Mathematics Jump To: Jump To: Program Offerings A.B. Minor Offering type A.B. Most first-year students and sophomores interested in science, engineering or finance take courses from the standard calculus and linear algebra sequence 103-104-201-202, which emphasizes concrete computations over more theoretical considerations. Note that 201 and 202 can be taken in either order.Students who are not prepared to begin with 103 may take 100, an introduction to calculus with pre-calculus review. Prospective economics majors can minimally fulfill their mathematics prerequisites with (100)-103-175. Note that 175 covers selected topics from 201, with biology and economics applications in mind. It is recommended that prospective math-track economics/finance majors take the standard sequence 103-104-201-202 instead of 175.More mathematically inclined students, especially prospective physics majors, may opt to replace 201-202 with 203-204, for greater emphasis on theory and more challenging computational problems.Prospective mathematics majors must take at least one course introducing formal mathematical argument and rigorous proofs. The recommended first-year sequence for prospective majors is 215-217-300. Prospective majors who already have substantial experience with university-level proof-based analysis courses may consider the accelerated sequence 216-218 instead. Other possible sequences for prospective majors include 210-215-217-300 or 214-204-203 and 203-204-215, although the latter two are relatively rare. Note that 203 and 204 can be taken in either order. Some of our successful majors became interested in mathematics as a major after taking courses at the 100 level. As an alternative to 104, and as a lead-in to MAT215, students may opt to take 210 in the fall, followed by 215 in the spring. Such students should consult with the junior adviser or the associate director of undergraduate studies as soon as possible to plan a course of study.Placement GuidelinesStudents who need placement advice about 214, 215 or 216 should consult the Department of Mathematics homepage (the "Undergraduate" tab has a section on placement) or contact Professor McConnell, the junior adviser.Students with little or no background in calculus, but with strong pre-calculus skills, should take 103. Alternatively, 100 offers intensive pre-calculus review, along with an introduction to the main ideas of calculus, as preparation for 103. To succeed in 104 or 175, a background in differential calculus at the level of a 5 on the BC Advanced Placement Examination is advised. Students with a stronger calculus preparation in differential and integral calculus, as well as infinite series, may opt to start in 201. Students who possess, in addition, a particularly strong interest in mathematics as well as an SAT mathematics score of at least 750 may opt for 203 or 210 or 214 or 215 or 216 instead.The placement workshops for incoming students at orientation are designed to help students consider all these issues in order to make a good initial self-placement in the mathematics curriculum, which can then be adjusted during drop/add if necessary. Goals for Student Learning Mathematics is a discipline inseparable from scientific and philosophical inquiry. The rigorous and logical thinking that characterizes mathematics is an essential tool for theory-building of any kind because its clarity and precision expose hidden assumptions, inner inconsistencies and deep structural similarities in problems that seem unrelated on the surface. Our courses cover a wide variety of well-established mathematical knowledge that is actively under development by today’s mathematicians and that offers fundamental tools for scientists and engineers of all kinds.Students begin their work in the department with a thorough training in rigorous logical reasoning and mathematical proofs in the context of analysis and linear algebra. Next, they complete a survey of the main areas of modern mathematics by completing core courses in real and complex analysis, in algebra and in geometry/topology or discrete mathematics. Then students are free to take courses exploring a wide variety of topics in both pure and applied mathematics to acquire a good general knowledge of the main areas of current mathematical work.In the independent work, students learn how to move beyond the classical knowledge found in textbooks to explore contemporary research literature through collaboration with their peers and with active researchers in mathematics or applied fields. Through this collaboration, students:Learn how to join a scholarly discussion in progress to orient themselves in a rapidly developing area of research.Build on their broad general knowledge of mathematics and logical reasoning skills in order to develop a working knowledge of a significant area of contemporary mathematics via the research literature.Learn to identify interesting problems they want to investigate and develop their own ideas about how to carry out those investigations.Learn to come up with a complete argument of their own, adapting and expanding ideas and techniques from various sources, as needed.Develop mastery of rigorous logical thinking by constructing complete and correct arguments.Develop mastery of clear mathematical exposition in a manner that allows and invites dialog with other scholars in the intended audience, where important definitions and theorems are clearly explained and contributions of other scholars are properly acknowledged.The program produces critical and creative thinkers with a broad general knowledge of contemporary mathematics and with the analytic and expository skills needed for collaborative problem-solving in any quantitative setting. Advanced Placement Incoming first-year students who report a score of 5 on the BC Advanced Placement Examination (or a 7 on the IC [higher level] math examination, or an A on the British A-level math exam) will receive one unit of advanced placement credit for MAT103. For more information, please consult the website of the Office of the Dean of the College. Prerequisites To major in mathematics, students need a strong foundation in linear algebra and analysis, in both one and several variables, as well as experience in understanding and writing rigorous mathematical proofs. Generally, prospective majors with a very strong background in calculus are strongly recommended to start in 215, followed by 217 and 300. Prospective majors with extensive prior experience with calculus and rigorous proofs can start in 216, followed by 218. Students with an interest in rigorous proofs may start in 210 or 214 and then continue to the 215-217-300 sequence.Students who are undecided, or students who are initially planning to major in a different quantitative discipline, may opt instead to begin their calculus and linear algebra work in the 203-204 or the 103-104-201-202 sequences, where proofs are not an emphasis. From these sequences, prospective mathematics majors should consult with the junior adviser or the associate director of undergraduate studies early on, in order to plan a program that includes as much of the 215-217-300 sequence as possible.Most majors begin taking courses at the 300 level by the second semester of sophomore year, in preparation for their junior independent work. The junior adviser meets with students individually to help them plan a course of study. A student’s path through the upper-division courses will naturally depend on their long-term goals and prior experience.Further information for prospective majors is available on the department home page. Program of Study Students must complete four core requirements:one course in real analysis (e.g., 300 or 320 or 325 or 425 or 385)one course in complex analysis (e.g., 330 or 335)one course in algebra (e.g., 340 or 345)one course in geometry or topology (e.g., 355 or 365 or 560)It is recommended that students complete some of these core requirements by the end of their sophomore year. Completing these core courses early allows for more options for junior and senior independent work.Note: One course in discrete mathematics (e.g., 375, 377 or 378) can replace the geometry/topology core requirement, if desired.In addition to the four core requirements, students must complete an additional four courses at the 300 level or higher, up to three of which may be cognate courses outside the mathematics department, with permission from the junior or senior advisers or the director of undergraduate studies. No more than two of the eight courses may be reading courses.The departmental grade (the average grade of the eight departmental courses) together with grades and reports on independent work form the basis on which honors and prizes are awarded upon graduation.Students should refer to Course Offerings to determine which courses are offered in a given term. Programs of study in various fields of pure mathematics and applied mathematics are available. Appropriate plans of study may be arranged for students interested in specialized related fields such as physics, the biological sciences, finance and computer science, for example. For students interested in these areas, a coherent program containing up to three courses in a cognate field may be approved. Independent Work All departmental students engage in independent work, supervised by a member of the department chosen in consultation with a departmental adviser. The junior year independent work generally consists of participating actively in a junior seminar in both the fall and spring semesters. Alternatively, a student may opt to replace one junior seminar with supervised reading in a special subject and then writing a paper based on that reading. The independent work in the senior year centers on writing a senior thesis. A substantial percentage of our majors work with faculty in other departments on their senior project. Senior Departmental Examination Each senior takes an oral examination based on the senior thesis and the broader subfield to which it contributes. A departmental committee conducts the examination in May. Offering type Minor The mathematics minor provides a structure that encourages students to explore mathematical ideas for their own sake in the open-minded, intellectually curious way that is the main goal of a liberal arts education. The program allows both for a minor in mathematics that supplements students’ work in their major field of concentration, and also for a minor in mathematics that allows students to explore new directions. A student who decides to minor in mathematics might be someone who wants to learn more math because they have always been curious and enthusiastic math learners, and they want to build a deeper understanding of the mathematical ideas that appear throughout their work in their chosen disciplines. We also welcome students from the humanities or social sciences who have enjoyed learning math and would like to continue to explore more advanced ideas in mathematics, whether or not they are directly relevant to their major work. Goals for Student Learning The goal of the minor in mathematics is to allow students to explore contemporary mathematics, either as a topic for its own sake or to gain a deeper knowledge of a specific area most relevant for the students’ work in their major field of concentration. In consultation with the minor program director of the mathematics department, the student should develop a plan that complements their work in their chosen field of concentration. The plan can be to gain a broad knowledge of mathematics, in which case we recommend courses in algebra, complex analysis and real analysis. Alternatively, students with a strong interest in a more specialized area of mathematics can choose a sequence of courses in that field. Prerequisites The recommended preparation for the minor is the same as for the major in math: knowledge of calculus (both single and several variable), linear algebra and familiarity with proofs. These criteria are satisfied by taking one each of the following: • MAT201/203/218 (several variable calculus);• MAT202/204/217 (linear algebra); and • MAT214/215/216/217/218 (familiarity with proofs). (Note that 217 and 218 each cover two of the recommended preparations.) With the prior approval of the minor program director, students may count one course taken in another department toward the minor prerequisites. These courses should be completed by the end of sophomore year. Admission to the Program The student should contact the minor program director of the mathematics department during the spring of sophomore year to have a course plan approved. Later admission to the Program is allowed during the junior year, though this offers less flexibility in the choice of the required courses. Program of Study In addition to completing the required prerequisite courses, the minor requires four mathematics department courses at the 300 level or higher. By permission, one of these can be a cognate if it fits into the student’s plan and is approved in advance. These courses cannot be counted for the student’s major or for other minors or certificates. In addition to the four mathematics department courses, one Junior Seminar in the mathematics department is required, which can be taken in the senior year. Exceptionally, if the plan of the student justifies it, the Junior Seminar can be replaced by a Junior Paper, which can be done in the senior year or as a summer project. Additional Information Normally, students may not pursue both the minor in PACM and the minor in MAT. However, students who believe they have a compelling curricular reason to pursue both may apply to the directors of both programs for permission to do so. Faculty Chair Igor Rodnianski Associate Chair János Kollár Director of Undergraduate Studies Jennifer M. Johnson (associate) János Kollár Director of Graduate Studies Lue Pan (associate) Chenyang Xu Professor Michael Aizenman Noga M. Alon Manjul Bhargava Sun-Yung A. Chang Maria Chudnovsky Fernando Codá Marques Peter Constantin Mihalis Dafermos Zeev Dvir Charles L. Fefferman David Gabai June E. Huh Alexandru D. Ionescu Nicholas M. Katz Sergiu Klainerman János Kollár Assaf Naor Peter Steven Ozsváth Igor Rodnianski Peter C. Sarnak Will Sawin Paul Seymour Amit Singer Christopher M. Skinner Allan M. Sly Zoltán Szabó Chenyang Xu Paul C. Yang Shou-Wu Zhang Assistant Professor Bjoern Bringmann Matija Bucic Marc Aurèle Tiberius Gilles Jonathan Hanselman Susanna Haziot Ana Menezes Lue Pan Ravi Shankar Jacob Shapiro Jakub Witaszek Ruobing Zhang Associated Faculty John P. Burgess, Philosophy René A. Carmona, Oper Res and Financial Eng Bernard Chazelle, Computer Science Hans P. Halvorson, Philosophy William A. Massey, Oper Res and Financial Eng Frans Pretorius, Physics Robert E. Tarjan, Computer Science Ramon van Handel, Oper Res and Financial Eng Instructor Louis Esser Tangli Ge Sepehr Hajebi Lili He Tongmu He Kimoi Kemboi Dmitry Krachun Hongyi Liu Anubhav Mukherjee Sung Gi Park Semon Rezchikov Joshua X. Wang Mingjia Zhang University Lecturer Jennifer M. Johnson Senior Lecturer Jonathan M. Fickenscher Mark W. McConnell Lecturer Fraser M. Binns Tatyana Chmutova Tatiana K. Howard Justin Lacini Tristan J. Leger Jennifer Li Andrew O'Desky Stan Palasek John T. Sheridan Sahana Vasudevan David Villalobos Liyang Yang Bogdan Zavyalov Visiting Professor Bhargav B. Bhatt Alex Kontorovich Visiting Lecturer with Rank of Professor Camillo De Lellis Helmut H. Hofer Aaron Naber Akshay Venkatesh For a full list of faculty members and fellows please visit the department or program website. Courses MAT 100 - Calculus Foundations Fall/Spring QCR Introduction to limits and derivatives as preparation for further courses in calculus. Fundamental functions (polynomials, rational functions, exponential, logarithmic, trigonometric) and their graphs will be also reviewed. Other topics include tangent and normal lines, linearization, computing area and rates of change. The emphasis will be on learning to think independently and creatively in the mathematical setting. Staff MAT 102 - Survey of Calculus Not offered this year QCR One semester survey of the major concepts and computational techniques of calculus including limits, derivatives and integrals. Emphasis on basic examples and applications of calculus including approximation, differential equations, rates of change and error estimation for students who will take no further calculus. Prerequisites: MAT100 or equivalent. Restrictions: Cannot receive course credit for both MAT103 and MAT102. Provides adequate preparation for MAT175. Three classes. Staff MAT 103 - Calculus I Fall/Spring QCR First semester of calculus. Topics include limits, continuity, the derivative, basic differentiation formulas and applications (curve-sketching, optimization, related rates), definite and indefinite integrals, the fundamental theorem of calculus. Prerequisite: MAT100 or equivalent. Staff MAT 104 - Calculus II Fall/Spring QCR Continuation of MAT103. Topics include techniques of integration, arclength, area, volume, convergence of series and improper integrals, L'Hopital's rule, power series and Taylor's theorem, introduction to differential equations and complex numbers. Prerequisite: MAT103 or equivalent. Staff MAT 175 - Mathematics for Economics/Life Sciences Fall/Spring QCR Survey of topics from multivariable calculus as preparation for future course work in economics or life sciences. Topics include basic techniques of integration, average value, vectors, partial derivatives, gradient, optimization of multivariable functions, and constrained optimization with Lagrange multipliers. Students preparing for math track econometrics and finance courses need MAT201/202 instead. Students who complete 175 can continue in 202 if they wish. Staff MAT 201 - Multivariable Calculus Fall/Spring QCR Vectors in the plane and in space, vector functions and motion, surfaces, coordinate systems, functions of two or three variables and their derivatives, maxima and minima and applications, double and triple integrals, vector fields, and Stokes's theorem. Prerequisite: 104 or equivalent. Staff MAT 202 - Linear Algebra with Applications Fall/Spring QCR Companion course to MAT201. Matrices, linear transformations, linear independence and dimension, bases and coordinates, determinants, orthogonal projection, least squares, eigenvectors and their applications to quadratic forms and dynamical systems. Prerequisite: MAT103 or equivalent. Staff MAT 203 - Advanced Vector Calculus Fall QCR Vector spaces, limits, derivatives of vector-valued functions, Taylor's formula, Lagrange multipliers, double and triple integrals, change of coordinates, surface and line integrals, generalizations of the fundamental theorem of calculus to higher dimensions. More abstract than 201 but more concrete than 216/218. Recommended for prospective physics majors and others with a strong interest in applied mathematics. Prerequisite: MAT104 or equivalent. D. Gabai MAT 204 - Advanced Linear Algebra with Applications Spring QCR Companion course to MAT203. Linear systems of equations, linear independence and dimension, linear transforms, determinants, (real and complex) eigenvectors and eigenvalues, orthogonality, spectral theorem, singular value decomposition, Jordan forms, other topics as time permits. More abstract than MAT202 but more concrete than MAT217. Recommended for prospective physics majors and others with a strong interest in applied mathematics. Prerequisite: MAT104 or equivalent. Staff MAT 214 - Numbers, Equations, and Proofs Fall QCR An introduction to classical number theory to prepare for higher-level courses in the department. Topics include Pythagorean triples and sums of squares, unique factorization, Chinese remainder theorem, arithmetic of Gaussian integers, finite fields and cryptography, arithmetic functions, and quadratic reciprocity. There will be a topic from more advanced or more applied number theory such as p-adic numbers, cryptography, and Fermat's Last Theorem. This course is suitable both for students preparing to enter the mathematics department and for non-majors interested in exposure to higher mathematics. Staff MAT 215 - Single Variable Analysis with an Introduction to Proofs Fall/Spring QCR An introduction to the mathematical discipline of analysis, to prepare for higher-level course work in the department. Topics include the rigorous epsilon-delta treatment of limits, convergence, and uniform convergence of sequences and series. Continuity, uniform continuity, and differentiability of functions. The Heine-Borel theorem, the Riemann integral, conditions for integrability of functions and term by term differentiation and integration of series of functions, Taylor's theorem. Staff MAT 217 - Honors Linear Algebra Spring QCR A rigorous course in linear algebra with an emphasis on proof rather than applications. Topics include vector spaces, linear transformations, inner product spaces, determinants, eigenvalues, the Cayley-Hamilton theorem, Jordan form, the spectral theorem for normal transformations, bilinear and quadratic forms. Staff MAT 218 - Multivariable Analysis and Linear Algebra II Spring QCR Continuation of Multivariable Analysis and Linear Algebra I (MAT 216) from the fall. A rigorous course in analysis with an emphasis on proof rather than applications. Topics include metric spaces, completeness, compactness, total derivatives, partial derivatives, inverse function theorem, implicit function theorem, Riemann integrals in several variables, Fubini. See the department website for details: http://www.math.princeton.edu. Staff MAT 300 - Multivariable Analysis I Fall QCR Continuation of Single Variable Analysis (MAT215) and Honors Linear Algebra (MAT217) needed to prepare for further work in differential geometry, analysis and topology. Calculus on manifolds: Introduces the concept of differentiable manifold, develops the notions of vector fields and differential forms, Stokes' theorem and the de Rham complex. The basic existence theorem in ODEs is used to prove the Frobenius theorem on integrability of plane fields. The intent is to provide the preparation for the courses in differential geometry and topology. Staff MAT 305 - Mathematical Logic Not offered this year QCR A development of logic from the mathematical viewpoint, including propositional and predicate calculus, consequence and deduction, truth and satisfaction, the Goedel completeness and incompleteness theorems. Applications to model theory, recursion theory, and set theory as time permits. Some underclass background in logic or in mathematics is recommended. Staff MAT 320 - Introduction to Real Analysis Fall QCR Introduction to real analysis, including the theory of Lebesgue measure and integration on the line and n-dimensional space and the theory of Fourier series. Prerequisite: MAT201 and MAT202 or equivalent. Staff MAT 323 - Topics in Mathematical Modeling (also APC 323) Not offered this year QCR Draws problems from the sciences and engineering for which mathematical models have been developed and analyzed to describe, understand and predict natural and man-made phenomena. Emphasizes model building strategies, analytical and computational methods, and how scientific problems motivate new mathematics. This interdisciplinary course in collaboration with Molecular Biology, Psychology and the Program in Neuroscience is directed toward upper class undergraduate students and first-year graduate students with knowledge of linear algebra and differential equations. Staff MAT 325 - Analysis I: Fourier Series and Partial Differential Equations Spring QCR Basic facts about Fourier Series, Fourier Transformations, and applications to the classical partial differential equations will be covered. Also Fast Fourier Transforms, Finite Fourier Series, Dirichlet Characters, and applications to properties of primes. Prerequisites: 215, 218, or permission of instructor. Staff MAT 330 - Complex Analysis with Applications Spring QCR The theory of functions of one complex variable, covering power series expansions, residues, contour integration, and conformal mapping. Although the theory will be given adequate treatment, the emphasis of this course is the use of complex analysis as a tool for solving problems. Prerequisite: MAT201 and MAT202 or equivalent. Staff MAT 335 - Analysis II: Complex Analysis Fall QCR Study of functions of a complex variable, with emphasis on interrelations with other parts of mathematics. Cauchy's theorems, singularities, contour integration, power series, infinite products. The gamma and zeta functions and the prime number theorem. Elliptic functions, theta functions, Jacobi's triple product and combinatorics. An overall view of Special Functions via the hypergeometric series. This course is the second semester of a four-semester sequence, but may be taken independently of the other semesters. Staff MAT 345 - Algebra I Fall QCR This course will cover the basics of symmetry and group theory, with applications. Topics include the fundamental theorem of finitely generated abelian groups, Sylow theorems, group actions, and the representation theory of finite groups, rings and modules. Staff MAT 346 - Algebra II Spring QCR Continuation of MAT345. Further develop knowledge of algebraic structures by exploring examples that connect to higher mathematics. Staff MAT 355 - Introduction to Differential Geometry Spring QCR Introduction to geometry of surfaces. Surfaces in Euclidean space, second fundamental form, minimal surfaces, geodesics, Gauss curvature, Gauss-Gonnet formula, uniformization of surfaces, elementary notions of contact geometry. Prerequisite: MAT218 or MAT300, or MAT203 or equivalent. Staff MAT 365 - Topology Fall QCR Introduction to point-set topology, the fundamental group, covering spaces, methods of calculation and applications. Prerequisite: MAT202 or 204 or 218 or equivalent. Z. Szabó MAT 375 - Introduction to Graph Theory (also COS 342) Spring QCR The fundamental theorems and algorithms of graph theory. Topics include: connectivity, matchings, graph coloring, planarity, the four-color theorem, extremal problems, network flows, and related algorithms. Prerequisite: MAT202 or 204 or 217 or equivalent. Staff MAT 377 - Combinatorial Mathematics (also APC 377) Fall QCR Combinatorics is the study of enumeration and structure of discrete objects. These structures are widespread throughout mathematics, including geometry, topology and algebra, as well as computer science, physics and optimization. This course will give an introduction to modern techniques in the field, and how they relate to objects such as polytopes, permutations and hyperplane arrangements. N. Alon MAT 378 - Theory of Games Spring QCR Games in extensive form, pure and behavioral strategies; normal form, mixed strategies, equilibrium points; coalitions, characteristic-function form, imputations, solution concepts; related topics and applications. Prerequisite: MAT202 or 204 or 217 or equivalent. MAT215 or equivalent is recommended. Staff MAT 385 - Probability Theory Fall QCR Sequence of independent trials, applications to number theory and analysis, Monte Carlo method. Markov chains, ergodic theorem for Markov chains. Entropy and McMillan theorem. Random walks, recurrence and non-recurrence; connection with the linear difference equations. Strong laws of large numbers, random series and products. Weak convergence of probability measures, weak Helly theorems, Fourier transforms of distributions. Limit theorems of probability theory. Prerequisite: MAT203 or 218 or equivalent. Staff MAT 393 - Mathematical Programming Not offered this year QCR Linear programs, duality, Dantzig's simplex method; theory of dual linear systems; matrix games, von Neumann's minimax theorem, simplex solution; algorithms for assignment, transport, flow; brief introduction to nonlinear programming. Staff MAT 419 - Topics in Geometry and Number Theory Spring QCR Topics at the intersection of number theory and geometry, including quadratic forms and elliptic curves. See Course Offerings listing for topic details. Prerequisites: MAT 215, 345, 346 or equivalent. Staff MAT 425 - Analysis III: Integration Theory and Hilbert Spaces Spring QCR The theory of Lebesgue integration in n-dimensional space. Differentiation theory. Hilbert space theory and applications to Fourier Transforms, and partial differential equations. Introduction to fractals. This course is the third semester of a four-semester sequence, but may be taken independently of the other semesters. Prerequisites: MAT215 or 218 or equivalent. Staff MAT 427 - Ordinary Differential Equations Not offered this year QCR Introduction to the study of ordinary differential equations; explicit solutions, general properties of solutions, and applications. Topics include explicit solutions of some non-linear equations in two variables by separation of variables and integrating factors, explicit solution of simultaneous linear equations with constant coefficients, explicit solution of some linear equations with variable forcing term by Laplace transform methods, geometric methods (description of the phase portrait), and the fundamental existence and uniqueness theorem. Staff MAT 429 - Topics in Analysis Not offered this year QCR Introduction to incompressible fluid dynamics. The course will give an introduction to the mathematical theory of the Euler equations, the fundamental partial differential equation arising in the study of incompressible fluids. We will discuss several topics in analysis that emerge in the study of these equations: Lebesgue and Sobolev spaces, distribution theory, elliptic PDEs, singular integrals, and Fourier analysis. Content varies from year to year. See Course Offerings listing for topic details. Staff MAT 449 - Topics in Algebra Fall QCR Topics in algebra selected from areas such as representation theory of finite groups and the theory of Lie algebras. Prerequisite: MAT 345 or MAT 346. Staff MAT 459 - Topics in Geometry Not offered this year QCR Elements of smooth manifold theory, tensors, and differential forms, Riemannian metrics, connection and curvature; selected applications to Hodge theory, curvature in topology and general relativity. Staff MAT 478 - Topics In Combinatorics Spring QCR This course will cover topics in Extremal Combinatorics including ones motivated by questions in other areas like Computer Science, Information Theory, Number Theory and Geometry. The subjects that will be covered include Graph powers, the Shannon capacity and the Witsenhausen rate of graphs, Szemeredi's Regularity Lemma and its applications in graph property testing and in the study of sets with no 3 term arithmetic progressions, the Combinatorial Nullstellensatz and its applications, the capset problem, Containers and list coloring, and related topics as time permits. N. Alon MAT 486 - Random Processes QCR Wiener measure. Stochastic differential equations. Markov diffusion processes. Linear theory of stationary processes. Ergodicity, mixing, central limit theorem for stationary processes. If time permits, the theory of products of random matrices and PDE with random coefficients will be discussed. Prerequisite: MAT385. Staff APC 199 - Math Alive (also MAT 199) Spring QCR An exploration of some of the mathematical ideas behind important modern applications, from banking and computing to listening to music. Intended for students who have not had college-level mathematics and are not planning to major in a mathematically based field. The course is organized in independent two-week modules focusing on particular applications, such as bar codes, CD-players, population models, and space flight. The emphasis is on ideas and mathematical reasoning, not on sophisticated mathematical techniques. Two 90-minute classes, one computer laboratory. Staff APC 350 - Introduction to Differential Equations (also MAT 322) Spring QCR This course will introduce the basic theory, models and techniques for ordinary and partial differential equations. Emphasis will be placed on the connection with other disciplines of science and engineering. We will try to strike a balance between the theoretical (e.g. existence and uniqueness issues, qualitative properties) and the more practical issues such as analytical and numerical approximations. Prerequisites: Multivariable calculus and linear algebra. M. Gilles COS 433 - Cryptography (also MAT 473) Fall An introduction to the theory of modern cryptography. Topics covered include private key and public key encryption schemes, digital signatures, pseudorandom generators and functions, zero-knowledge proofs, and some advanced topics. Prerequisites: COS 240 is a required prerequisite or equivalent proof-based mathematical maturity. Two lectures. A. Lombardi COS 487 - Theory of Computation (also MAT 407) Not offered this year Studies the limits of computation by identifying tasks that are either inherently impossible to compute, or impossible to compute within the resources available. Introduces students to computability and decidability, Godel's incompleteness theorem, computational complexity, NP-completeness, and other notions of intractability. This course also surveys the status of the P versus NP question. Additional topics may include: interactive proofs, hardness of computing approximate solutions, cryptography, and quantum computation. Two lectures, one precept. Prerequisite: 240 or 341, or instructor's permission. Staff COS 488 - Introduction to Analytic Combinatorics (also MAT 474) Not offered this year Analytic Combinatorics aims to enable precise quantitative predictions of the properties of large combinatorial structures. The theory has emerged over recent decades as essential both for the scientific analysis of algorithms in computer science and for the study of scientific models in many other disciplines. This course combines motivation for the study of the field with an introduction to underlying techniques, by covering as applications the analysis of numerous fundamental algorithms from computer science. The second half of the course introduces Analytic Combinatorics, starting from basic principles. Staff EGR 191 - An Integrated Introduction to Engineering, Mathematics, Physics (also MAT 191/PHY 191) Not offered this year SEL Taken concurrently with EGR/MAT/PHY 192. An integrated course that covers the material of PHY 103 and MAT 201 with the emphasis on applications to engineering. Physics topics include: mechanics with applications to fluid mechanics, wave phenomena, and thermodynamics. The lab revolves around a single project to build, launch, and analyze the flight dynamics of water-propelled rockets. One lecture, three preceptorials, one three-hour laboratory. P. Meyers EGR 192 - An Integrated Introduction to Engineering, Mathematics, Physics (also APC 192/MAT 192/PHY 192) Not offered this year QCR Taken concurrently with EGR/MAT/PHY 191. An integrated course that covers the material of PHY 103 and MAT 201 with the emphasis on applications to engineering. Math topics include: vector calculus; partial derivatives and matrices; line integrals; simple differential equations; surface and volume integrals; and Green's, Stokes's, and divergence theorems. One lecture, two preceptorials. C. Kelleher MAE 305 - Mathematics in Engineering I (also CBE 305/EGR 305/MAT 391) Fall/Spring QCR An introduction to ordinary differential equations. Use of numerical methods. Equations of a single variable and systems of linear equations. Method of undermined coefficients and method of variation of parameters. Series solutions. Use of eigenvalues and eigenvectors. Laplace transforms. Nonlinear equations and stability; phase portraits. Partial differential equations via separation of variables. Sturm-Liouville theory. Three lectures. Prerequisites: MAT 201 or 203, and MAT 202 or 204. E. Yariv MAE 306 - Mathematics in Engineering II (also MAT 392) Spring Solution of partial differential equations. Complex variable methods. Characteristics, orthogonal functions, and integral transforms. Cauchy-Riemann conditions and analytic functions, mapping, the Cauchy integral theorem, and the method of residues with application to inversion of transforms. Applications to diffusion, wave and Laplace equations in fluid mechanics and electrostatics. Three lectures, one preceptorial. Prerequisite: 305, MAT 301 or equivalent. M. Haataja ORF 309 - Probability and Stochastic Systems (also EGR 309/MAT 380) Fall/Spring An introduction to probability and its applications. Topics include: basic principles of probability; Lifetimes and reliability, Poisson processes; random walks; Brownian motion; branching processes; Markov chains. Prerequisite: MAT 201, 203, 216, or instructor's permission. Three lectures, one precept. M. Cerenzia PHI 323 - Set Theory (also MAT 306) Not offered this year QCR This course deals with topics chosen from recursion theory, proof theory, and model theory. In recent years the course has most often given an introduction to recursion theory with applications to formal systems. Two 90-minute classes. Prerequisite: 312 or instructor's permission. Staff PHY 403 - Mathematical Methods of Physics (also MAT 493) Not offered this year QCR Mathematical methods and techniques that are essential for modern theoretical physics. Topics such as group theory, Lie algebras, and differential geometry are discussed and applied to concrete physical problems. Special attention will be given to mathematical techniques that originated in physics, such as functional integration and current algebras. Three classes. Prerequisite: MAT 330 or instructor's permission. Staff
