Complete Course List

1.0 [max 3.0 cr] cr; Prereq: ! math grad student in good standing or #;
S-N or Aud
Fall, Spring, Every Year
New approaches to teaching/learning, issues in mathematics education, components/expectations of a college mathematics professor.

MATH 8141 - Applied Logic

A-F or Aud
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

A-F or Aud
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

3.0 cr; Prereq: 5166 or #;
A-F or Aud
Periodically
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.

MATH 8152 - Axiomatic Set Theory

3.0 cr; Prereq: 8151 or #;
A-F or Aud
Periodically
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.
Course Number and Name Section Location Term Instructor
8152 Axiomatic Set Theory 1 Vincent Hall 301 Spring 2018 William Messing

MATH 8166 - Recursion Theory

3.0 cr; Prereq: Math grad student or #;
A-F or Aud
Periodically

MATH 8167 - Recursion Theory

3.0 cr; Prereq: 8166;
A-F or Aud
Spring
Sample topics: complexity theory, recursive analysis, generalized recursion theory, analytical hierarchy, constructive ordinals.

MATH 8172 - Model Theory

3.0 cr; Prereq: Math grad student or #;
A-F or Aud
Periodically
Interplay of formal theories, their models. Elementary equivalence, elementary extensions, partial isomorphisms. Lowenheim-Skolem theorems, compactness theorems, preservation theorems. Ultraproducts.

MATH 8173 - Model Theory

3.0 cr; Prereq: 8172 or #;
A-F or Aud
Periodically
Types of elements. Prime models, homogeneity, saturation, categoricity in power. Forking.

MATH 8190 - Topics in Logic

A-F or Aud
Fall, Spring, Periodically

MATH 8201 - General Algebra

3.0 cr; Prereq: 4xxx algebra or equiv or #;
A-F or Aud
Fall, Every Year
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.

MATH 8202 - General Algebra

3.0 cr; Prereq: 8201 or #;
A-F or Aud
Spring, Every Year
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.
Course Number and Name Section Location Term Instructor
8202 General Algebra 1 Vincent Hall 364 Spring 2018 Peter Webb
3.0 cr; Prereq: 8202 or #;
A-F or Aud
Fall, Spring, Periodically
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.
3.0 cr; Prereq: 8207 or #;
A-F or Aud
Periodically
Applications of Eisenstein series: special values and analytic continuation and functional equations of L-functions. Trace formulas. Applications of representation theory. Computations.
Course Number and Name Section Location Term Instructor
8208 Theory of Modular Forms and L-Functions 1 Vincent Hall 2 Spring 2018 Paul Garrett
3.0 cr; Prereq: 8202 or #;
A-F or Aud
Fall
Selected topics.
3.0 cr; Prereq: 8211 or #;
A-F or Aud
Periodically
Selected topics.
Course Number and Name Section Location Term Instructor
8212 Commutative and Homological Algebra 1 Vincent Hall 301 Spring 2018 Christine Berkesch Zamaere

MATH 8245 - Group Theory

3.0 cr; Prereq: 8202 or #;
A-F or Aud
Fall, Every Year
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.

MATH 8246 - Group Theory

3.0 cr; Prereq: 8245 or #;
A-F or Aud
Fall, Spring
Representation and character theory, simple groups, free groups and products, presentations, extensions, Schur multipliers.
3.0 cr; Prereq: 8202 or #;
A-F or Aud
Periodically
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.
3.0 cr; Prereq: 8251 or #;
A-F or Aud
Periodically
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.

MATH 8253 - Algebraic Geometry

3.0 cr; Prereq: 8202 or #;
A-F or Aud
Fall
Curves, surfaces, projective space, affine and projective varieties. Rational maps. Blowing-up points. Zariski topology. Irreducible varieties, divisors.

MATH 8254 - Algebraic Geometry

3.0 cr; Prereq: 8253 or #;
A-F or Aud
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.
Course Number and Name Section Location Term Instructor
8254 Algebraic Geometry 1 Vincent Hall 313 Spring 2018 Alexander Voronov
1.0 - 3.0 [max 12.0 cr] cr; Prereq: Math 8201, Math 8202;
A-F or Aud
Fall, Spring, Every Year, Periodically
3.0 cr; Prereq: 8302 or #;
A-F or Aud
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.
3.0 cr; Prereq: 8271 or #;
A-F or Aud
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.
Course Number and Name Section Location Term Instructor
8272 Lie Groups and Lie Algebras 1 Vincent Hall 213 Spring 2018 Kai-Wen Lan
1.0 - 3.0 [max 12.0 cr] cr; Prereq: #;
A-F or Aud
Periodically
Course Number and Name Section Location Term Instructor
8280 Topics in Number Theory 1 Vincent Hall 364 Spring 2018 Dihua Jiang

MATH 8300 - Topics in Algebra

1.0 - 3.0 [max 12.0 cr] cr; Prereq: Grad math major or #;
A-F or Aud
Fall, Every Year, Periodically
Selected topics.
Course Number and Name Section Location Term Instructor
8300 Topics in Algebra 1 Vincent Hall 209 Spring 2018 Gennady Lyubeznik
3.0 cr; Prereq: [Some point-set topology, algebra] or #;
A-F or Aud
Fall, Every Year
Classification of compact surfaces, fundamental group/covering spaces. Homology group, basic cohomology. Application to degree of a map, invariance of domain/dimension.
3.0 cr; Prereq: 8301 or #;
A-F or Aud
Spring, Every Year
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.
Course Number and Name Section Location Term Instructor
8302 Manifolds and Topology 1 Vincent Hall 207 Spring 2018 Craig Westerland

MATH 8306 - Algebraic Topology

3.0 cr; Prereq: 8301 or #;
A-F or Aud
Periodically
Singular homology, cohomology theory with coefficients. Eilenberg-Stenrod axioms, Mayer-Vietoris theorem.

MATH 8307 - Algebraic Topology

3.0 cr; Prereq: 8306 or #;
A-F or Aud
Periodically
Basic homotopy theory, cohomology rings with applications. Time permitting: fibre spaces, cohomology operations, extra-ordinary cohomology theories.
Course Number and Name Section Location Term Instructor
8307 Algebraic Topology 1 Vincent Hall 209 Spring 2018 Tyler Lawson

MATH 8333 - FTE: Master's

Periodically
No description

MATH 8360 - Topics in Topology

1.0 - 3.0 [max 12.0 cr] cr; Prereq: 8301 or #;
A-F or Aud
Fall, Spring, Periodically
Selected topics.

MATH 8365 - Riemannian Geometry

3.0 cr; Prereq: 8301 or basic point-set topology or #;
A-F or Aud
Fall, Every Year
Riemannian metrics, curvature. Bianchi identities, Gauss-Bonnet theorem, Meyers's theorem, Cartan-Hadamard theorem.

MATH 8366 - Riemannian Geometry

3.0 cr; Prereq: 8365 or #;
A-F or Aud
Spring, Every Year
Gauss, Codazzi equations. Tensor calculus, Hodge theory, spinors, global differential geometry, applications.
1.0 - 3.0 [max 12.0 cr] cr; Prereq: 8301 or 8365;
A-F or Aud
Fall, Spring, Every Year, Periodically
Current research in Differential Geometry.
1.0 - 3.0 [max 12.0 cr] cr; Prereq: 8301, 8365;
A-F or Aud
Fall, Spring
Current research.
3.0 cr; Prereq: 4xxx partial differential equations or #;
A-F or Aud
Periodically
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.
3.0 cr; Prereq: 8595 or #;
A-F or Aud
Periodically
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.
3.0 cr; Prereq: [5xxx numerical analysis, some computer experience] or #;
A-F or Aud
Fall, Every Year
Mathematical models from physical, biological, social systems. Emphasizes industrial applications. Modeling of deterministic/probabilistic, discrete/continuous processes; methods for analysis/computation.
3.0 cr; Prereq: 8597 or #;
A-F or Aud
Periodically
Techniques for analysis of mathematical models. Asymptotic methods; design of simulation and visualization techniques. Specific computation for models arising in industrial problems.
1.0 - 3.0 [max 12.0 cr] cr; Prereq: 8601;
A-F or Aud
Periodically
Current research.
3.0 cr; Prereq: 4xxx numerical analysis and applied linear algebra or #;
A-F or Aud
Fall, Every Year
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.
3.0 cr; Prereq: 5xxx numerical analysis of partial differential equations or #;
A-F or Aud
Periodically
Equations of continuity/motion. Kinematics. Bernoulli's theorem, stream function, velocity potential. Applications of conformal mapping.
3.0 cr; Prereq: 8431 or #;
Periodically
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.
3.0 cr; Prereq: [4xxx analysis, 4xxx applied linear algebra] or #;
Fall, Every Year
Approximation of functions, numerical integration. Numerical methods for elliptic partial differential equations, including finite element methods, finite difference methods, and spectral methods. Grid generation.
3.0 cr; Prereq: 8441 or #; 5477-5478 recommended for engineering and science grad students;
Spring, Every Year
Numerical methods for integral equations, parabolic partial differential equations, hyperbolic partial differential equations. Monte Carlo methods.
Course Number and Name Section Location Term Instructor
8442 Numerical Analysis and Scientific Computing 1 Vincent Hall 207 Spring 2018 Bernardo Cockburn

MATH 8444 - FTE: Doctoral

Periodically
3.0 cr; Prereq: 4xxx numerical analysis, 4xxx partial differential equations or #;
A-F or Aud
Fall, Spring, Every Year
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.
3.0 cr; Prereq: 8445 or #;
A-F or Aud
Spring, Every Year
Numerical methods for parabolic equations (e.g., heat equations). Methods for elasticity, fluid mechanics, electromagnetics. Applications to specific computations.
Course Number and Name Section Location Term Instructor
8446 Numerical Analysis of Differential Equations 1 Vincent Hall 364 Spring 2018 Douglas Arnold
1.0 - 3.0 [max 12.0 cr] cr; Prereq: Grad math major or #;
A-F or Aud
Fall, Spring, Every Year, Periodically
Selected topics.
Course Number and Name Section Location Term Instructor
8450 Topics in Numerical Analysis 1 Vincent Hall 213 Spring 2018 Mitchell Luskin
A-F or Aud
Fall, Spring, Periodically
3.0 cr; Prereq: 4xxx ODE or #;
A-F or Aud
Fall, Every Year
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.
3.0 cr; Prereq: 8501 or #;
A-F or Aud
Spring, Every Year
Selected topics: 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.
3.0 cr; Prereq: 8501 or #;
A-F or Aud
Periodically
Basic bifurcation theory, Hopf bifurcation, and method averaging. Silnikov bifurcations. Singular perturbations. Higher order bifurcations. Applications.
3.0 cr; Prereq: 5525 or 8502 or #;
A-F or Aud
Periodically
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.
3.0 cr; Prereq: 5587 or #;
A-F or Aud
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.
1.0 - 3.0 [max 12.0 cr] cr; Prereq: 8502;
A-F or Aud
Fall, Spring
Current research.
1.0 - 3.0 [max 3.0 cr] cr; Prereq: 8502;
A-F or Aud
Fall, Spring, Periodically
A-F or Aud
Fall, Spring, Every Year, Periodically
3.0 cr; Prereq: 8502 or #;
A-F or Aud
Fall, Every Year
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.
3.0 cr; Prereq: 8571 or #;
A-F or Aud
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.
Course Number and Name Section Location Term Instructor
8572 Theory of Evolutionary Equations 1 Vincent Hall 206 Spring 2018 Arnd Scheel
1.0 - 3.0 [max 12.0 cr] cr; Prereq: 8572 or #;
A-F or Aud
Periodically
3.0 cr; Prereq: 4xxx applied mathematics or #;
A-F or Aud
Periodically
Metric spaces, continuity, completeness, contraction mappings, compactness. Normed linear spaces, continuous linear transformations. Hilbert spaces, orthogonality, projections.
3.0 cr; Prereq: 8581 or #;
A-F or Aud
Periodically
Fourier theory. Self-adjoint, compact, unbounded linear operators. Spectral analysis, eigenvalue-eigenvector problem, spectral theorem, operational calculus.
3.0 cr; Prereq: [Some 5xxx PDE, 8601] or #;
A-F or Aud
Fall, Every Year
Classification of partial differential equations/characteristics. Laplace, wave, heat equations. Some mixed problems.
3.0 cr; Prereq: 8583 or #;
A-F or Aud
Spring, Every Year
Fundamental solutions/distributions, Sobolev spaces, regularity. Advanced elliptic theory (Schauder estimates, Garding's inequality). Hyperbolic systems.
Course Number and Name Section Location Term Instructor
8584 Theory of Partial Differential Equations 1 Vincent Hall 209 Spring 2018 Mikhail Safonov
1.0 - 3.0 [max 3.0 cr] cr; Prereq: 8602;
A-F or Aud
Fall, Spring, Every Year, Periodically
Research topics.
Fall, Spring, Every Year
Offered for one yr or one semester as circumstances warrant. Topics vary. For details, contact instructor.

MATH 8601 - Real Analysis

3.0 cr; Prereq: 5616 or #;
A-F or Aud
Fall, Every Year
Set theory/fundamentals. Axiom of choice, measures, measure spaces, Borel/Lebesgue measure, integration, fundamental convergence theorems, Riesz representation.

MATH 8602 - Real Analysis

3.0 cr; Prereq: 8601 or #;
A-F or Aud
Spring, Every Year
Radon-Nikodym, Fubini theorems. C(X). Lp spaces (introduction to metric, Banach, Hilbert spaces). Stone-Weierstrass theorem. Basic Fourier analysis. Theory of differentiation.
Course Number and Name Section Location Term Instructor
8602 Real Analysis 1 Vincent Hall 207 Spring 2018 Paul Garrett
3.0 [max 12.0 cr] cr; Prereq: 8602 or #;
A-F or Aud
Periodically
Current research.
3.0 cr; Prereq: 5616 or #;
Fall, Every Year
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.
3.0 cr; Prereq: 8651 or #;
Spring, Every Year
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.
Course Number and Name Section Location Term Instructor
8652 Theory of Probability Including Measure Theory 1 Ford Hall 151 Spring 2018 Wei-Kuo Chen
3.0 cr; Prereq: 8651 or 8602 or #;
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.
3.0 cr; Prereq: 8654 or 8659 or #;
Fall, Every Year
Stochastic integration with respect to martingales, Ito's formula, applications to business models, filtering, and stochastic control theory.

MATH 8659 - Stochastic Processes

3.0 cr; Prereq: 8652 or #;
Fall, Every Year
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.

MATH 8660 - Topics in Probability

Fall, Spring, Every Year, Periodically
Periodically
No description

MATH 8668 - Combinatorial Theory

A-F or Aud
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

3.0 cr; Prereq: 8668 or #;
A-F or Aud
Spring, Odd Years
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.
Course Number and Name Section Location Term Instructor
8669 Combinatorial Theory 1 Vincent Hall 213 Spring 2018 Gregg Musiker
1.0 - 3.0 [max 12.0 cr] cr; Prereq: Grad math major or #;
A-F or Aud
Fall, Spring, Every Year, Periodically
Selected topics.

MATH 8701 - Complex Analysis

3.0 cr; Prereq: 5616 or #;
A-F or Aud
Fall, Every Year
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.

MATH 8702 - Complex Analysis

3.0 cr; Prereq: 8701 or #;
A-F or Aud
Spring, Every Year
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).
Course Number and Name Section Location Term Instructor
8702 Complex Analysis 1 Vincent Hall 301 Spring 2018 Albert Marden
Periodically
No description
1.0 - 3.0 [max 12.0 cr] cr; Prereq: 8702 or #;
A-F or Aud
Periodically
Current research.

MATH 8801 - Functional Analysis

3.0 cr; Prereq: 8602 or #;
A-F or Aud
Fall, Every Year
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.

MATH 8802 - Functional Analysis

3.0 cr; Prereq: 8801 or #;
A-F or Aud
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.
Course Number and Name Section Location Term Instructor
8802 Functional Analysis 1 Vincent Hall 364 Spring 2018 Peter Polacik
Periodically
No description

MATH 8990 - Topics in Mathematics

1.0 - 6.0 [max 24.0 cr] cr; Prereq: #;
S-N or Aud
Fall, Spring, Every Year
Readings, research.

MATH 8991 - Independent Study

1.0 - 6.0 [max 24.0 cr] cr; Prereq: #;
S-N or Aud
Spring, Summer, Every Year
Individually directed study.

MATH 8992 - Directed Reading

1.0 - 6.0 [max 24.0 cr] cr; Prereq: #;
S-N or Aud
Fall, Spring, Every Year
Individually directed reading.

MATH 8993 - Directed Study

1.0 - 6.0 [max 24.0 cr] cr; Prereq: #;
S-N or Aud
Spring, Every Year
Individually directed study.

MATH 8994 - Topics at the IMA

1.0-3.0 cr; Prereq: None;
Fall, Spring, Every Year
Current research at IMA.