Applied Mathematics

MATH UN1003 COLLEGE ALGEBRA-ANLYTC GEOMTRY. 3.00 points.

Prerequisites: score of 550 on the mathematics portion of the SAT completed within the last year, or the appropriate grade on the General Studies Mathematics Placement Examination. For students who wish to study calculus but do not know analytic geometry. Algebra review, graphs and functions, polynomial functions, rational functions, conic sections, systems of equations in two variables, exponential and logarithmic functions, trigonometric functions and trigonometric identities, applications of trigonometry, sequences, series, and limits

MATH UN1101 CALCULUS I. 3.00 points.

Prerequisites: (see Courses for First-Year Students). Functions, limits, derivatives, introduction to integrals, or an understanding of pre-calculus will be assumed. (SC)

MATH UN1102 CALCULUS II. 3.00 points.

Prerequisites: MATH UN1101 or the equivalent.
Prerequisites: MATH UN1101 or the equivalent. Methods of integration, applications of the integral, Taylors theorem, infinite series. (SC)

MATH UN1201 CALCULUS III. 3.00 points.

Prerequisites: MATH UN1101 or the equivalent
Prerequisites: MATH UN1101 or the equivalent Vectors in dimensions 2 and 3, complex numbers and the complex exponential function with applications to differential equations, Cramers rule, vector-valued functions of one variable, scalar-valued functions of several variables, partial derivatives, gradients, surfaces, optimization, the method of Lagrange multipliers. (SC)

MATH UN1202 CALCULUS IV. 3.00 points.

Prerequisites: MATH UN1102 and MATH UN1201 or the equivalent
Prerequisites: MATH UN1102 and MATH UN1201 or the equivalent Multiple integrals, Taylor's formula in several variables, line and surface integrals, calculus of vector fields, Fourier series. (SC)

MATH UN1205 ACCELERATED MULTIVARIABLE CALC. 4.00 points.

Prerequisites: (MATH UN1101 and MATH UN1102)
Prerequisites: (MATH UN1101 and MATH UN1102) Vectors in dimensions 2 and 3, vector-valued functions of one variable, scalar-valued functions of several variables, partial derivatives, gradients, optimization, Lagrange multipliers, double and triple integrals, line and surface integrals, vector calculus. This course is an accelerated version of MATH UN1201 - MATH UN1202. Students taking this course may not receive credit for MATH UN1201 and MATH UN1202

MATH UN1207 HONORS MATHEMATICS A. 4.00 points.

Prerequisites: (see Courses for First-Year Students). The second term of this course may not be taken without the first. Multivariable calculus and linear algebra from a rigorous point of view. Recommended for mathematics majors. Fulfills the linear algebra requirement for the major. (SC)

MATH UN1208 HONORS MATHEMATICS B. 4.00 points.

Prerequisites: (see Courses for First-Year Students).
Prerequisites: (see Courses for First-Year Students). The second term of this course may not be taken without the first. Multivariable calculus and linear algebra from a rigorous point of view. Recommended for mathematics majors. Fulfills the linear algebra requirement for the major. (SC)

MATH UN2000 INTRO TO HIGHER MATHEMATICS. 3.00 points.

Introduction to understanding and writing mathematical proofs. Emphasis on precise thinking and the presentation of mathematical results, both in oral and in written form. Intended for students who are considering majoring in mathematics but wish additional training. CC/GS: Partial Fulfillment of Science Requirement. BC: Fulfillment of General Education Requirement: Quantitative and Deductive Reasoning (QUA)

MATH UN2005 INTRODUCTION TO MATHEMATICS PROOFS. 0.00 points.

This is a seminar course that covers the basics of mathematical proofs and in particular the epsilon-delta argument in single variable calculus. Students who have little experience with mathematical proofs are strongly encouraged to take this course concurrently with Honors Math, Into to Modern Algebra, or Intro to Modern Analysis

MATH BC2006 COMBINATORICS. 3.00 points.

MATH UN2010 LINEAR ALGEBRA. 3.00 points.

Prerequisites: MATH UN1201 or the equivalent.
Matrices, vector spaces, linear transformations, eigenvalues and eigenvectors, canonical forms, applications. (SC)

MATH UN2015 Linear Algebra and Probability. 3.00 points.

Linear algebra with a focus on probability and statistics. The course covers the standard linear algebra topics: systems of linear equations, matrices, determinants, vector spaces, bases, dimension, eigenvalues and eigenvectors, the Spectral Theorem and singular value decompositions. It also teaches applications of linear algebra to probability, statistics and dynamical systems giving a background sufficient for higher level courses in probability and statistics. The topics covered in the probability theory part include conditional probability, discrete and continuous random variables, probability distributions and the limit theorems, as well as Markov chains, curve fitting, regression, and pattern analysis. The course contains applications to life sciences, chemistry, and environmental life sciences. No a prior i background in the life sciences is assumed. This course is best suited for students who wish to focus on applications and practical approaches to problem solving. It is recommended to students majoring in engineering, technology, life sciences, social sciences, and economics. Math majors, joint majors, and math concentrators must take MATH UN2010 Linear Algebra, which focuses on linear algebra concepts and foundations that are needed for upper-level math courses. MATH UN2015 (Linear Algebra and Probability) does NOT replace MATH UN2010 (Linear Algebra) as prerequisite requirements of math courses. Students may not receive full credit for both courses MATH UN2010 and MATH UN2015

MATH UN2030 ORDINARY DIFFERENTIAL EQUATIONS. 3.00 points.

Prerequisites: MATH UN1102 and MATH UN1201 or the equivalent.
Prerequisites: MATH UN1102 and MATH UN1201 or the equivalent. Special differential equations of order one. Linear differential equations with constant and variable coefficients. Systems of such equations. Transform and series solution techniques. Emphasis on applications

MATH UN2500 ANALYSIS AND OPTIMIZATION. 3.00 points.

Prerequisites: MATH UN1102 and MATH UN1201 or the equivalent and MATH UN2010.
Prerequisites: MATH UN1102 and MATH UN1201 or the equivalent and MATH UN2010. Mathematical methods for economics. Quadratic forms, Hessian, implicit functions. Convex sets, convex functions. Optimization, constrained optimization, Kuhn-Tucker conditions. Elements of the calculus of variations and optimal control. (SC)

MATH UN3007 COMPLEX VARIABLES. 3.00 points.

Prerequisites: MATH UN1202 An elementary course in functions of a complex variable.
Prerequisites: MATH UN1202 An elementary course in functions of a complex variable. Fundamental properties of the complex numbers, differentiability, Cauchy-Riemann equations. Cauchy integral theorem. Taylor and Laurent series, poles, and essential singularities. Residue theorem and conformal mapping.(SC)

MATH UN3020 NUMBER THEORY AND CRYPTOGRAPHY. 3.00 points.

Prerequisites: one year of calculus.
Prerequisites: one year of calculus. Prerequisite: One year of Calculus. Congruences. Primitive roots. Quadratic residues. Contemporary applications

MATH UN3025 MAKING, BREAKING CODES. 3.00 points.

Prerequisites: (MATH UN1101 and MATH UN1102 and MATH UN1201) and and MATH UN2010.
Prerequisites: (MATH UN1101 and MATH UN1102 and MATH UN1201) and and MATH UN2010. A concrete introduction to abstract algebra. Topics in abstract algebra used in cryptography and coding theory

MATH UN3028 PARTIAL DIFFERENTIAL EQUATIONS. 3.00 points.

Prerequisites: MATH UN3027 and MATH UN2010 or the equivalent
Prerequisites: (MATH UN2010 and MATH UN2030) or the equivalent introduction to partial differential equations. First-order equations. Linear second-order equations; separation of variables, solution by series expansions. Boundary value problems

MATH UN3050 DISCRETE TIME MODELS IN FINANC. 3.00 points.

Prerequisites: (MATH UN1102 and MATH UN1201) or (MATH UN1101 and MATH UN1102 and MATH UN1201) and MATH UN2010 Recommended: MATH UN3027 (or MATH UN2030 and SIEO W3600).
Prerequisites: (MATH UN1102 and MATH UN1201) or (MATH UN1101 and MATH UN1102 and MATH UN1201) and MATH UN2010 Recommended: MATH UN3027 (or MATH UN2030 and SIEO W3600). Elementary discrete time methods for pricing financial instruments, such as options. Notions of arbitrage, risk-neutral valuation, hedging, term-structure of interest rates

MATH UN3386 DIFFERENTIAL GEOMETRY. 3.00 points.

Prerequisites: MATH UN1202 or the equivalent.

Local and global differential geometry of submanifolds of Euclidiean 3-space. Frenet formulas for curves. Various types of curvatures for curves and surfaces and their relations. The Gauss-Bonnet theorem.

MATH UN3901 SUPERVISED READINGS I. 1.00-3.00 points.

Prerequisites: The written permission of the faculty member who agrees to act as sponsor (sponsorship limited to full-time instructors on the staff list), as well as the permission of the Director of Undergraduate Studies. The written permission must be deposited with the Director of Undergraduate Studies before registration is completed. Guided reading and study in mathematics. A student who wishes to undertake individual study under this program must present a specific project to a member of the staff and secure his or her willingness to act as sponsor. Written reports and periodic conferences with the instructor. Supervising Readings do NOT count towards major requirements, with the exception of an advanced written approval by the DUS

MATH UN3902 SUPERVISED READINGS II. 1.00-3.00 points.

Prerequisites: The written permission of the faculty member who agrees to act as sponsor (sponsorship limited to full-time instructors on the staff list), as well as the permission of the Director of Undergraduate Studies. The written permission must be deposited with the Director of Undergraduate Studies before registration is completed. Guided reading and study in mathematics. A student who wishes to undertake individual study under this program must present a specific project to a member of the staff and secure his or her willingness to act as sponsor. Written reports and periodic conferences with the instructor. Supervising Readings do NOT count towards major requirements, with the exception of an advanced written approval by the DUS

MATH UN3951 UNDERGRADUATE SEMINARS I. 3.00 points.

Prerequisites: Two years of calculus, at least one year of additional mathematics courses, and the director of undergraduate studies' permission.
Prerequisites: Two years of calculus, at least one year of additional mathematics courses, and the director of undergraduate studies permission. The subject matter is announced at the start of registration and is different in each section. Each student prepares talks to be given to the seminar, under the supervision of a faculty member or senior teaching fellow

MATH UN3952 UNDERGRADUATE SEMINARS II. 3.00 points.

Prerequisites: two years of calculus, at least one year of additional mathematics courses, and the director of undergraduate studies' permission.
Prerequisites: two years of calculus, at least one year of additional mathematics courses, and the director of undergraduate studies' permission. The subject matter is announced at the start of registration and is different in each section. Each student prepares talks to be given to the seminar, under the supervision of a faculty member or senior teaching fellow. Prerequisite: two years of calculus, at least one year of additional mathematics courses, and the director of undergraduate studies' permission

MATH UN3994 SENIOR THESIS IN MATHEMATICS I. 4.00 points.

Majors in Mathematics are offered the opportunity to write an honors senior thesis under the guidance of a faculty member. Interested students should contact a faculty member to determine an appropriate topic, and receive written approval from the faculty advisor and the Director of Undergraduate Studies (faculty sponsorship is limited to full-time instructors on the staff list). Research is conducted primarily during the fall term; the final paper is submitted to the Director of Undergraduate Studies during the subsequent spring term. MATH UN3994 SENIOR THESIS IN MATHEMATICS I must be taken in the fall term, during which period the student conducts primary research on the agreed topic. An optional continuation course MATH UN3995 SENIOR THESIS IN MATHEMATICS II is available during the spring. The second term of this sequence may not be taken without the first. Registration for the spring continuation course has no impact on the timeline or outcome of the final paper. Sections of SENIOR THESIS IN MATHEMATICS I and II do NOT count towards the major requirements, with the exception of an advanced written approval by the DUS

MATH UN3995 SENIOR THESIS IN MATHEMATICS II. 2.00 points.

Majors in Mathematics are offered the opportunity to write an honors senior thesis under the guidance of a faculty member. Interested students should contact a faculty member to determine an appropriate topic, and receive written approval from the faculty advisor and the Director of Undergraduate Studies (faculty sponsorship is limited to full-time instructors on the staff list). Research is conducted primarily during the fall term; the final paper is submitted to the Director of Undergraduate Studies during the subsequent spring term. MATH UN3994 SENIOR THESIS IN MATHEMATICS I must be taken in the fall term, during which period the student conducts primary research on the agreed topic. An optional continuation course MATH UN3995 SENIOR THESIS IN MATHEMATICS II is available during the spring. The second term of this sequence may not be taken without the first. Registration for the spring continuation course has no impact on the timeline or outcome of the final paper. Sections of SENIOR THESIS IN MATHEMATICS I and II do NOT count towards the major requirements, with the exception of an advanced written approval by the DUS

MATH GU4007 ANALYTIC NUMBER THEORY. 3.00 points.

Prerequisites: MATH UN3007
Prerequisites: MATH UN3007 A one semeser course covering the theory of modular forms, zeta functions, L -functions, and the Riemann hypothesis. Particular topics covered include the Riemann zeta function, the prime number theorem, Dirichlet characters, Dirichlet L-functions, Siegel zeros, prime number theorem for arithmetic progressions, SL (2, Z) and subgroups, quotients of the upper half-plane and cusps, modular forms, Fourier expansions of modular forms, Hecke operators, L-functions of modular forms

MATH GU4032 FOURIER ANALYSIS. 3.00 points.

Prerequisites: three terms of calculus and linear algebra or four terms of calculus.
Prerequisites: three terms of calculus and linear algebra or four terms of calculus. Prerequisite: three terms of calculus and linear algebra or four terms of calculus. Fourier series and integrals, discrete analogues, inversion and Poisson summation formulae, convolution. Heisenberg uncertainty principle. Stress on the application of Fourier analysis to a wide range of disciplines

MATH GU4041 INTRO MODERN ALGEBRA I. 3.00 points.

Prerequisites: MATH UN1102 and MATH UN1202 and MATH UN2010 or the equivalent
Prerequisites: MATH UN1102 and MATH UN1202 and MATH UN2010 or the equivalent. The second term of this course may not be taken without the first. Groups, homomorphisms, normal subgroups, the isomorphism theorems, symmetric groups, group actions, the Sylow theorems, finitely generated abelian groups

MATH GU4042 INTRO MODERN ALGEBRA II. 3.00 points.

Prerequisites: MATH UN1102 and MATH UN1202 and MATH UN2010 or the equivalent.
Prerequisites: MATH UN1102 and MATH UN1202 and MATH UN2010 or the equivalent. The second term of this course may not be taken without the first. Rings, homomorphisms, ideals, integral and Euclidean domains, the division algorithm, principal ideal and unique factorization domains, fields, algebraic and transcendental extensions, splitting fields, finite fields, Galois theory

MATH GU4043 ALGEBRAIC NUMBER THEORY. 3.00 points.

Prerequisites: MATH GU4041 and MATH GU4042 or the equivalent
Prerequisites: MATH GU4041 and MATH GU4042 or the equivalent Algebraic number fields, unique factorization of ideals in the ring of algebraic integers in the field into prime ideals. Dirichlet unit theorem, finiteness of the class number, ramification. If time permits, p-adic numbers and Dedekind zeta function

MATH GU4044 REPRESENTATNS OF FINITE GROUPS. 3.00 points.

Prerequisites: MATH UN2010 and MATH GU4041 or the equivalent.
Prerequisites: MATH UN2010 and MATH GU4041 or the equivalent. Finite groups acting on finite sets and finite dimensional vector spaces. Group characters. Relations with subgroups and factor groups. Arithmetic properties of character values. Applications to the theory of finite groups: Frobenius groups, Hall subgroups and solvable groups. Characters of the symmetric groups. Spherical functions on finite groups

MATH GU4045 ALGEBRAIC CURVES. 3.00 points.

Prerequisites: (MATH GU4041 and MATH GU4042) and MATH UN3007
Prerequisites: (MATH GU4041 and MATH GU4042) and MATH UN3007 Plane curves, affine and projective varieties, singularities, normalization, Riemann surfaces, divisors, linear systems, Riemann-Roch theorem

MATH GU4051 TOPOLOGY. 3.00 points.

Prerequisites: (MATH UN1202 and MATH UN2010) and rudiments of group theory (e.g., MATH GU4041). MATH UN1208 or MATH GU4061 is recommended, but not required.
Prerequisites: (MATH UN1202 and MATH UN2010) and rudiments of group theory (e.g. MATH GU4041). MATH UN1208 or MATH GU4061 is recommended, but not required. Metric spaces, continuity, compactness, quotient spaces. The fundamental group of topological space. Examples from knot theory and surfaces. Covering spaces

MATH GU4052 INTRODUCTION TO KNOT THEORY. 3.00 points.

CC/GS: Partial Fulfillment of Science Requirement

Prerequisites: MATH GU4051 Topology and / or MATH GU4061 Introduction To Modern Analysis I (or equivalents). Recommended (can be taken concurrently): MATH UN2010 linear algebra, or equivalent.
Prerequisites: MATH GU4051 Topology and / or MATH GU4061 Introduction To Modern Analysis I (or equivalents). Recommended (can be taken concurrently): MATH UN2010 linear algebra, or equivalent. The study of algebraic and geometric properties of knots in R＾3, including but not limited to knot projections and Reidemeisters theorm, Seifert surfaces, braids, tangles, knot polynomials, fundamental group of knot complements. Depending on time and student interest, we will discuss more advanced topics like knot concordance, relationship to 3-manifold topology, other algebraic knot invariants

MATH GU4053 INTRO TO ALGEBRAIC TOPOLOGY. 3.00 points.

Prerequisites: MATH UN2010 and MATH GU4041 and MATH GU4051
Prerequisites: MATH UN2010 and MATH GU4041 and MATH GU4051 The study of topological spaces from algebraic properties, including the essentials of homology and the fundamental group. The Brouwer fixed point theorem. The homology of surfaces. Covering spaces

MATH GU4061 INTRO MODERN ANALYSIS I. 3.00 points.

Prerequisites: MATH UN1202 or the equivalent, and MATH UN2010. The second term of this course may not be taken without the first.
Prerequisites: MATH UN1202 or the equivalent, and MATH UN2010. The second term of this course may not be taken without the first. Real numbers, metric spaces, elements of general topology, sequences and series, continuity, differentiation, integration, uniform convergence, Ascoli-Arzela theorem, Stone-Weierstrass theorem

MATH GU4062 INTRO MODERN ANALYSIS II. 3.00 points.

Prerequisites: MATH UN1202 or the equivalent, and MATH UN2010. The second term of this course may not be taken without the first.
Prerequisites: MATH UN1202 or the equivalent, and MATH UN2010. The second term of this course may not be taken without the first. Power series, analytic functions, Implicit function theorem, Fubini theorem, change of variables formula, Lebesgue measure and integration, function spaces

MATH GU4065 HONORS COMPLEX VARIABLES. 3.00 points.

Prerequisites: (MATH UN1207 and MATH UN1208) or MATH GU4061
Prerequisites: (MATH UN1207 and MATH UN1208) or MATH GU4061 A theoretical introduction to analytic functions. Holomorphic functions, harmonic functions, power series, Cauchy-Riemann equations, Cauchy's integral formula, poles, Laurent series, residue theorem. Other topics as time permits: elliptic functions, the gamma and zeta function, the Riemann mapping theorem, Riemann surfaces, Nevanlinna theory

MATH GU4081 INTRO-DIFFERENTIABLE MANIFOLDS. 3.00 points.

Prerequisites: (MATH GU4051 or MATH GU4061) and MATH UN2010
Prerequisites: (MATH GU4051 or MATH GU4061) and MATH UN2010 Concept of a differentiable manifold. Tangent spaces and vector fields. The inverse function theorem. Transversality and Sards theorem. Intersection theory. Orientations. Poincare-Hopf theorem. Differential forms and Stokes theorem

MATH GU4155 PROBABILITY THEORY. 3.00 points.

Prerequisites: MATH GU4061 or MATH UN3007
Prerequisites: MATH GU4061 or MATH UN3007 A rigorous introduction to the concepts and methods of mathematical probability starting with basic notions and making use of combinatorial and analytic techniques. Generating functions. Convergence in probability and in distribution. Discrete probability spaces, recurrence and transience of random walks. Infinite models, proof of the law of large numbers and the central limit theorem. Markov chains

MATH GU4156 ADVANCED PROBABILITY THEORY. 3.00 points.

This course will cover advance topics in probability, including: the theory of martingales in discrete and in continuous time; Brownian motion and its properties, stochastic integration, ordinary and partial stochastic differential equations; Applications to optimal filtering, stopping, control, and finance; Continuous-time Markov chains, systems of interacting particles, relative entropy dissipation, notions of information theory; Electrical networks, random walks on graphs and groups, percolation

MATH GU4391 INTRO TO QUANTUM MECHANICS. 3.00 points.

This course will focus on quantum mechanics, paying attention to both the underlying mathematical structures as well as their physical motivations and consequences. It is meant to be accessible to students with no previous formal training in quantum theory. The role of symmetry, groups and representations will be stressed

MATH GU4392 INTRO TO QUANTUM MECHANICS II. 3.00 points.

Continuation of GU4391. This course will focus on quantum mechanics, paying attention to both the underlying mathematical structures as well as their physical motivations and consequences. It is meant to be accessible to students with no previous formal training in quantum theory. The role of symmetry, groups and representations will be stressed.