Twin Cities campus

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Twin Cities Campus

Mathematics Minor

School of Mathematics
College of Science and Engineering
Link to a list of faculty for this program.
Contact Information
School of Mathematics, University of Minnesota, 127 Vincent Hall, 206 Church Street SE, Minneapolis, MN 55455 (612-624-6391, fax: 612-624-6702)
Email: mathgrad@umn.edu
Website: https://cse.umn.edu/math
  • Program Type: Graduate minor related to major
  • Requirements for this program are current for Fall 2022
  • Length of program in credits (master's): 6
  • Length of program in credits (doctoral): 12
  • This program does not require summer semesters for timely completion.
Special areas of research include ordinary and partial differential equations; probability; real, complex, harmonic, functional, and numerical analysis; differential and algebraic geometry; topology; number theory; commutative algebra; group theory; logic; combinatorics; mathematical physics; and applied and industrial mathematics, mathematical biology, and dynamical systems.
Program Delivery
  • via classroom (the majority of instruction is face-to-face)
Prerequisites for Admission
Special Application Requirements:
Students interested in the minor are strongly encouraged to confer with their major field advisor and director of graduate studies, and the Mathematics director of graduate studies regarding feasibility and requirements.
For an online application or for more information about graduate education admissions, see the General Information section of this website.
Program Requirements
Use of 4xxx courses towards program requirements is not permitted.
Courses offered on both the A-F and S/N grading basis must be taken A-F, with a minimum grade of B earned for each course. The minimum cumulative GPA for the minor is 3.00. Consult the Mathematics director of graduate studies in advance for course approval.
Minor Coursework (6 to 12 credits)
Master's students select at least 6 credits from two semester courses and doctoral students select at least 12 credits from four semester courses to meet coursework requirements. Courses are selected in consultation with the advisor and the Mathematics director of graduate study.
MATH 8141 - Applied Logic (3.0 cr)
MATH 8142 - Applied Logic (3.0 cr)
MATH 8151 - Axiomatic Set Theory (3.0 cr)
MATH 8152 - Axiomatic Set Theory (3.0 cr)
MATH 8166 - Recursion Theory (3.0 cr)
MATH 8167 - Recursion Theory (3.0 cr)
MATH 8172 - Model Theory (3.0 cr)
MATH 8173 - Model Theory (3.0 cr)
MATH 8190 - Topics in Logic (1.0-3.0 cr)
MATH 8201 - General Algebra (3.0 cr)
MATH 8202 - General Algebra (3.0 cr)
MATH 8207 - Theory of Modular Forms and L-Functions (3.0 cr)
MATH 8208 - Theory of Modular Forms and L-Functions (3.0 cr)
MATH 8211 - Commutative and Homological Algebra (3.0 cr)
MATH 8212 - Commutative and Homological Algebra (3.0 cr)
MATH 8245 - Group Theory (3.0 cr)
MATH 8246 - Group Theory (3.0 cr)
MATH 8251 - Algebraic Number Theory (3.0 cr)
MATH 8252 - Algebraic Number Theory (3.0 cr)
MATH 8253 - Algebraic Geometry (3.0 cr)
MATH 8254 - Algebraic Geometry (3.0 cr)
MATH 8270 - Topics in Algebraic Geometry (1.0-3.0 cr)
MATH 8271 - Lie Groups and Lie Algebras (3.0 cr)
MATH 8272 - Lie Groups and Lie Algebras (3.0 cr)
MATH 8280 - Topics in Number Theory (1.0-3.0 cr)
MATH 8300 - Topics in Algebra (1.0-3.0 cr)
MATH 8301 - Manifolds and Topology (3.0 cr)
MATH 8302 - Manifolds and Topology (3.0 cr)
MATH 8306 - Algebraic Topology (3.0 cr)
MATH 8307 - Algebraic Topology (3.0 cr)
MATH 8360 - Topics in Topology (1.0-3.0 cr)
MATH 8365 - Riemannian Geometry (3.0 cr)
MATH 8366 - Riemannian Geometry (3.0 cr)
MATH 8370 - Topics in Differential Geometry (1.0-3.0 cr)
MATH 8380 - Topics in Advanced Geometry (1.0-3.0 cr)
MATH 8385 - Calculus of Variations and Minimal Surfaces (3.0 cr)
MATH 8386 - Calculus of Variations and Minimal Surfaces (3.0 cr)
MATH 8387 - Mathematical Modeling of Industrial Problems (3.0 cr)
MATH 8388 - Mathematical Modeling of Industrial Problems (3.0 cr)
MATH 8390 - Topics in Mathematical Physics (1.0-3.0 cr)
MATH 8401 - Mathematical Modeling and Methods of Applied Mathematics (3.0 cr)
MATH 8402 - Mathematical Modeling and Methods of Applied Mathematics (3.0 cr)
MATH 8431 - Mathematical Fluid Mechanics (3.0 cr)
MATH 8432 - Mathematical Fluid Mechanics (3.0 cr)
MATH 8441 - Numerical Analysis and Scientific Computing (3.0 cr)
MATH 8442 - Numerical Analysis and Scientific Computing (3.0 cr)
MATH 8445 - Numerical Analysis of Differential Equations (3.0 cr)
MATH 8446 - Numerical Analysis of Differential Equations (3.0 cr)
MATH 8450 - Topics in Numerical Analysis (1.0-3.0 cr)
MATH 8470 - Topics in Mathematical Theory of Continuum Mechanics (1.0-3.0 cr)
MATH 8501 - Differential Equations and Dynamical Systems I (3.0 cr)
MATH 8502 - Differential Equations and Dynamical Systems II (3.0 cr)
MATH 8503 - Bifurcation Theory in Ordinary Differential Equations (3.0 cr)
MATH 8505 - Applied Dynamical Systems and Bifurcation Theory I (3.0 cr)
MATH 8506 - Applied Dynamical Systems and Bifurcation Theory II (3.0 cr)
MATH 8520 - Topics in Dynamical Systems (1.0-3.0 cr)
MATH 8530 - Topics in Ordinary Differential Equations (1.0-3.0 cr)
MATH 8540 - Topics in Mathematical Biology (1.0-3.0 cr)
MATH 8571 - Theory of Evolutionary Equations (3.0 cr)
MATH 8572 - Theory of Evolutionary Equations (3.0 cr)
MATH 8580 - Topics in Evolutionary Equations (1.0-3.0 cr)
MATH 8581 - Applications of Linear Operator Theory (3.0 cr)
MATH 8582 - Applications of Linear Operator Theory (3.0 cr)
MATH 8583 - Theory of Partial Differential Equations (3.0 cr)
MATH 8584 - Theory of Partial Differential Equations (3.0 cr)
MATH 8590 - Topics in Partial Differential Equations (1.0-3.0 cr)
MATH 8600 - Topics in Advanced Applied Mathematics (1.0-3.0 cr)
MATH 8601 - Real Analysis (3.0 cr)
MATH 8602 - Real Analysis (3.0 cr)
MATH 8640 - Topics in Real Analysis (3.0 cr)
MATH 8641 - Spatial Ecology (3.0 cr)
MATH 8651 - Theory of Probability Including Measure Theory (3.0 cr)
MATH 8652 - Theory of Probability Including Measure Theory (3.0 cr)
MATH 8654 - Fundamentals of Probability Theory and Stochastic Processes (3.0 cr)
MATH 8655 - Stochastic Calculus with Applications (3.0 cr)
MATH 8659 - Stochastic Processes (3.0 cr)
MATH 8660 - Topics in Probability (1.0-3.0 cr)
MATH 8668 - Combinatorial Theory (3.0 cr)
MATH 8669 - Combinatorial Theory (3.0 cr)
MATH 8680 - Topics in Combinatorics (1.0-3.0 cr)
MATH 8701 - Complex Analysis (3.0 cr)
MATH 8702 - Complex Analysis (3.0 cr)
MATH 8790 - Topics in Complex Analysis (1.0-3.0 cr)
MATH 8801 - Functional Analysis (3.0 cr)
MATH 8802 - Functional Analysis (3.0 cr)
MATH 8990 - Topics in Mathematics (1.0-6.0 cr)
MATH 8991 - Independent Study (1.0-6.0 cr)
MATH 8992 - Directed Reading (1.0-6.0 cr)
MATH 8993 - Directed Study (1.0-6.0 cr)
MATH 8994 - Topics at the IMA (1.0-3.0 cr)
Program Sub-plans
Students are required to complete one of the following sub-plans.
Students may not complete the program with more than one sub-plan.
Masters
Doctoral
 
More program views..
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MATH 8141 - Applied Logic
Credits: 3.0 [max 3.0]
Grading Basis: A-F or Aud
Typically offered: Periodic Fall & Spring
Applying techniques of mathematical logic to other areas of mathematics and computer science. Sample topics: complexity of computation, computable analysis, unsolvability of diophantine problems, program verification, database theory.
MATH 8142 - Applied Logic
Credits: 3.0 [max 3.0]
Grading Basis: A-F or Aud
Typically offered: Periodic Spring
Applying techniques of mathematical logic to other areas of mathematics, computer science. Complexity of computation, computable analysis, unsolvability of diophantine problems, program verification, database theory.
MATH 8151 - Axiomatic Set Theory
Credits: 3.0 [max 3.0]
Grading Basis: A-F or Aud
Typically offered: Periodic Fall
Axiomatic development of basic properties of ordinal/cardinal numbers, infinitary combinatorics, well founded sets, consistency of axiom of foundation, constructible sets, consistency of axiom of choice and of generalized continuum hypothesis. prereq: 5166 or instr consent
MATH 8152 - Axiomatic Set Theory
Credits: 3.0 [max 3.0]
Grading Basis: A-F or Aud
Typically offered: Periodic Fall
Notion of forcing, generic extensions, forcing with finite partial functions, independence of continuum hypothesis, forcing with partial functions of infinite cardinalities, relationship between partial orderings and Boolean algebras, Boolean-valued models, independence of axiom of choice. prereq: 8151 or instr consent
MATH 8166 - Recursion Theory
Credits: 3.0 [max 3.0]
Grading Basis: A-F or Aud
Typically offered: Periodic Fall
Analysis of concept of computability, including various equivalent definitions. Primitive recursive, recursive, partial recursive functions. Oracle Turing machines. Kleene Normal Form Theorem. Recursive, recursively enumerable sets. Degrees of unsolvability. Arithmetic hierarchy. prereq: Math grad student or instr consent
MATH 8167 - Recursion Theory
Credits: 3.0 [max 3.0]
Grading Basis: A-F or Aud
Typically offered: Periodic Spring
Sample topics: complexity theory, recursive analysis, generalized recursion theory, analytical hierarchy, constructive ordinals. prereq: 8166
MATH 8172 - Model Theory
Credits: 3.0 [max 3.0]
Grading Basis: A-F or Aud
Typically offered: Periodic Fall
Interplay of formal theories, their models. Elementary equivalence, elementary extensions, partial isomorphisms. Lowenheim-Skolem theorems, compactness theorems, preservation theorems. Ultraproducts. prereq: Math grad student or instr consent
MATH 8173 - Model Theory
Credits: 3.0 [max 3.0]
Grading Basis: A-F or Aud
Typically offered: Periodic Fall
Types of elements. Prime models, homogeneity, saturation, categoricity in power. Forking. prereq: 8172 or instr consent
MATH 8190 - Topics in Logic
Credits: 1.0 -3.0 [max 12.0]
Grading Basis: A-F or Aud
Typically offered: Periodic Fall & Spring
Offered for one year or one semester as circumstances warrant.
MATH 8201 - General Algebra
Credits: 3.0 [max 3.0]
Grading Basis: A-F or Aud
Typically offered: Every Fall
Groups through Sylow, Jordan-H[o]lder theorems, structure of finitely generated Abelian groups. Rings and algebras, including Gauss theory of factorization. Modules, including projective and injective modules, chain conditions, Hilbert basis theorem, and structure of modules over principal ideal domains. prereq: 4xxx algebra or equiv or instr consent
MATH 8202 - General Algebra
Credits: 3.0 [max 3.0]
Grading Basis: A-F or Aud
Typically offered: Every Spring
Classical field theory through Galois theory, including solvable equations. Symmetric, Hermitian, orthogonal, and unitary form. Tensor and exterior algebras. Basic Wedderburn theory of rings; basic representation theory of groups. prereq: 8201 or instr consent
MATH 8207 - Theory of Modular Forms and L-Functions
Credits: 3.0 [max 3.0]
Grading Basis: A-F or Aud
Typically offered: Periodic Fall
Zeta and L-functions, prime number theorem, Dirichlet's theorem on primes in arithmetic progressions, class number formulas; Riemann hypothesis; modular forms and associated L-function; Eisenstein series; Hecke operators, Poincar[e] series, Euler products; Ramanujan conjectures; Theta series and quadratic forms; waveforms and L-functions. prereq: 8202 or instr consent
MATH 8208 - Theory of Modular Forms and L-Functions
Credits: 3.0 [max 3.0]
Grading Basis: A-F or Aud
Typically offered: Periodic Fall
Applications of Eisenstein series: special values and analytic continuation and functional equations of L-functions. Trace formulas. Applications of representation theory. Computations. prereq: 8207 or instr consent
MATH 8211 - Commutative and Homological Algebra
Credits: 3.0 [max 3.0]
Grading Basis: A-F or Aud
Typically offered: Periodic Fall
Selected topics. prereq: 8202 or instr consent
MATH 8212 - Commutative and Homological Algebra
Credits: 3.0 [max 3.0]
Grading Basis: A-F or Aud
Typically offered: Periodic Fall
Selected topics. prereq: 8211 or instr consent
MATH 8245 - Group Theory
Credits: 3.0 [max 3.0]
Grading Basis: A-F or Aud
Typically offered: Every Fall
Permutations, Sylow's theorems, representations of groups on groups, semi-direct products, solvable and nilpotent groups, generalized Fitting subgroups, p-groups, co-prime action on p-groups. prereq: 8202 or instr consent
MATH 8246 - Group Theory
Credits: 3.0 [max 3.0]
Grading Basis: A-F or Aud
Typically offered: Periodic Fall & Spring
Representation and character theory, simple groups, free groups and products, presentations, extensions, Schur multipliers. prereq: 8245 or instr consent
MATH 8251 - Algebraic Number Theory
Credits: 3.0 [max 3.0]
Grading Basis: A-F or Aud
Typically offered: Periodic Fall
Algebraic number fields and algebraic curves. Basic commutative algebra. Completions: p-adic fields, formal power series, Puiseux series. Ramification, discriminant, different. Finiteness of class number and units theorem. prereq: 8202 or instr consent
MATH 8252 - Algebraic Number Theory
Credits: 3.0 [max 3.0]
Grading Basis: A-F or Aud
Typically offered: Periodic Fall
Zeta and L-functions of global fields. Artin L-functions. Hasse-Weil L-functions. Tchebotarev density. Local and global class field theory. Reciprocity laws. Finer theory of cyclotomic fields. prereq: 8251 or instr consent
MATH 8253 - Algebraic Geometry
Credits: 3.0 [max 3.0]
Grading Basis: A-F or Aud
Typically offered: Periodic Fall
Curves, surfaces, projective space, affine and projective varieties. Rational maps. Blowing-up points. Zariski topology. Irreducible varieties, divisors. prereq: 8202 or instr consent
MATH 8254 - Algebraic Geometry
Credits: 3.0 [max 3.0]
Grading Basis: A-F or Aud
Typically offered: Periodic Spring
Sheaves, ringed spaces, and schemes. Morphisms. Derived functors and cohomology, Serre duality. Riemann-Roch theorem for curves, Hurwitz's theorem. Surfaces: monoidal transformations, birational transformations. prereq: 8253 or instr consent
MATH 8270 - Topics in Algebraic Geometry
Credits: 1.0 -3.0 [max 12.0]
Grading Basis: A-F or Aud
Typically offered: Every Fall & Spring
N/A prereq: Math 8201, Math 8202; offered for one year or one semester as circumstances warrant
MATH 8271 - Lie Groups and Lie Algebras
Credits: 3.0 [max 3.0]
Grading Basis: A-F or Aud
Typically offered: Periodic Fall
Definitions and basic properties of Lie groups and Lie algebras; classical matrix Lie groups; Lie subgroups and their corresponding Lie subalgebras; covering groups; Maurer-Cartan forms; exponential map; correspondence between Lie algebras and simply connected Lie groups; Baker-Campbell-Hausdorff formula; homogeneous spaces. prereq: 8302 or instr consent
MATH 8272 - Lie Groups and Lie Algebras
Credits: 3.0 [max 3.0]
Grading Basis: A-F or Aud
Typically offered: Periodic Spring
Solvable and nilpotent Lie algebras and Lie groups; Lie's and Engels's theorems; semisimple Lie algebras; cohomology of Lie algebras; Whitehead's lemmas and Levi's theorem; classification of complex semisimple Lie algebras and compact Lie groups; representation theory. prereq: 8271 or instr consent
MATH 8280 - Topics in Number Theory
Credits: 1.0 -3.0 [max 12.0]
Grading Basis: A-F or Aud
Typically offered: Periodic Fall & Spring
Various topics in Number Theory.
MATH 8300 - Topics in Algebra
Credits: 1.0 -3.0 [max 12.0]
Grading Basis: A-F or Aud
Typically offered: Every Fall & Spring
Selected topics. prereq: Grad math major or instr consent; offered as one yr or one sem crse as circumstances warrant
MATH 8301 - Manifolds and Topology
Credits: 3.0 [max 3.0]
Grading Basis: A-F or Aud
Typically offered: Every Fall
Classification of compact surfaces, fundamental group/covering spaces. Homology group, basic cohomology. Application to degree of a map, invariance of domain/dimension. prereq: [Some point-set topology, algebra] or instr consent
MATH 8302 - Manifolds and Topology
Credits: 3.0 [max 3.0]
Grading Basis: A-F or Aud
Typically offered: Every Spring
Smooth manifolds, tangent spaces, embedding/immersion, Sard's theorem, Frobenius theorem. Differential forms, integration. Curvature, Gauss-Bonnet theorem. Time permitting: de Rham, duality in manifolds. prereq: 8301 or instr consent
MATH 8306 - Algebraic Topology
Credits: 3.0 [max 3.0]
Grading Basis: A-F or Aud
Typically offered: Periodic Fall
Singular homology, cohomology theory with coefficients. Eilenberg-Stenrod axioms, Mayer-Vietoris theorem. prereq: 8301 or instr consent
MATH 8307 - Algebraic Topology
Credits: 3.0 [max 3.0]
Grading Basis: A-F or Aud
Typically offered: Periodic Fall
Basic homotopy theory, cohomology rings with applications. Time permitting: fibre spaces, cohomology operations, extra-ordinary cohomology theories. prereq: 8306 or instr consent
MATH 8360 - Topics in Topology
Credits: 1.0 -3.0 [max 12.0]
Grading Basis: A-F or Aud
Typically offered: Periodic Fall & Spring
Selected topics. prereq: 8301 or instr consent; offered as one yr or one sem crse as circumstances warrant
MATH 8365 - Riemannian Geometry
Credits: 3.0 [max 3.0]
Grading Basis: A-F or Aud
Typically offered: Every Fall
Riemannian metrics, curvature. Bianchi identities, Gauss-Bonnet theorem, Meyers's theorem, Cartan-Hadamard theorem. prereq: 8301 or basic point-set topology or instr consent
MATH 8366 - Riemannian Geometry
Credits: 3.0 [max 3.0]
Grading Basis: A-F or Aud
Typically offered: Every Spring
Gauss, Codazzi equations. Tensor calculus, Hodge theory, spinors, global differential geometry, applications. prereq: 8365 or instr consent
MATH 8370 - Topics in Differential Geometry
Credits: 1.0 -3.0 [max 12.0]
Grading Basis: A-F or Aud
Typically offered: Every Fall & Spring
Current research in Differential Geometry. prereq: 8301 or 8365; offered for one yr or one sem as circumstances warrant
MATH 8380 - Topics in Advanced Geometry
Credits: 1.0 -3.0 [max 12.0]
Grading Basis: A-F or Aud
Typically offered: Periodic Fall & Spring
Current research. prereq: 8301, 8365
MATH 8385 - Calculus of Variations and Minimal Surfaces
Credits: 3.0 [max 3.0]
Grading Basis: A-F or Aud
Typically offered: Periodic Fall
Comprehensive exposition of calculus of variations and its applications. Theory for one-dimensional problems. Survey of typical problems. Necessary conditions. Sufficient conditions. Second variation, accessory eigenvalue problem. Variational problems with subsidiary conditions. Direct methods. prereq: 4xxx partial differential equations or instr consent
MATH 8386 - Calculus of Variations and Minimal Surfaces
Credits: 3.0 [max 3.0]
Grading Basis: A-F or Aud
Typically offered: Periodic Fall
Theory of multiple integrals. Geometrical differential equations, i.e., theory of minimal surfaces and related structures (surfaces of constant or prescribed mean curvature, solutions to variational integrals involving surface curvatures), all extremals for variational problems of current interest as models for interfaces in real materials. prereq: 8595 or instr consent
MATH 8387 - Mathematical Modeling of Industrial Problems
Credits: 3.0 [max 3.0]
Grading Basis: A-F or Aud
Typically offered: Every Fall
Mathematical models from physical, biological, social systems. Emphasizes industrial applications. Modeling of deterministic/probabilistic, discrete/continuous processes; methods for analysis/computation. prereq: [5xxx numerical analysis, some computer experience] or instr consent
MATH 8388 - Mathematical Modeling of Industrial Problems
Credits: 3.0 [max 3.0]
Grading Basis: A-F or Aud
Typically offered: Periodic Fall
Techniques for analysis of mathematical models. Asymptotic methods; design of simulation and visualization techniques. Specific computation for models arising in industrial problems. prereq: 8597 or instr consent
MATH 8390 - Topics in Mathematical Physics
Credits: 1.0 -3.0 [max 12.0]
Grading Basis: A-F or Aud
Typically offered: Periodic Fall
Current research. prereq: 8601; offered for one yr or one sem as circumstances warrant
MATH 8401 - Mathematical Modeling and Methods of Applied Mathematics
Credits: 3.0 [max 3.0]
Grading Basis: A-F or Aud
Typically offered: Every Fall
Dimension analysis, similarity solutions, linearization, stability theory, well-posedness, and characterization of type. Fourier series and integrals, wavelets, Green's functions, weak solutions and distributions. prereq: 4xxx numerical analysis and applied linear algebra or instr consent
MATH 8402 - Mathematical Modeling and Methods of Applied Mathematics
Credits: 3.0 [max 3.0]
Grading Basis: A-F or Aud
Typically offered: Every Spring
Calculus of variations, integral equations, eigenvalue problems, spectral theory. Perturbation, asymptotic methods. Artificial boundary conditions, conformal mapping, coordinate transformations. Applications to specific modeling problems. prereq: 8401 or instr consent
MATH 8431 - Mathematical Fluid Mechanics
Credits: 3.0 [max 3.0]
Grading Basis: A-F or Aud
Typically offered: Periodic Fall
Equations of continuity/motion. Kinematics. Bernoulli's theorem, stream function, velocity potential. Applications of conformal mapping. prereq: 5xxx numerical analysis of partial differential equations or instr consent
MATH 8432 - Mathematical Fluid Mechanics
Credits: 3.0 [max 3.0]
Typically offered: Periodic Fall
Plane flow of gas, characteristic method, hodograph method. Singular surfaces, shock waves, shock layers. Viscous flow, Navier-Stokes equations, exact solutions. Uniqueness, stability, existence theorems. prereq: 8431 or instr consent
MATH 8441 - Numerical Analysis and Scientific Computing
Credits: 3.0 [max 3.0]
Typically offered: Every Fall
Approximation of functions, numerical integration. Numerical methods for elliptic partial differential equations, including finite element methods, finite difference methods, and spectral methods. Grid generation. prereq: [4xxx analysis, 4xxx applied linear algebra] or instr consent
MATH 8442 - Numerical Analysis and Scientific Computing
Credits: 3.0 [max 3.0]
Typically offered: Every Spring
Numerical methods for integral equations, parabolic partial differential equations, hyperbolic partial differential equations. Monte Carlo methods. prereq: 8441 or instr consent; 5477-5478 recommended for engineering and science grad students
MATH 8445 - Numerical Analysis of Differential Equations
Credits: 3.0 [max 3.0]
Grading Basis: A-F or Aud
Typically offered: Every Fall
Finite element and finite difference methods for elliptic boundary value problems (e.g., Laplace's equation) and solution of resulting linear systems by direct and iterative methods. prereq: 4xxx numerical analysis, 4xxx partial differential equations or instr consent
MATH 8446 - Numerical Analysis of Differential Equations
Credits: 3.0 [max 3.0]
Grading Basis: A-F or Aud
Typically offered: Every Spring
Numerical methods for parabolic equations (e.g., heat equations). Methods for elasticity, fluid mechanics, electromagnetics. Applications to specific computations. prereq: 8445 or instr consent
MATH 8450 - Topics in Numerical Analysis
Credits: 1.0 -3.0 [max 12.0]
Grading Basis: A-F or Aud
Typically offered: Every Fall & Spring
Selected topics. prereq: Grad math major or instr consent; offered as one year or one semester course as circumstances warrant
MATH 8470 - Topics in Mathematical Theory of Continuum Mechanics
Credits: 1.0 -3.0 [max 12.0]
Grading Basis: A-F or Aud
Typically offered: Periodic Fall & Spring
Offered for one year or one semester as circumstances warrant.
MATH 8501 - Differential Equations and Dynamical Systems I
Credits: 3.0 [max 3.0]
Grading Basis: A-F or Aud
Typically offered: Every Fall
Existence, uniqueness, continuity, and differentiability of solutions. Linear theory and hyperbolicity. Basics of dynamical systems. Local behavior near a fixed point, a periodic orbit, and a homoclinic or heteroclinic orbit. Perturbation theory. prereq: 4xxx ODE or instr consent
MATH 8502 - Differential Equations and Dynamical Systems II
Credits: 3.0 [max 3.0]
Grading Basis: A-F or Aud
Typically offered: Every Spring
Stable, unstable, and center manifolds. Normal hyperbolicity. Nonautonomous dynamics and skew product flows. Invariant manifolds and quasiperiodicity. Transversality and Melnikov method. Approximation dynamics. Morse-Smale systems. Coupled oscillators and network dynamics. prereq: 8501 or instr consent
MATH 8503 - Bifurcation Theory in Ordinary Differential Equations
Credits: 3.0 [max 3.0]
Grading Basis: A-F or Aud
Typically offered: Periodic Fall
Basic bifurcation theory, Hopf bifurcation, and method averaging. Silnikov bifurcations. Singular perturbations. Higher order bifurcations. Applications. prereq: 8501 or instr consent
MATH 8505 - Applied Dynamical Systems and Bifurcation Theory I
Credits: 3.0 [max 3.0]
Grading Basis: A-F or Aud
Typically offered: Periodic Fall
Static/Hopf bifurcations, invariant manifold theory, normal forms, averaging, Hopf bifurcation in maps, forced oscillations, coupled oscillators, chaotic dynamics, co-dimension 2 bifurcations. Emphasizes computational aspects/applications from biology, chemistry, engineering, physics. prereq: 5525 or 8502 or instr consent
MATH 8506 - Applied Dynamical Systems and Bifurcation Theory II
Credits: 3.0 [max 3.0]
Grading Basis: A-F or Aud
Typically offered: Periodic Fall
Background on analysis in Banach spaces, linear operator theory. Lyapunov-Schmidt reduction, static bifurcation, stability at a simple eigenvalue, Hopf bifurcation in infinite dimensions invariant manifold theory. Applications to hydrodynamic stability problems, reaction-diffusion equations, pattern formation, and elasticity. prereq: 5587 or instr consent
MATH 8520 - Topics in Dynamical Systems
Credits: 1.0 -3.0 [max 12.0]
Grading Basis: A-F or Aud
Typically offered: Periodic Fall & Spring
Current research. prereq: 8502
MATH 8530 - Topics in Ordinary Differential Equations
Credits: 1.0 -3.0 [max 3.0]
Grading Basis: A-F or Aud
Typically offered: Periodic Fall & Spring
Offered for one year or one semester as circumstances warrant. prereq: 8502
MATH 8540 - Topics in Mathematical Biology
Credits: 1.0 -3.0 [max 12.0]
Grading Basis: A-F or Aud
Typically offered: Every Fall & Spring
Offered for one year or one semester as circumstances warrant.
MATH 8571 - Theory of Evolutionary Equations
Credits: 3.0 [max 3.0]
Grading Basis: A-F or Aud
Typically offered: Every Fall
Infinite dimensional dynamical systems, global attractors, existence and robustness. Linear semigroups, analytic semigroups. Linear and nonlinear reaction diffusion equations, strong and weak solutions, well-posedness of solutions. prereq: 8502 or instr consent
MATH 8572 - Theory of Evolutionary Equations
Credits: 3.0 [max 3.0]
Grading Basis: A-F or Aud
Typically offered: Periodic Spring
Dynamics of Navier-Stokes equations, strong/weak solutions, global attractors. Chemically reacting fluid flows. Dynamics in infinite dimensions, unstable manifolds, center manifolds perturbation theory. Inertial manifolds, finite dimensional structures. Dynamical theories of turbulence. prereq: 8571 or instr consent
MATH 8580 - Topics in Evolutionary Equations
Credits: 1.0 -3.0 [max 12.0]
Grading Basis: A-F or Aud
Typically offered: Periodic Fall
N/A prereq: 8572 or instr consent; offered for one yr or one semester as circumstances warrant
MATH 8581 - Applications of Linear Operator Theory
Credits: 3.0 [max 3.0]
Grading Basis: A-F or Aud
Typically offered: Periodic Fall
Metric spaces, continuity, completeness, contraction mappings, compactness. Normed linear spaces, continuous linear transformations. Hilbert spaces, orthogonality, projections. prereq: 4xxx applied mathematics or instr consent
MATH 8582 - Applications of Linear Operator Theory
Credits: 3.0 [max 3.0]
Grading Basis: A-F or Aud
Typically offered: Periodic Fall
Fourier theory. Self-adjoint, compact, unbounded linear operators. Spectral analysis, eigenvalue-eigenvector problem, spectral theorem, operational calculus. prereq: 8581 or instr consent
MATH 8583 - Theory of Partial Differential Equations
Credits: 3.0 [max 3.0]
Grading Basis: A-F or Aud
Typically offered: Every Fall
Classification of partial differential equations/characteristics. Laplace, wave, heat equations. Some mixed problems. prereq: [Some 5xxx PDE, 8601] or instr consent
MATH 8584 - Theory of Partial Differential Equations
Credits: 3.0 [max 3.0]
Grading Basis: A-F or Aud
Typically offered: Every Spring
Fundamental solutions/distributions, Sobolev spaces, regularity. Advanced elliptic theory (Schauder estimates, Garding's inequality). Hyperbolic systems. prereq: 8583 or instr consent
MATH 8590 - Topics in Partial Differential Equations
Credits: 1.0 -3.0 [max 3.0]
Grading Basis: A-F or Aud
Typically offered: Every Fall & Spring
Research topics. prereq: 8602; offered for one yr or one sem as circumstances warrant
MATH 8600 - Topics in Advanced Applied Mathematics
Credits: 1.0 -3.0 [max 12.0]
Typically offered: Every Fall & Spring
Offered for one yr or one semester as circumstances warrant. Topics vary. For details, contact instructor.
MATH 8601 - Real Analysis
Credits: 3.0 [max 3.0]
Grading Basis: A-F or Aud
Typically offered: Every Fall
Set theory/fundamentals. Axiom of choice, measures, measure spaces, Borel/Lebesgue measure, integration, fundamental convergence theorems, Riesz representation. prereq: 5616 or instr consent
MATH 8602 - Real Analysis
Credits: 3.0 [max 3.0]
Grading Basis: A-F or Aud
Typically offered: Every Spring
Radon-Nikodym, Fubini theorems. C(X). Lp spaces (introduction to metric, Banach, Hilbert spaces). Stone-Weierstrass theorem. Basic Fourier analysis. Theory of differentiation. prereq: 8601 or instr consent
MATH 8640 - Topics in Real Analysis
Credits: 3.0 [max 12.0]
Grading Basis: A-F or Aud
Typically offered: Periodic Fall
Current research. prereq: 8602 or instr consent; offered for one year or one semester as circumstances warrant
MATH 8641 - Spatial Ecology
Credits: 3.0 [max 3.0]
Grading Basis: S-N or Aud
Typically offered: Periodic Fall
Introduction: role of space in population dynamics and interspecific interaction; includes single species and multispecies models, deterministic and stochastic theory, different modeling approaches, effects of implicit/explicit space on competition, pattern formation, stability diversity and invasion. Recent literature. Computer lab. prereq: Two semesters calculus, theoretical population ecology or four semesters more robust calculus, course in statistics or probability or instr consent
MATH 8651 - Theory of Probability Including Measure Theory
Credits: 3.0 [max 3.0]
Typically offered: Every Fall
Probability spaces. Distributions/expectations of random variables. Basic theorems of Lebesque theory. Stochastic independence, sums of independent random variables, random walks, filtrations. Probability, moment generating functions, characteristic functions. Laws of large numbers. prereq: 5616 or instr consent
MATH 8652 - Theory of Probability Including Measure Theory
Credits: 3.0 [max 3.0]
Typically offered: Every Spring
Conditional distributions and expectations, convergence of sequences of distributions on real line and on Polish spaces, central limit theorem and related limit theorems, Brownian motion, martingales and introduction to other stochastic sequences. prereq: 8651 or instr consent
MATH 8654 - Fundamentals of Probability Theory and Stochastic Processes
Credits: 3.0 [max 3.0]
Typically offered: Periodic Spring
Review of basic theorems of probability for independent random variables; introductions to Brownian motion process, Poisson process, conditioning, Markov processes, stationary processes, martingales, super- and sub-martingales, Doob-Meyer decomposition. prereq: 8651 or 8602 or instr consent
MATH 8655 - Stochastic Calculus with Applications
Credits: 3.0 [max 3.0]
Typically offered: Every Fall
Stochastic integration with respect to martingales, Ito's formula, applications to business models, filtering, and stochastic control theory. prereq: 8654 or 8659 or instr consent
MATH 8659 - Stochastic Processes
Credits: 3.0 [max 3.0]
Typically offered: Every Fall
In-depth coverage of various stochastic processes and related concepts, such as Markov sequences and processes, renewal sequences, exchangeable sequences, stationary sequences, Poisson point processes, Levy processes, interacting particle systems, diffusions, and stochastic integrals. prereq: 8652 or instr consent
MATH 8660 - Topics in Probability
Credits: 1.0 -3.0 [max 12.0]
Typically offered: Every Fall & Spring
Offered for one year or one semester as circumstances warrant.
MATH 8668 - Combinatorial Theory
Credits: 3.0 [max 3.0]
Grading Basis: A-F or Aud
Typically offered: Periodic Fall
Basic enumeration, including sets and multisets, permutation statistics, inclusion-exclusion, integer/set partitions, involutions and Polya theory. Partially ordered sets, including lattices, incidence algebras, and Mobius inversion. Generating functions.
MATH 8669 - Combinatorial Theory
Credits: 3.0 [max 3.0]
Grading Basis: A-F or Aud
Typically offered: Spring Even Year
Further topics in enumeration, including symmetric functions, Schensted correspondence, and standard tableaux; non-enumerative combinatorics, including graph theory and coloring, matching theory, connectivity, flows in networks, codes, and extremal set theory. prereq: 8668 or instr consent
MATH 8680 - Topics in Combinatorics
Credits: 1.0 -3.0 [max 12.0]
Grading Basis: A-F or Aud
Typically offered: Every Fall & Spring
Selected topics. prereq: Grad math major or instr consent; offered as one yr or one sem crse as circumstances warrant
MATH 8701 - Complex Analysis
Credits: 3.0 [max 3.0]
Grading Basis: A-F or Aud
Typically offered: Every Fall
Foundations of holomorphic functions of one variable; relation to potential theory, complex manifolds, algebraic geometry, number theory. Cauchy's theorems, Poisson integral. Singularities, series, product representations. Hyperbolic geometry, isometries. Covering surfaces, Riemann-Hurwitz formula. Schwarz-Christoffel polygonal functions. Residues. prereq: 5616 or instr consent
MATH 8702 - Complex Analysis
Credits: 3.0 [max 3.0]
Grading Basis: A-F or Aud
Typically offered: Every Spring
Riemann mapping, uniformization, Dirichlet problem. Dirichlet principle, Green's functions, harmonic measures. Approximation theory. Complex analysis on tori (elliptic functions, modular functions, conformal moduli). Complex dynamical systems (Julia sets, Mandelbrot set). prereq: 8701 or instr consent
MATH 8790 - Topics in Complex Analysis
Credits: 1.0 -3.0 [max 12.0]
Grading Basis: A-F or Aud
Typically offered: Periodic Fall
Current research. prereq: 8702 or instr consent; offered for one yr or one sem as circumstances warrant
MATH 8801 - Functional Analysis
Credits: 3.0 [max 3.0]
Grading Basis: A-F or Aud
Typically offered: Every Fall
Motivation in terms of specific problems (e.g., Fourier series, eigenfunctions). Theory of compact operators. Basic theory of Banach spaces (Hahn-Banach, open mapping, closed graph theorems). Frechet spaces. prereq: 8602 or instr consent
MATH 8802 - Functional Analysis
Credits: 3.0 [max 3.0]
Grading Basis: A-F or Aud
Typically offered: Periodic Spring
Spectral theory of operators, theory of distributions (generalized functions), Fourier transformations and applications. Sobolev spaces and pseudo-differential operators. C-star algebras (Gelfand-Naimark theory) and introduction to von Neumann algebras. prereq: 8801 or instr consent
MATH 8990 - Topics in Mathematics
Credits: 1.0 -6.0 [max 24.0]
Grading Basis: S-N or Aud
Typically offered: Every Fall & Spring
Readings, research. prereq: instr consent
MATH 8991 - Independent Study
Credits: 1.0 -6.0 [max 24.0]
Typically offered: Every Fall, Spring & Summer
Individually directed study. prereq: instr consent
MATH 8992 - Directed Reading
Credits: 1.0 -6.0 [max 24.0]
Grading Basis: S-N or Aud
Typically offered: Every Fall & Spring
Individually directed reading. prereq: instr consent
MATH 8993 - Directed Study
Credits: 1.0 -6.0 [max 24.0]
Grading Basis: S-N or Aud
Typically offered: Every Spring
Individually directed study. prereq: instr consent
MATH 8994 - Topics at the IMA
Credits: 1.0 -3.0 [max 6.0]
Typically offered: Every Fall & Spring
Current research at IMA.