determine the cohomology of a product space.  


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  • MATH 743 - Homotopy Theory

    (3 units)
    Introduction to the basic techniques of homotopy theory, with applications to topology and geometry. Properties and computations involving higher homotopy groups; the general theory of fibrations and fiber bundles.

    Maximum units a student may earn: 3

    Prerequisite(s): MATH 731; MATH 741. Corequisite(s): MATH 742.

    Grading Basis: Graded
    Units of Lecture: 3 units: 150-minutes per week X 15 weeks
    Offered: Every Spring - Odd Years

    Student Learning Outcomes
    Upon completion of this course, students will be able to:
    1. perform basic computations of homotopy groups via the Homotopy Excision Theorem.
    2. compute homotopy groups via the long exact sequence associated to a Serre fibration.
    3. characterize obstructions to lifting and extension problems in topology via Postnikov towers and obstruction classes.


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  • MATH 744 - Differential Topology

    (3 units)
    Introduction to differential topology, including smooth manifolds and smooth maps, tangent and cotangent bundle, differential forms, and integration on manifolds, Whitney’s Embedding Theorem, Sard’s Theorem, and Stokes’ Theorem.

    Prerequisite(s): MATH 311 . Corequisite: MATH 731 .

    Grading Basis: Graded
    Units of Lecture: 3
    Offered: Every Fall - Odd Years

    Student Learning Outcomes
    Upon completion of this course, students will be able to:
    1. demonstrate an understanding of smooth manifolds and differentiable maps through proofs and examples.
    2. appropriately apply foundational theorems in differential topology, including Sard’s Theorem, Whitney’s Embedding Theorem, and Stokes’ Theorem.
    3. appropriately apply the concepts of the tangent bundle, the cotangent bundle, and the derived exterior algebra bundles on manifolds, and master integration and manipulation of differential forms.


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  • MATH 745 - Riemannian Geometry

    (3 units)
    This introductory course to Riemannian geometry shall cover Riemannian metrics, affine and Riemannian connections, geodesics and geodesic flow, the Riemann and Ricci curvature tensors, and sectional and scalar curvatures.

    Prerequisite(s): MATH 744 .

    Grading Basis: Graded
    Units of Lecture: 3
    Offered: Every Spring - Even Years

    Student Learning Outcomes
    Upon completion of this course, students will be able to:
    1. demonstrate an understanding of Riemannian metrics and affine connections on differentiable manifolds.
    2. work with geodesics on Riemannian manifolds, and use their properties to solve concrete problems, such as finding distance minimizing paths.
    3. show a mastery of computing and applying the Riemannian and Ricci tensors, and sectional and scalar curvatures.


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  • MATH 751 - Operations Research I

    (3 units)
    Theoretical optimization theory; linear programming, simplex method convexity, duality theory, integer and fractional programming algorithms, Kuhn-Tucker theory, linear complementary problem, Lemke’s algorithm.

    Grading Basis: Graded
    Units of Lecture: 3
    Offered: Every Fall - Even Years

    Student Learning Outcomes
    Upon completion of this course, students will be able to:
    1. demonstrate understanding of and facility with all of the mathematical details of the simplex algorithm for solving linear programs, duality theory, and a linear algebraic approach to sensitivity analysis for linear programs, including being able to formally prove results in these areas.
    2. demonstrate understanding of and facility with extensions of the simplex algorithm to solve nonlinear problems, such as integer programming, fractional programming, and linear complementarity problems.
    3. demonstrate understanding of the Kuhn Tucker conditions for nonlinear programming, how they are related to optimality conditions for certain optimization problems, and how to use them to solve certain convex and non-convex problems.


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  • MATH 752 - Operations Research II

    (3 units)
    A survey of stochastic operations research: decision analysis, reliability theory, Markov chains, queuing theory, stochastic inventory theory.

    Grading Basis: Graded
    Units of Lecture: 3
    Offered: Every Fall - Odd Years

    Student Learning Outcomes
    Upon completion of this course, students will be able to:
    1. set up, model, and solve real world problems from the business and non-business world as problems in decision analysis, reliability theory, queuing theory, or stochastic inventory theory.
    2. demonstrate knowledge of the mathematical theories of: decision analysis, risk theory, reliability theory (including models with a true time component), Markov Chains, birth-death processes (especially queues), and stochastic inventory theory.


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  • MATH 761 - Methods in Applied Math I

    (3 units)
    Finite dimensional Vector Spaces, Function Spaces, Integral Equations, Greens Functions, Differential Operators.

    Grading Basis: Graded
    Units of Lecture: 3
    Student Learning Outcomes
    Upon completion of this course, students will be able to:
    1. demonstrate understanding of finite dimensional vector spaces and function spaces through proofs and examples.
    2. determine analytical solutions for certain classes of linear integral equations, and apply Greens function methodologies to solve the problems formulated in the area of partial differential equations.
    3. set up, analyze and solve problems arising in various application domains using methodologies they will study in this course.


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  • MATH 762 - Methods in Applied Math II

    (3 units)
    Calculus of Variations, Complex Variables, Transform and Spectral Theory, Partial Differential Equations, Asymptotics, Perturbation Theory.

    Grading Basis: Graded
    Units of Lecture: 3
    Student Learning Outcomes
    Upon completion of this course, students will be able to:
    1. demonstrate ability to solve problems using calculus of variations and complex variables methodology.
    2. demonstrate competence at the advanced level in application of spectral and integral transform methodologies, and perturbation theory to analysis and approximating of solutions of partial differential equations.
    3. apply these methodologies to solving practically important problems from various applied fields and interpret the solutions using analytical and numerical approaches.


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  • MATH 773 - Topics in Algebra

    (3 units)
    Variable content chosen from such topics as theory of equations, number theory, and groups and permutations.

    Maximum units a student may earn: 9

    Grading Basis: Graded
    Units of Lecture: 3
    Student Learning Outcomes
    Upon completion of this course, students will be able to:
    1. demonstrate an advanced level of competency in Algebra.


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  • MATH 774 - Topics in Geometry and Analysis

    (3 units)
    Variable content chosen from such topics as plane algebraic curves, theory of surfaces, pseudo-Euclidean spaces. May be repeated when course content differs.

    Maximum units a student may earn: 9

    Grading Basis: Graded
    Units of Lecture: 3
    Student Learning Outcomes
    Upon completion of this course, students will be able to:
    1. demonstrate an advanced level of competency in geometry and analysis.


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  • MATH 775 - Advanced Study of Topics in Probability

    (3 units)
    Variable content based on faculty and student interests. May be repeated when course content differs.

    Maximum units a student may earn: 9

    Grading Basis: Graded
    Units of Lecture: 3
    Student Learning Outcomes
    Upon completion of this course, students will be able to:
    1. demonstrate understanding of the methods discussed in class.
    2. apply methods discussed in class to research questions.
    3. articulate the relationship of methods discussed in class to the broader field of probability and statistics.


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  • MATH 777 - Seminar in Teaching Mathematics and Statistics

    (2 units)
    Prepares graduate students to become successful Teaching Assistants in mathematics and statistics courses. The activities, discussions, and work are designed to be useful and practical during the first semester of teaching and beyond. Topics include fostering student engagement, selecting appropriate mathematical tasks, formative assessment, reflective instruction, lesson planning, preparing to be a peer mentor, building a teaching CV, and documenting teaching practices.

    Maximum units a student may earn: 2

    Grading Basis: Satisfactory/Unsatisfactory
    Units of Lecture: 2
    Offered: Every Fall

    Student Learning Outcomes
    Upon completion of this course, students will be able to:
    1. participate in professional development activities related to mathematics teaching assistants’ duties.
    2. discuss instructional strategies for teaching mathematics that can be used by teaching assistants in classroom, group, or individual learning settings.
    3. reflect on how these new strategies and skills will be implemented into their current teaching assistant duties and future instructional responsibilities.
    4. document their teaching and prepare a CV and teaching statement.


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  • MATH 778 - Topics in Applied Math for High School Teachers

    (1 to 3 units)
    Intended for students of the MATM program and inservice training for active secondary teachers. May be repeated when course content differs.

    Maximum units a student may earn: 9

    Grading Basis: Graded
    Student Learning Outcomes
    Upon completion of this course, students will be able to:
    1. demonstrate an advanced level of competency in applied mathematics.


    Click here for course scheduling information. | Check course textbook information
  

  • MATH 780 - Topics in Advanced Mathematics

    (1 to 3 units)
    Variable content chosen from such topics as mathematical methods in applied science, manifold theory, functional analysis, or geometric methods in ODE theory. May be repeated when course content differs.

    Maximum units a student may earn: 9

    Grading Basis: Graded
    Student Learning Outcomes
    Upon completion of this course, students will be able to:
    1. demonstrate an advanced level of competency in mathematics.


    Click here for course scheduling information. | Check course textbook information
  

  • MATH 786 - Cooperative Game Theory

    (3 units)
    Shapley-Bondareva Theorem, convex games, market games, assignment games, permutation games, Shapley value, power indices, multilinear extensions, ordinal preference matching games, NTU games, Scarf Theorem.

    Grading Basis: Graded
    Units of Lecture: 3
    Offered: Every Spring - Even Years

    Student Learning Outcomes
    Upon completion of this course, students will be able to:
    1. demonstrate understanding of the main theories (and proofs) from classical TU cooperative game theory, including the Shapley-Bondareva theorem, and the theory of convex games, market games, and the Shapley Value, as well as the Scarf theorem for NTU games.
    2. demonstrate understanding of the main theories (and proofs) from the broad area of matching games, including the assignment game, permutation game, houseswapping game, and marriage game.
    3. read a research paper (in the area of cooperative game theory) from a journal and present it in a lucid manner.


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  • MATH 793 - Independent Study

    (1 to 3 units)
    Individual study conducted under the direction of a faculty member. Limited to 6 credits except under special circumstances.

    Maximum units a student may earn: 6

    Grading Basis: Graded
    Offered: Every Fall, Spring, and Summer

    Student Learning Outcomes
    Upon completion of this course, students will be able to:
    1. demonstrate understanding of the methods discussed in class.
    2. apply methods discussed in class to research questions.
    3. articulate the relationship of methods discussed in class to the broader field of probability and statistics.


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  • MATH 794 - Research in Mathematical Sciences

    (.5 units)
    This course provides an overview of the current research topics and methods in the mathematical sciences. It covers professional preparation and presentation of mathematical and scientific research results. Maximum of 2 units.

    Maximum units a student may earn: 2

    Grading Basis: Graded
    Units of Lecture: .5
    Offered: Every Fall

    Student Learning Outcomes
    Upon completion of this course, students will be able to:


    Click here for course scheduling information. | Check course textbook information
  

  • MATH 795 - Comprehensive Examination

    (1 unit)
    Course is used by graduate programs to administer comprehensive examinations either as an end of program comprehensive examination or as a qualifying examination for doctoral candidates prior to being advanced to candidacy.

    Grading Basis: Satisfactory/Unsatisfactory
    Units of Independent Study: 1
    Offered: Every Fall and Spring

    Student Learning Outcomes
    Upon completion of this course, students will be able to:
    1. demonstrate comprehensive knowledge in the broad area of their graduate training.


    Click here for course scheduling information. | Check course textbook information
  
  
  

  • MATH 899 - Graduate Advisement

    (1 to 4 units)
    Provides access to faculty for continued consultation and advisement. No grade is filed and credits may not be applied to any degree requirements. Limited to 8 credits (2 semester) enrollment. For non-thesis master’s degree students only.

    Maximum units a student may earn: 8

    Grading Basis: Satisfactory/Unsatisfactory
    Units of Independent Study: X
    Student Learning Outcomes
    Upon completion of this course, students will be able to:


    Click here for course scheduling information. | Check course textbook information

Mechanical Engineering
  

  • ME 203 - Introduction to Computer Methods for Engineers

    (3 units)
    Introduction to algorithm development and software for analysis of engineering problems and design. Computer skills development, file and data management, graphics and numerical methods, spreadsheets.

    Prerequisite(s): MATH 182  with a “C” or better. Corequisite(s): MATH 283 .

    Grading Basis: Graded
    Units of Lecture: 2
    Units of Laboratory/Studio: 1
    Offered: Every Fall and Spring

    Student Learning Outcomes
    Upon completion of this course, students will be able to:
    1. identify, formulate, and solve complex engineering problems by applying principles of engineering, science, and mathematics.


    Click here for course scheduling information. | Check course textbook information
  

  • ME 242 - Dynamics

    (3 units)
    Kinematics and kinetics of particles and rigid bodies in two and three dimensions; relative motion; work and energy; impulse and momentum.

    Prerequisite(s): ENGR 241  with a “C” or better. Corequisite(s): MATH 283 .

    Grading Basis: Graded
    Units of Lecture: 3
    Offered: Every Fall and Spring

    Student Learning Outcomes
    Upon completion of this course, students will be able to:
    1. identify, formulate, and solve complex engineering problems by applying principles of engineering, science, and mathematics.


    Click here for course scheduling information. | Check course textbook information
  

  • ME 303 - Applied Numerical Methods

    (3 units)
    Introduces numerical methods commonly used in engineering applications.

    Prerequisite(s): CS 135  or ME 203 ; MATH 285  with a “C” or better.

    Grading Basis: Graded
    Units of Lecture: 3
    Offered: Every Fall and Spring

    Student Learning Outcomes
    Upon completion of this course, students will be able to:
    1. identify, formulate, and solve complex engineering problems by applying principles of engineering, science, and mathematics.


    Click here for course scheduling information. | Check course textbook information
  

  • ME 310 - System Analysis and Design

    (4 units)
    Mathematical modeling and response analysis of linear mechanical and electrical systems. Introduction to experimental modeling.

    Prerequisite(s): ME 242  with a “C” or better; ME 303 .

    Grading Basis: Graded
    Units of Lecture: 3
    Units of Laboratory/Studio: 1
    Offered: Every Fall and Spring

    Student Learning Outcomes
    Upon completion of this course, students will be able to:
    1. identify, formulate, and solve complex engineering problems by applying principles of engineering, science, and mathematics.


    Click here for course scheduling information. | Check course textbook information
  

  • ME 311 - Engineering Thermodynamics I

    (3 units)
    Principles of engineering thermodynamics. A study of the first and second laws, entropy, ideal and real gases, and second-law analysis of engineering systems. (CHE 311 and ME 311 are cross-listed; credit may be earned in one of the two.)

    Prerequisite(s): PHYS 181  with a “C” or better and CHEM 121A  with a “C” or better or CHEM 201 . Corequisite(s): PHYS 181  for CHE majors.

    Grading Basis: Graded
    Units of Lecture: 3
    Offered: Every Fall and Spring

    Student Learning Outcomes
    Upon completion of this course, students will be able to:
    1. identify, formulate, and solve complex engineering problems by applying principles of engineering, science, and mathematics.


    Click here for course scheduling information. | Check course textbook information
  

  • ME 312 - Engineering Thermodynamics II

    (3 units)
    Continuation of ME 311  covering power and refrigeration cycles, gas mixtures, thermodynamics relations, combustion and thermodynamics of high-speed flow.

    Prerequisite(s): ME 311  with a “C” or better.

    Grading Basis: Graded
    Units of Lecture: 3
    Offered: Every Spring

    Student Learning Outcomes
    Upon completion of this course, students will be able to:
    1. acquire and apply new knowledge as needed, using appropriate learning strategies.


    Click here for course scheduling information. | Check course textbook information
  

  • ME 314 - Introduction to Heat Transfer

    (3 units)
    Conduction, convection and radiation relationships are applied to engineering problems. Analytical, numerical and graphical solutions are demonstrated.

    Prerequisite(s): ENGR 360 ; ME 203  or CS 135 ; ME 311  with a “C” or better. 

    Grading Basis: Graded
    Units of Lecture: 3
    Offered: Every Fall and Spring

    Student Learning Outcomes
    Upon completion of this course, students will be able to:
    1. identify, formulate, and solve complex engineering problems by applying principles of engineering, science, and mathematics.


    Click here for course scheduling information. | Check course textbook information
  

  • ME 322 - Instrumentation

    (4 units)
    Theory, use and design of instruments for static and dynamic measurements and controls. Experimental data analysis.

    Prerequisite(s): CEE 372  with a “C” or better; EE 220L . Corequisite(s): ME 314 .

    Grading Basis: Graded
    Units of Lecture: 3
    Units of Laboratory/Studio: 1
    Offered: Every Fall and Spring

    Student Learning Outcomes
    Upon completion of this course, students will be able to:
    1. communicate effectively with a range of audiences.
    2. develop and conduct appropriate experimentation, analyze and interpret data, and use engineering judgment to draw conclusions.


    Click here for course scheduling information. | Check course textbook information
  

  • ME 350 - Computer Aided Design and Manufacturing

    (3 units)
    Topics include CAD; drafting standards; GD&T; CNC machining; 3D printing; introduction to manufacturing processes; fasteners, gears, belts. A hands-on group design project is required.

    Prerequisite(s): MATH 283  with “C” or better. Corequisite(s): ENGR 100 .

    Grading Basis: Graded
    Units of Lecture: 2
    Units of Laboratory/Studio: 1
    Offered: Every Fall and Spring

    Student Learning Outcomes
    Upon completion of this course, students will be able to:
    1. apply engineering design to produce solutions that meet specified needs with consideration of public health, safety, and welfare, as well as global, cultural, social, environmental, and economic factors.


    Click here for course scheduling information. | Check course textbook information
  

  • ME 351 - Mechanical Design

    (4 units)
    Design of machine elements. Computer simulation and analysis are emphasized.

    Prerequisite(s): CEE 372  with a “C” or better; ME 350 ; MSE 250 .

    Grading Basis: Graded
    Units of Lecture: 3
    Units of Laboratory/Studio: 1
    Offered: Every Fall and Spring

    Student Learning Outcomes
    Upon completion of this course, students will be able to:
    1. identify, formulate, and solve complex engineering problems by applying principles of engineering, science, and mathematics.


    Click here for course scheduling information. | Check course textbook information
  

  • ME 354 - Introduction to Manufacturing Processes

    (3 units)
    Introduction to a variety of conventional and advanced manufacturing processes including metal casting; bulk deformation, powder metallurgy/ceramics; polymers/plastics, composites, and micromanufacturing.

    Corequisite(s): MSE 250 .

    Grading Basis: Graded
    Units of Lecture: 2
    Units of Laboratory/Studio: 1
    Offered: Every Fall and Spring

    Student Learning Outcomes
    Upon completion of this course, students will be able to:
    1. acquire and apply new knowledge as needed, using appropriate learning strategies.


    Click here for course scheduling information. | Check course textbook information
  

  • ME 380 - Fluid Dynamics for Mechanical Engineers

    (3 units)
    Properties of fluids, hydrostatics, conservation laws, dimensional analysis, similarity, Bernoulli, pipe flow, drag, lift and boundary-layer theory.

    Prerequisite(s): MATH 283  with a “C” or better; MATH 285  with a “C” or better; ME 242  with a “C” or better.

    Grading Basis: Graded
    Units of Lecture: 3
    Offered: Every Fall and Spring

    Student Learning Outcomes
    Upon completion of this course, students will be able to:
    1. identify, formulate, and solve complex engineering problems by applying principles of engineering, science, and mathematics.


    Click here for course scheduling information. | Check course textbook information
  

  • ME 410 - Introduction to System Control

    (3 units)
    Mathematics of linear systems and their control.

    Prerequisite(s): ME 310 .

    Grading Basis: Graded
    Units of Lecture: 3
    Offered: Every Fall and Spring

    Student Learning Outcomes
    Upon completion of this course, students will be able to:
    1. identify, formulate, and solve complex engineering problems by applying principles of engineering, science, and mathematics.


    Click here for course scheduling information. | Check course textbook information
  

  • ME 411 - Comparative Biomechanics

    (3 units)
    This course will review bio-material properties, muscles, scaling laws, animal sensory systems and central pattern generators. Students will learn about jumping fleas, sliding slugs, flying flies, gripping geckos, trotting turkeys, and so much more. (BIOL 411 and ME 411 are cross-listed; credits may be earned in one of the two.)

    Prerequisite(s): MATH 181  with a “C” or better; ME 311  with a “C’ or better or BIOL 316  with a “C” or better.

    Grading Basis: Graded
    Units of Lecture: 3
    Offered: Every Fall - Odd Years

    Student Learning Outcomes
    Upon completion of this course, students will be able to:
    1. acquire and apply new knowledge as needed, using appropriate learning strategies.


    Click here for course scheduling information. | Check course textbook information
 

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