Advance your understanding of mathematics while tackling complex challenges that shape our world. Michigan Tech's Mathematical Sciences MS and PhD programs provide opportunities to specialize in computational and applied mathematics, discrete mathematics, and pure mathematics while working alongside faculty engaged in cutting-edge research. From developing advanced computational models and solving differential equations to exploring graph theory, coding theory, and number theory, students gain the analytical and problem-solving skills sought by industry, government, and academia. Small class sizes, personalized faculty mentorship, and hands-on research experiences prepare graduates to lead innovation in data-driven and technology-focused fields.
Master's Degree
Students can pursue a Master's Degree in Mathematical Sciences with an emphasis in discrete mathematics, pure mathematics, or computational and applied mathematics. Applicants are not required to have an undergraduate degree in mathematics. Each concentration area has a set of required courses.
Degree Options
There are three different options under which the MS in mathematical sciences can be earned: thesis, report, or coursework. Regardless of the option, students must complete the core courses in their chosen concentration.
This option requires a research thesis prepared under the supervision of the advisor. The thesis describes a research investigation and its results. The scope of the research topic for the thesis should be defined in such a way that a full-time student could complete the requirements for a master’s degree in 12 months or three semesters following the completion of coursework by regularly scheduling graduate research credits.
The minimum requirements are as follows:
| Option Parts | Credits |
|---|---|
| Coursework (minimum) | 20 Credits |
| Thesis research | 6-10 Credits |
| Total (minimum) | 30 Credits |
| Distribution | Credits |
|---|---|
| 5000-6000 series (minimum) | 12 Credits |
| 3000-4000 (maximum) | 12 Credits |
Programs may have stricter requirements and may require more than the minimum number of credits listed here.
This option requires a report describing the results of an independent study project. The scope of the research topic should be defined in such a way that a full-time student could complete the requirements for a master’s degree in 12 months or three semesters following the completion of coursework by regularly scheduling graduate research credits.
Of the minimum total of 30 credits, at least 24 must be earned in coursework other than the project:
| Option Parts | Credits |
|---|---|
| Coursework (minimum) | 24 Credits |
| Report | 2-6 Credits |
| Total (minimum) | 30 Credits |
| Distribution | Credits |
|---|---|
| 5000-6000 series (minimum) | 12 Credits |
| 3000-4000 (maximum) | 12 Credits |
Programs may have stricter requirements and may require more than the minimum number of credits listed here.
This option requires a minimum of 30 credits be earned through coursework. A limited number of research credits may be used with the approval of the advisor, department, and Graduate School. See degree requirements for more information.
A graduate program may require an oral or written examination before conferring the degree and may require more than the minimum credits listed here:
| Distribution | Credits |
|---|---|
| 5000-6000 series (minimum) | 18 Credits |
| 3000-4000 (maximum) | 12 Credits |
Concentrations—Core Courses and Electives
All MS students must choose one of four concentrations and complete the core coursework
in that concentration. Note: It is important to recognize that many of these courses
are offered only in alternate years. Students must plan carefully to complete the
MS degree in the expected two academic years.
Computational and Applied Mathematics Concentration
Core Courses
Functional analytic basis of modern numerical analysis. Linear spaces, including Sobolev space theory, linear operators, approximation theory, and applications to Fourier analysis, fixed point theorems, iterative methods, finite difference methods, etc.
Qualitative theory of solutions of ordinary differential equations, including existence, uniqueness, and continuous dependence; theory of linear equations; solution of constant coefficient systems; phase plane analysis; design and analysis of numerical methods.
Theory of partial differential equations. Covers classification, appropriate boundary conditions and initial conditions, PDEs of mathematical physics, characteristics, Green`s functions, and variational principles.
Design and analysis of algorithms for problems in linear algebra. Covers floating point arithmetic, condition numbers, error analysis, solution of linear systems (direct and iterative methods), eigenvalue problems, least squares, and singular value decomposition. Includes the use of appropriate software including high performance computational libraries.
Analysis and design of algorithms for the numerical solution of partial differential equations.
Elective Courses
Choose two courses.
A graduate-level study of the Lebesgue integral including its comparison with the Riemann integral; the Lebesgue measure, measurable functions and measurable sets. Integrable functions, the monotone convergence theorem, the dominated convergence theorem, and Fatou's lemma.
Numerical solution of unconstrained and constrained optimization problems and nonlinear equations. Topics include optimality conditions, local convergence of Newton and Quasi-Newton methods, line search and trust region globalization techniques, quadratic penalty and augmented Lagrangian methods for equality-constrained problems, logarithmic barrier method for inequality-constrained problems, and Sequential Quadratic Programming.
Topics will vary with instructor, but will cover areas in computational and applied mathematics.
Advanced topics in applied mathematics.
Advanced topics in computational mathematics.
Discrete Mathematics
Core Courses
Review of basic graph theory followed by one or more advanced topics which may include topological graph theory, algebraic graph theory, graph decomposition or graph coloring.
Methods for the construction of different combilateral structures such as difference sets, symmetric designs, projective geometries, orthogonal latin squares, transversal designs, steiner systems and tournaments.
Basic concepts, motivation from information transmission, finite fields, bounds, optimal codes, projective spaces, duality and orthogonal arrays, important families of codes, MacWilliams' identities, applications.
Theory of finite groups, their actions and applications. Review of basic group theory (Sylow theorems). Simple groups and group actions (transitivity). Symmetric and alternating groups, linear groups and more general classical groups. Applications: finite fields, designs, finite geometries.
Elective Courses
Choose two courses.
Basic algorithmic and computational methods used in the solution of fundamental combinatorial problems. Topics may include but are not limited to backtracking, hill-climbing, combinatorial optimization, linear and integer programming, and network analysis.
Topics will vary with instructor but will emphasize real world applications of discrete mathematics.
Introduction to polynomial rings, finite fields and field extensions. Review of basic notions concerning rings, polynomials and power series. General theory of finite and algebraic field extensions. The basics of Galois theory (field extensions and their Galois groups).
Introduction to commutative algebra and combinatorial algebra. A first description of research issues is also given. Topics include: commutative rings (quotients, morphisms; prime, maximal ideals); modules, Noetherian, artinian rings; combinatorial algebra (gradings, monomials, Hilbert functions, resolutions, level, Gorenstein algebras).
Topics may include, but not limited to, unique factorization, elementary estimates on the distribution of prime numbers, congruences, Chinese remainder theorem, primitive roots, n-th powers modulo an integer, quadratic residues, quadratic reciprocity, quadratic characters, Gauss sums, and finite fields.
Advanced topics in design theory.
Advanced topics in coding theory.
Advanced topics in combinatorics, algebra, or number theory.
Advanced topics in algebra.
Pure Mathematics
Core Courses
Review of basic graph theory followed by one or more advanced topics which may include topological graph theory, algebraic graph theory, graph decomposition or graph coloring.
Theory of finite groups, their actions and applications. Review of basic group theory (Sylow theorems). Simple groups and group actions (transitivity). Symmetric and alternating groups, linear groups and more general classical groups. Applications: finite fields, designs, finite geometries.
Additional Core Courses
Choose one course.
Qualitative theory of solutions of ordinary differential equations, including existence, uniqueness, and continuous dependence; theory of linear equations; solution of constant coefficient systems; phase plane analysis; design and analysis of numerical methods.
Theory of partial differential equations. Covers classification, appropriate boundary conditions and initial conditions, PDEs of mathematical physics, characteristics, Green`s functions, and variational principles.
Elective Courses
Choose four courses.
Methods for the construction of different combilateral structures such as difference sets, symmetric designs, projective geometries, orthogonal latin squares, transversal designs, steiner systems and tournaments.
Basic concepts, motivation from information transmission, finite fields, bounds, optimal codes, projective spaces, duality and orthogonal arrays, important families of codes, MacWilliams' identities, applications.
Introduction to polynomial rings, finite fields and field extensions. Review of basic notions concerning rings, polynomials and power series. General theory of finite and algebraic field extensions. The basics of Galois theory (field extensions and their Galois groups).
Introduction to commutative algebra and combinatorial algebra. A first description of research issues is also given. Topics include: commutative rings (quotients, morphisms; prime, maximal ideals); modules, Noetherian, artinian rings; combinatorial algebra (gradings, monomials, Hilbert functions, resolutions, level, Gorenstein algebras).
Topics may include, but not limited to, unique factorization, elementary estimates on the distribution of prime numbers, congruences, Chinese remainder theorem, primitive roots, n-th powers modulo an integer, quadratic residues, quadratic reciprocity, quadratic characters, Gauss sums, and finite fields.
Functional analytic basis of modern numerical analysis. Linear spaces, including Sobolev space theory, linear operators, approximation theory, and applications to Fourier analysis, fixed point theorems, iterative methods, finite difference methods, etc.
Advanced topics in design theory.
Advanced topics in coding theory.
Advanced topics in combinatorics, algebra, or number theory.
Advanced topics in algebra.
Review of discrete probability, probability measures, random variables, distribution functions, expectation as a Lebesgue-Stieltjes integral, independence, modes of convergence, laws of large numbers and iterated logarithms, characteristic functions, central limit theorems, conditional expectation, martingales, introduction to stochastic processes.
General advice for MS students
1
You should find an advisor by the end of your first semester if possible, and no later than the end of your second semester.
2
You will take the qualifying examination during orientation week. If you do not pass, you should try to take it again in your second semester. Students are limited to three attempts to pass the exam.
3
The graduate school requires MS students to submit certain forms documenting their
progress through the degree requirements (for example, completion of necessary courses,
passage of required exams, scheduling of an oral defense, etc.).
PhD Program
The doctoral program has two areas of concentration for Mathematical Sciences: computational and applied mathematics and discrete mathematics. The doctoral program requires advanced coursework (beyond the master's degree) and successful completion of the Qualifying and Comprehensive Examinations. Students must demonstrate the ability to independently conduct research. Doctoral students work closely with a major advisor and must have their research proposal and dissertation approved by their graduate committee.
Overview of Program Requirements
To complete a doctoral degree, students must complete the following milestones:
- Complete all coursework and research credits (see credit requirements below)
- Pass Qualifying Examination
- Pass Research Proposal Examination
- Prepare and Submit Approved Dissertation
- Pass Final Oral Defense
The minimum credit requirements are as follows:
| Degrees | Credits |
|---|---|
| MS-PhD (minimum) | 30 Credits |
| BS-PhD (minimum) | 60 Credits |
Individual programs may have higher standards and students are expected to know their program's requirements. See the Doctor of Philosophy Requirements website for more information about PhD milestones and related timelines.
List of PhD degree requirements
It is important to note that this list is not chronological; indeed, not all students will complete the requirements in the same order.
1
Choose a concentration and complete the core MS coursework in that concentration.
Computational and Applied Mathematics students develop expertise in the theory and application of ordinary and partial differential equations, linear algebra, and computational methods.
Core courses:
- MA5501 Theoretical Numerical Analysis
- MA5510 ODEs
- MA5565 PDEs
- MA5627 Numerical Linear Algebra
- MA5629 Numerical PDEs
Elective courses (choose two):
Students of Discrete Mathematics study design and coding theory, graph theory number theory, and algebra.
Core courses:
Elective courses (choose two):
- MA5201 Combinatorial Algorithms
- MA5280 Topics in Applied Combinatorics
- MA5302 Algebra II
- MA5320 Commutative Algebra
- MA5360 Number Theory
- MA6222 Advanced Topics in Design Theory
- MA6231 Advanced Topics in Coding Theory
- MA6280 Advanced Topics in Combinatorics, Algebra, or Number Theory
- MA6300 Advanced Topics in Algebra
2
Find an advisor no later than the end of your second regular semester and form a PhD dissertation committee.
- Note that the committee must include one faculty member from another department.
- See also: Process for changing your advisor(s) below.
3
- Complete at least two 6000-level courses in your concentration.
- Students in Discrete Mathematics can use MA5280 as a 6000-level course.
- Students in Applied and Computational Mathematics can use MA5580 as a 6000-level course.
- MA5280 and MA5580 are repeatable up to 4 times.
4
Complete the "breadth" requirement by taking two graduate level courses in other concentrations.
5
Pass the qualifying examination. This is a written exam covering advanced undergraduate material; it must be passed by the end of the third semester (summer semesters do not count) in the PhD program.
6
Pass the comprehensive examination. This multi-part exam covers graduate course work; it must be passed by the end of the sixth semester in the PhD program (summer semesters do not count).
7
Recommended: Present a dissertation proposal to the satisfaction of your dissertation committee.
- Note: Depending on your committee, this proposal may be written or oral. Check with your advisor.
8
Write a dissertation detailing the results of a substantial and original research project.
9
Defend the dissertation with a public presentation and examination by your committee.
10
The graduate school requires PhD students to submit certain forms documenting their
progress through the degree requirements (for example, completion of necessary courses,
passage of required exams, scheduling of an oral defense, etc.).