Solving equations and inequalities; function properties and graphs with transformations; inverse functions; linear, quadratic, polynomial, rational, exponential and logarithmic functions with applications; systems of equations.

Trigonometric functions with applications; graphs with transformations; inverse functions; fundamental identities and angle formulas; solving equations; solving triangles with applications; polar coordinate system; vectors.

Function properties and graphs; inverse functions; linear, quadratic, polynomial, rational, exponential and logarithmic functions, with applications; systems of equations; partial fraction decomposition; conics; trigonometric functions and graphs; inverse trigonometric functions; fundamental identities and angle formulas; solving equations and triangles with applications; polar coordinate system; vectors.

Mathematical approaches to contemporary problems, handling of data and optimization using basic concepts from algebra, geometry and discrete mathematics.

Logic; the algebra of logic, computers, and number systems; networks and combinatorics; probability and statistics.

Cardinality and integers; decimal representation and the number line; number sense; open ended problem solving strategies; expressions and equation solving; primes, factors, and proofs; ratio and proportion; written communication of mathematics.

Geometry and measurement in two and three dimensions; similarity; congruence; Pythagorean Theorem; written communication of mathematics.

Differential and integral calculus of algebraic, logarithmic, and exponential functions; applications to business and economics, such as maximum-minimum problems, marginal analysis, and exponential growth models.

Limits, derivatives, and integrals of algebraic, exponential, logarithmic, and trigonometric functions, with applications.

Same as Math 1550, but with special honors emphasis for qualified students.

Techniques of integration, parametric equations, analytic geometry, polar coordinates, infinite series, vectors in low dimensions; introduction to differential equations and partial derivatives.

Same as MATH 1552, with special honors emphasis for qualified students.

Designed for biological science majors. Techniques of integration, introduction to differential equations, stability of equilibrium points, elementary linear algebra, elements of multivariable calculus, systems of differential equations.

Topics selected from formal logic, set theory, counting, discrete probability, graph theory, and number theory. Emphasis on reading and writing rigorous mathematics.

Topics: Haar wavelets, multiresolution analysis, and applications to imaging and signal processing. Emphasis on reading and writing rigorous mathematical proofs through linear algebra and wavelet transforms.

The mathematical topics covered are fundamental in mathematical analysis, and are chosen from the area of discrete dynamical systems. These topics include precise definitions of limits, continuity, and stability properties of fixed points and cycles. Quadratic maps and their bifurcations are studied in detail, and metric spaces are introduced as the natural abstraction to explore deeper properties of symbolic dynamics, chaos, and fractals.

Three-dimensional analytic geometry, partial derivatives, multiple integrals.

Same as MATH 2057, with special honors emphasis for qualified students.

Use of computers for investigating, solving, and documenting mathematical problems; numerical, symbolic, and graphical manipulation of mathematical constructs discussed in MATH 1550, 1552, and 2057.

Ordinary differential equations; emphasis on solving linear differential equations.

Ordinary differential equations, Laplace transforms, linear algebra, and Fourier series; physical applications stressed.

Systems of linear equations, vector spaces, linear transformations, matrices, determinants.

Introduction to first order differential equations, linear differential equations with constant coefficients, and systems of differential equations; vector spaces, linear transformations, matrices, determinants, linear dependence, bases, systems of equations, eigenvalues, eigenvectors, and Laplace transforms.

Development of a connected, balanced view of mathematics; interrelationship of patterns, relations, and functions; applications of algebraic reasoning in mathematical situations and structures using contextual, numeric, graphic, and symbolic representations; written communication of mathematics.

Current standards for middle and high school mathematics and the mathematics certification exam. Students will prepare and present middle and/or high school mathematics lessons that incorporate this content and appropriate use of technology.

Using problem-based learning, technology, and exploring in depth relationships between various areas of mathematics, students strengthen mathematical understandings of core concepts taught at the secondary level. Connections between secondary and college mathematics are investigated. Various topics from new math standards for functions and statistics are included.

Measurement of interest (including accumulated and present value factors), annuities certain, yield rates, amortization schedules and sinking funds, bonds and related securities, derivative instruments, and hedging and investment strategies.

Introduction to probability, emphasizing concrete problems and applications; random variables, expectation, conditional probability, law of large numbers, central limit theorem, stochastic processes.

Instruction and practice in solving a wide variety of mathematical and logical problems as seen in the Putnam competition.

The foundations of geometry, including work in Euclidean and non-Euclidean geometries.

Students will be mentored by a calculus instructor and will participate in the planning and instruction of a recitation section for a calculus course. Skills and topics for teaching Calculus AP will be included.

Provides opportunities for students to consolidate their mathematical knowledge, and to obtain a perspective on the meaning and significance of that knowledge. Course work will emphasize communication skills, including reading, writing, and speaking mathematics.

Finite algebraic structures relevant to computers: groups, graphs, groups and computer design, group codes, semigroups, finite-state machines.

Construction, development, and study of mathematical models for real situations; basic examples, model construction, Markov chain models, models for linear optimization, selected case studies.

Basic methods and techniques for solving optimization problems; n-dimensional geometry and convex sets; classical and search optimization of functions of one and several variables; linear, nonlinear, and integer programming.

Ordinary differential equations, with attention to theory.

Completeness of the real line, Bolzano-Weierstrass theorem and Heine-Borel theorem; continuous functions including uniform convergence and completeness of C[a,b]; Riemann integration and the Darboux Criterion.

Derivative, including uniform convergence, the mean value theorem, and Taylor's Theorem; absolute and uniform convergence of series, completeness of sequence spaces, dual spaces; real analytic functions; functions of bounded variation, the Stieltjes integral, and the dual of C[a,b].

Topology of n-dimensional space, differential calculus in n-dimensional space, inverse and implicit function theorems.

Analytic functions, integration, power series, residues, and conformal mapping.

Vector analysis; solution of partial differential equations by the method of separation of variables; introduction to orthogonal functions including Bessel functions.

Examples and classification of two-dimensional manifolds, covering spaces, the Brouwer theorem, and other selected topics.

Actuarial models for insurance and annuities. Severity-of-loss and frequency-of-loss models, aggregate models, risk models, empirical estimation.

Actuarial models for insurance and annuities. Statistical estimation procedures, credibility theory, and pricing and reserving.

Survival models and their estimation. Distribution of the time-to-death random variable and its significance for insurance and annuity functions, net premiums, and reserves.

Parametric survival models with multiple-life states; life insurance and annuity premium calculations; reserving and profit measures; participating insurances, pension plans, and retirement benefits.

Measurement of interest (including accumulated and present value factors), annuities certain, yield rates, amortization schedules and sinking funds, bonds and related securities, derivative instruments, and hedging and investment strategies.
Last offered fall 2018; replaced by Math 3050 beginning fall 2019.

Statistical inference including hypothesis testing, confidence intervals, estimators, and goodness-of-fit.

Markov chains, Poisson process, and Brownian motion.

Gaussian elimination and LU factorization, tridiagonal systems, vector and matrix norms, singular value decomposition, condition number, least squares problem, QR factorization, iterative methods, power methods for eigenvalues and eigenvectors, applications.

An introduction to numerical methods in basic analysis, including root-finding, polynomial interpolation, numerical integration and differentiation, splines and wavelets.

Numerical solutions to initial value problems and boundary value problems for ordinary and partial differential equations.

Vector spaces, linear transformations, determinants, eigenvalues and eigenvectors, and topics such as inner product space and canonical forms.

Rigorous development of the real numbers, sets, relations, product spaces, order and cardinality.

Fundamental concepts of undirected and directed graphs, trees, connectivity and traversability, planarity, colorability, network flows, matching theory and applications.

Topics selected from permutations and combinations, generating functions, principle of inclusion and exclusion, configurations and designs, matching theory, existence problems, applications.

Divisibility, Euclidean algorithm, prime numbers, congruences, and topics such as Chinese remainder theorem and sums of integral squares.

Elementary properties of sets, relations, mappings, integers; groups, subgroups, normal subgroups, quotient groups, homomorphisms, automorphisms, and permutation groups; elementary properties of rings.

Ideals in rings, factorization in polynomial rings, unique factorization and Euclidean domains, field extensions, splitting fields, finite fields, Galois theory.

Fourier analysis on the real line, the integers, and finite cyclic groups; the fast Fourier transform; generalized functions; attention to modern applications and computational methods.

First-order partial differential equations and systems, canonical second-order linear equations, Green's functions, method of characteristics, properties of solutions, and applications.

Sturm-Liouville problems, orthogonal functions (Bessel, Laguerre, Legendre, Hermite), orthogonal expansions including Fourier series, recurrence relations and generating functions, gamma and beta functions, Chebychev polynomials, and other topics.

This course will have substantial mathematical content; topics such as early Greek mathematics, from Euclid to Archimedes; algebra and number theory from Diophantus to the present; the calculus of Newton and Leibniz; the renewed emphasis on rigor and axiomatic foundations in the 19th and 20th centuries; interactions of mathematics with technology and the natural sciences; biographies of significant mathematicians.

This course is intended to provide opportunities for students to learn about mathematical research in a vertically integrated learning and research community. Undergraduate students, graduate students, post-doctoral researchers and faculty may work together as a unit to learn and create new mathematics. Possible formats include group reading and exposition, group research projects, and written and oral presentations. Undergraduate students may have a research capstone experience or write an honors thesis as part of this course.

Mathematics selected from nationally recognized curriculum standards for the elementary grades, treated with attention to depth and the specific needs of teachers.

Mathematics selected from nationally recognized curriculum standards for the middle grades, treated with attention to depth and the specific needs of teachers.

Mathematics selected from nationally recognized curriculum standards for high school, treated with attention to depth and the specific needs of teachers.

Topics of interest to teachers of secondary school mathematics.

Practical training in the teaching of undergraduate mathematics; how to write mathematics for publication; other issues relating to mathematical exposition.

Practical training in the written and oral presentation of mathematical papers; the teaching of mathematics and the uses of technology in the mathematics classroom.

Groups: Group actions and Sylow Theorems, finitely generated abelian groups; rings and modules: PIDs, UFDs, finitely generated modules over a PID, applications to Jordan canonical form, exact sequences.

Fields: algebraic, transcendental, normal, separable field extensions; Galois theory, simple and semisimple algebras, Wedderburn theorem, group representations, Maschke’s theorem, multilinear algebra.

Commutative rings and modules, prime ideals, localization, noetherian rings, primary decomposition, integral extensions and Noether normalization, the Nullstellensatz, dimension, flatness, graded rings, Hilbert polynomial, valuations, regular rings, homological dimension, depth, completion, Cohen-Macaulay modules.

Topics in number theory, such as algebraic integers, ideal class group, Galois theory of prime ideals, cyclotomic fields, class field theory, Gauss sums, quadratic fields, local fields, elliptic curves, L-functions and Dirichlet series, modular forms, Dirichlet's theorem and the Prime Number theorem, Diophantine equations, Circle method.

Topics in algebraic geometry, such as affine and projective varieties, morphisms and rational mappings, nonsingular varieties, sheaves and schemes, sheaf cohomology, algebraic curves and surfaces, elliptic curves, toric varieties, real algebraic geometry.

Representations of finite groups, group algebras, character theory, induced representations, Frobenius reciprocity, Lie algebras and their structure theory, classification of semisimple Lie algebras, universal enveloping algebras and the PBW theorem, highest weight representations, Verma modules, and finite-dimensional representations.

Modules over a ring, projective and injective modules and resolutions, abelian categories, functors and derived functors, Tor and Ext, homological dimension of rings and modules, spectral sequences, and derived categories.

Advanced topics such as commutative rings, homological algebra, algebraic curves, or algebraic geometry.

Advanced topics such as algebraic number theory, algebraic semigroups, quadratic forms, or algebraic K-theory.

Abstract measure and integration theory with application to Lebesgue measure on the real line and Euclidean space.

Existence and uniqueness theorems, approximation methods, linear equations, linear systems, stability theory; other topics such as boundary value problems.

Finite difference methods; finite element methods; iterative methods; methods of parallel computing; applications to the sciences and engineering.

Banach spaces and their generalizations; Baire category, Banach-Steinhaus, open mapping, closed graph, and Hahn-Banach theorems; duality in Banach spaces, weak topologies; other topics such as commutative Banach algebras, spectral theory, distributions, and Fourier transforms.

Theory of holomorphic functions of one complex variable; path integrals, power series, singularities, mapping properties, normal families, other topics.

Probability spaces, random variables and expectations, independence, convergence concepts, laws of large numbers, convergence of series, law of iterated logarithm, characteristic functions, central limit theorem, limiting distributions, martingales.

Brownian motion, basic stochastic calculus, applications to finance.

Wiener process, stochastic integrals, stochastic differential equations.

Lie groups, Lie algebras, subgroups, homomorphisms, the exponential map. Also topics in finite and infinite dimensional representation theory.

Fourier series; Fourier transform; windowed Fourier transform or short-time Fourier transform; the continuous wavelet transform; discrete wavelet transform; multiresolution analysis; construction of wavelets.

Advanced topics such as topological vector spaces, Banach algebras, operator theory, or nonlinear functional analysis

Advanced topics in the mathematics of material science, including mathematical techniques for the design of optimal structural materials, solution of problems in fracture mechanics, design of photonic band gap materials, and solution of basic problems in the theory of superconductivity.

Sobolev spaces. Theory of second order scalar elliptic equations: existence, uniqueness and regularity. Additional topics such as: Direct methods of the calculus of variations, parabolic equations, eigenvalue problems.

Problems of existence and enumeration in the study of arrangements of elements into sets; combinations and permutations; other topics such as generating functions, recurrence relations, inclusion-exclusion, Polya’s theorem, graphs and digraphs, combinatorial designs, incidence matrices, partially ordered sets, matroids, finite geometries, Latin squares, difference sets, matching theory.

Matchings and coverings, connectivity, planar graphs, colorings, flows, Hamilton graphs, Ramsey theory, topological graph theory, graph minors.

Advanced topics such as combinatorics, graph theory, automata theory, or optimization.

Basic notions of general topology, with emphasis on Euclidean and metric spaces, continuous and differentiable functions, inverse function theorem and its consequences. (This is the catalog description only: Be sure to compare with the current syllabus which includes the fundamental group.)

Theory of the fundamental group and covering spaces including the Seifert-Van Kampen theorem; universal covering space; classification of covering spaces; selected areas from algebraic or general topology.

Basic concepts of homology, cohomology, and homotopy theory.

Manifolds, vector fields, vector bundles, transversality, Riemannian geometry, other topics.

Advanced topics such as advanced algebraic topology, transformation groups, surgery theory, sheaf theory, or fiber bundles.

Gaussian elimination: LU and Cholesky factorizations; Least squares problem: QR factorization and Householder algorithm, backward stability, singular value decomposition and conditioning; Iterative methods: Jacobi, Gauss-Seidel and conjugate gradient; Eigenproblems: power methods and QR algorithm.

"S"/"U" grading.

"S"/"U" grading.