Applied Mathematics. Applied Mathematics Concentration Requirements. Chair. Standard program for the Sc.B. degree. Applied Mathematics 1

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1 Applied Mathematics Applied Mathematics Chair Yan Guo Associate Chair Kavita Ramanan The Division of Applied Mathematics at Brown University is one of the most prominent departments at Brown, and is also one of the oldest and strongest of its type in the country. The Division of Applied Mathematics is a world renowned center of research activity in a wide spectrum of traditional and modern mathematics. It explores the connections between mathematics and its applications at both the research and educational levels. The principal areas of research activities are ordinary, functional, and partial differential equations: stochastic control theory; applied probability, statistics and stochastic systems theory; neuroscience and computational molecular biology; numerical analysis and scientific computation; and the mechanics of solids, materials science and fluids. The effort in virtually all research ranges from applied and algorithmic problems to the study of fundamental mathematical questions. The Division emphasizes applied mathematics as a unifying theme. To facilitate cooperation among faculty and students, some research programs are partly organized around interdepartmental research centers. These centers facilitate funding and cooperative research in order to maintain the highest level of research and education in the Division. It is this breadth and the discovery from mutual collaboration which marks the great strength and uniqueness of the Division of Applied Mathematics at Brown. For additional information, please visit the department's website: Applied Mathematics Concentration Requirements The concentration in Applied Mathematics allows students to investigate the mathematics of problems arising in the physical, life and social sciences as well as in engineering. The basic mathematical skills of Applied Mathematics come from a variety of sources, which depend on the problems of interest: the theory of ordinary and partial differential equations, matrix theory, statistical sciences, probability and decision theory, risk and insurance analysis, among others. Applied Mathematics appeals to people with a variety of different interests, ranging from those with a desire to obtain a good quantitative background for use in some future career, to those who are interested in the basic techniques and approaches in themselves. The standard concentration leads to either the A.B. or Sc.B. degree. Students may also choose to pursue a joint program with biology, computer science or economics. The undergraduate concentration guide is available here ( applied-mathematics/undergraduate). Both the A.B. and Sc.B. concentrations in Applied Mathematics require certain basic courses to be taken, but beyond this there is a great deal of flexibility as to which areas of application are pursued. Students are encouraged to take courses in applied mathematics, mathematics and one or more of the application areas in the natural sciences, social sciences or engineering. Whichever areas are chosen should be studied in some depth. Standard program for the A.B. degree. Prerequisites MATH 0090 & MATH 000 Introductory Calculus, Part I and Introductory Calculus, Part II Or their equivalent Program Ten additional semester courses approved by the Division of Applied Mathematics. These classes must include: MATH 080 Intermediate Calculus MATH 050 Linear Algebra APMA 0350 & APMA 0360 Applied Ordinary Differential Equations and Applied Partial Differential Equations I 3 Select one course on programming from the following: 4 APMA 0090 Introduction to Mathematical Modeling APMA 060 Introduction to Scientific Computing CSCI 0040 Introduction to Scientific Computing and Problem Solving CSCI 050 Introduction to Object-Oriented Programming and Computer Science CSCI 070 Computer Science: An Integrated Introduction Five additional courses, of which four should be chosen from 5 the 000-level courses taught by the Division of Applied Mathematics. Total Credits 0 Substitution of alternate courses for the specific requirements is subject to approval by the division. Concentrators are urged to consider MATH 0540 as an alternative to MATH APMA 0330, APMA 0340 will sometimes be accepted as substitutes for APMA 0350, APMA Concentrators are urged to complete their introductory programming course before the end of their sophomore year. Standard program for the Sc.B. degree. Program Eighteen approved semester courses in mathematics, applied mathematics, engineering, the natural or social sciences. These classes must include: MATH 0090 Introductory Calculus, Part I & MATH 000 and Introductory Calculus, Part II MATH 080 Intermediate Calculus MATH 050 Linear Algebra APMA 0350 & APMA 0360 Applied Ordinary Differential Equations and Applied Partial Differential Equations I 3 Select one senior seminar from the APMA 930 or APMA 940 series, or an approved equivalent. Select one course on programming from the following: 4 APMA 0090 Introduction to Mathematical Modeling APMA 060 Introduction to Scientific Computing CSCI 0040 Introduction to Scientific Computing and Problem Solving CSCI 050 Introduction to Object-Oriented Programming and Computer Science CSCI 070 Computer Science: An Integrated Introduction Ten additional courses, of which six should be chosed from 0 the 000-level or higher level courses taught by the Division of Applied Mathematics. Total Credits Substitution of alternate courses for the specific requirements is subject to approval by the division. Concentrators are urged to consider MATH 0540 as an alternative to MATH 050. APMA 0330, APMA 0340 will sometimes be accepted as substitutes for APMA 0350, APMA Concentrators are urged to complete their introductory programming course before the end of their sophomore year. Applied Mathematics

2 Applied Mathematics Applied Mathematics-Biology Concentration Requirements The Applied Math - Biology concentration recognizes that mathematics is essential to address many modern biological problems in the post genomic era. Specifically, high throughput technologies have rendered vast new biological data sets that require novel analytical skills for the most basic analyses. These technologies are spawning a new "datadriven" paradigm in the biological sciences and the fields of bioinformatics and systems biology. The foundations of these new fields are inherently mathematical, with a focus on probability, statistical inference, and systems dynamics. These mathematical methods apply very broadly in many biological fields including some like population growth, spread of disease, that predate the genomics revolution. Nevertheless, the application of these methods in areas of biology from molecular genetics to evolutionary biology has grown very rapidly in with the availability of vast amounts of genomic sequence data. Required coursework in this program aims at ensuring expertise in mathematical and statistical sciences, and their application in biology. The students will focus in particular areas of biology. The program culminates in a senior capstone experience that pairs student and faculty in creative research collaborations. Standard program for the Sc.B. degree Required coursework in this program aims at ensuring expertise in mathematical and statistical sciences, and their application in biology. The students will focus in particular areas of biology. The program culminates in a senior capstone experience that pairs student and faculty in creative research collaborations. Applied Math Biology concentrators are prepared for careers in medicine, public health, industry and academic research. Required Courses: Students are required to take all of the following courses. MATH 0090 Introductory Calculus, Part I MATH 000 Introductory Calculus, Part II or MATH 070 Advanced Placement Calculus MATH 080 Intermediate Calculus (or equivalent placement) MATH 050 Linear Algebra or MATH 0540 Honors Linear Algebra CHEM 0330 Equilibrium, Rate, and Structure PHYS 0030 Basic Physics A or PHYS 0050 Foundations of Mechanics Select one of the following sequences: APMA 0350 & APMA 0360 Applied Ordinary Differential Equations and Applied Partial Differential Equations I APMA 0330 & APMA 0340 Methods of Applied Mathematics I, II and Methods of Applied Mathematics I, II APMA 650 Statistical Inference I or APMA 655 Statistical Inference I APMA 070 Quantitative Models of Biological Systems APMA 080 Inference in Genomics and Molecular Biology BIOL 000 The Foundation of Living Systems (or equivalent) Additional Courses In addition to required courses listed above, students must take the following: Two additional courses in Applied Math or Biology. At least one of these must be a directed research course, e.g. a senior seminar or independent study in Applied Math or a directed research/independent study in Biology. For example: A course from the APMA 930 series A course from the APMA 940 series APMA 970 Independent Study BIOL 950 Directed Research/Independent Study BIOL 960 Directed Research/Independent Study We strongly recommend that Applied Mathematics-Biology concentrators take one of the following programming courses on or before their first semester as a concentrator: APMA 060, CSCI 0040, CSCI 050, CSCI 070, CSCI 090, CLPS Those who do can use it as their second Applied Math or Biology course. Four classes in the biological sciences agreed upon by the student and advisor. These four courses should form a cohesive grouping ina specific area of emphasis, at least two of which should be at the 000-level. Some example groupings are below: Areas of Emphasis and Suggested Courses: Some areas of possible emphasis for focusing of elective courses are listed below. Given the large number of course offerings in the biosciences and neuroscience, students are free to explore classes in these areas that are not listed below. However, all classes must be approved by the concentration advisor. Biochemistry BIOL 080 Biochemistry BIOL 70 Advanced Biochemistry CHEM 0350/0360 Organic Chemistry CHEM 30 Chemical Biology Biotechnology and Physiology BIOL 0800 Principles of Physiology BIOL 00 Cell Physiology and Biophysics and/or appropriate bioengineering courses, such as: BIOL 090 Polymer Science for Biomaterials BIOL 0 Biomaterials BIOL 40 Tissue Engineering BIOL 50 Stem Cell Engineering BIOL 0 Synthetic Biological Systems Ecology, Evolution, and Genetics BIOL 040 Invertebrate Zoology & BIOL 0480 and Evolutionary Biology BIOL 040 Principles of Ecology & BIOL 0430 and The Evolution of Plant Diversity BIOL 0470 Genetics BIOL 40 Experimental Design in Ecology BIOL 430 Population Genetics BIOL 465 Human Population Genomics BIOL 540 Molecular Genetics Neuroscience APMA 040 Mathematical Methods in the Brain Sciences Neurosciences courses: See academics/neuroscience/undergraduate/neuroscienceconcentration-requirements BIOL 00 Cell Physiology and Biophysics BIOL 0 Topics in Signal Transduction BIOL 90 Synaptic Transmission and Plasticity Total Credits 8 Students whose independent study is expected to be in an experimental field are strongly encouraged to take APMA 660, which covers experimental design and the analysis of variance (ANOVA), a method commonly used in the analysis of experimental data. Honors Requirements and Process: Honors in the Applied Math-Biology concentration is based primarily upon an in-depth, original research project carried out under the guidance of a Brown (and usually Applied Math 4 Applied Mathematics

3 Applied Mathematics 3 or BioMed) affiliated faculty advisor. Projects must be conducted for no less than two full semesters, and student smust register for credit for the project via APMA 970 or BIOL 950/BIOL 960 or similar independent study courses. The project culminates in the writing of a thesis which is reviewed by the thesis advisor and a second reader. It is essential that the student have one advisor from the biological sciences and one in Applied Mathematics. The thesis work must be presented in the form of an oral presentation (arranged with the primary thesis advisor) or posted at the annual Undergraduate Research Day in either Applied Mathematics or Biology. For information on registering for BIOL 950/BIOL 960, please see undergraduate-research Excellence in grades within the concentration as well as a satisfactory evaluation by the advsors are also required for Honors. The student's grades must place them within the upper 0% of their cohort, in accordance with the university policy on honors. Honors recipients typically maintain a Grade Point Average of 3.4 or higher in the concentration. However, in the case of outstanding independent research as demonstrated in the thesis and supported by the Thesis Committee, candidates with a GPA between 3.0 an 3.4 will be considered and are encouraged to apply. The deadline for applying to graduate with honors in the concentration are the same as those of the biology concentrations. However, students in the joint concentration must inform the undergraduate chair in Applied Mathematics of their intention to apply for honors by these dates. Applied Mathematics-Computer Science Concentration Requirements The Sc.B. concentration in Applied Math-Computer Science provides a foundation of basic concepts and methodology of mathematical analysis and computation and prepares students for advanced work in computer science, applied mathematics, and scientific computation. Concentrators must complete courses in mathematics, applied math, computer science, and an approved English writing course. While the concentration in Applied Math-Computer Science allows students to develop the use of quantitative methods in thinking about and solving problems, knowledge that is valuable in all walks of life, students who have completed the concentration have pursued graduate study, computer consulting and information industries, and scientific and statistical analysis careers in industry or government. This degree offers a standard track and a professional track. Requirements for the Standard Track of the Sc.B. degree. Prerequisites - two semesters of Calculus, for example MATH 0090 & MATH 000 Introductory Calculus, Part I and Introductory Calculus, Part II MATH 070 Advanced Placement Calculus Concentration Requirements (7 courses) Core-Math: MATH 080 Intermediate Calculus or MATH 0350 Honors Calculus MATH 050 Linear Algebra or MATH 0540 Honors Linear Algebra or CSCI 0530 Directions: The Matrix in Computer Science Core-Applied Mathematics: APMA 0350 Applied Ordinary Differential Equations APMA 0360 Applied Partial Differential Equations I APMA 70 Introduction to Computational Linear Algebra or APMA 80 Introduction to Numerical Solution of Differential Equations Core-Computer Science: Select one of the following Series: Series A CSCI 050 & CSCI 060 Series B CSCI 070 & CSCI 080 Introduction to Object-Oriented Programming and Computer Science and Introduction to Algorithms and Data Structures Computer Science: An Integrated Introduction and Computer Science: An Integrated Introduction Series C CSCI 090 Accelerated Introduction to Computer Science (and an additional CS course not otherwise used to satisfy a concentration requirement; (this course may be CSCI 080, an intermediate-level CS course, or a 000-level course) ) Select three of the following intermediate-level courses, one of which must be math-oriented and one systems-oriented: CSCI 00 Introduction to Discrete Structures and Probability (math) CSCI 030 Introduction to Software Engineering (systems) CSCI 0330 Introduction to Computer Systems CSCI 00 Theory of Computation Three 000-level Computer Science courses. These three courses must include a pair of courses with a coherent theme. A list of approved pairs may be found at the approved-pairs web page. You are not restricted to the pairs on this list, but any pair not on the list must be approved by the director of undergraduate studies. Three 000-level Applied Mathematics courses approved by the concentration advisor, of which two should constitute a standard sequence or address a common theme. Typical sequences include: APMA 00/0 and APMA 650 or 655/660. A capstone course: a one-semester course, normally taken in the student's last undergraduate year, in which the student (or group of students) use a significant portion of their undergraduate education, broadly interpreted, in studying some current topic in depth, to produce a culminating artifact such as a paper or software project. Note: CSCI 450 may be used either as a math-oriented core course or as an advanced course. CSCI 450 was formerly known as CSCI 450: they are the same course and hence only one may be taken for credit. Applied Math 650 or Applied Math 655 may be used in place of CSCI 450. However, concentration credit will be given for only one of Applied Math 650, 655 and CSCI 450. Total Credits 7 Requirements for the Professional Track of the Sc.B. degree. The requirements for the professional track include all those of the standard track, as well as the following: Students must complete two two-to-four-month full-time professional experiences, doing work that is related to their concentration programs. Such work is normally done within an industrial organization, but may also be at a university under the supervision of a faculty member. On completion of each professional experience, the student must write and upload to ASK a reflective essay about the experience addressing the following prompts, to be approved by the student's concentration advisor: Which courses were put to use in your summer's work? Which topics, in particular, were important? In retrospect, which courses should you have taken before embarking on your summer experience? What are the topics from these courses that would have helped you over the summer if you had been more familiar with them? Applied Mathematics 3

4 4 Applied Mathematics Are there topics you should have been familiar with in preparation for your summer experience, but are not taught at Brown? What are these topics? What did you learn from the experience that probably could not have been picked up from course work? Is the sort of work you did over the summer something you would like to continue doing once you graduate? Explain. Would you recommend your summer experience to other Brown students? Explain. Applied Mathematics-Economics Concentration Requirements The Applied Mathematics-Economics concentration is designed to reflect the mathematical and statistical nature of modern economic theory and empirical research. This concentration has two tracks. The first is the advanced economics track, which is intended to prepare students for graduate study in economics. The second is the mathematical finance track, which is intended to prepare students for graduate study in finance, or for careers in finance or financial engineering. Both tracks have A.B. degree versions and Sc.B. degree versions, as well as a Professional track option. Standard Program for the A.B. degree (Advanced Economics track): Prerequisites: MATH 000 Introductory Calculus, Part II MATH 050 Linear Algebra Course Requirements: Applied Mathematics Requirements (a) APMA 0350 & APMA 0360 Applied Ordinary Differential Equations and Applied Partial Differential Equations I Select one of the following: APMA 060 Introduction to Scientific Computing (preferred) CSCI 0040 Introduction to Scientific Computing and Problem Solving (preferred) CSCI 050 Introduction to Object-Oriented Programming and Computer Science CSCI 070 Computer Science: An Integrated Introduction Select one of the following: APMA 00 Operations Research: Probabilistic Models APMA 0 Operations Research: Deterministic Models APMA 650 Statistical Inference I or APMA 655 Statistical Inference I (b) Select one of the following: APMA 00 Operations Research: Probabilistic Models APMA 0 Operations Research: Deterministic Models APMA 660 Statistical Inference II APMA 670 Statistical Analysis of Time Series APMA 680 Nonparametric Statistics APMA 690 Computational Probability and Statistics APMA 700 The Mathematics of Insurance APMA 740 Recent Applications of Probability and Statistics MATH 00 Analysis: Functions of One Variable Economics Requirements: ECON 30 Intermediate Microeconomics (Mathematical) 3 ECON 0 Intermediate Macroeconomics ECON 630 Econometrics I Two 000-level courses from the "mathematical-economics" group: 4 ECON 70 Welfare Economics and Social Choice Theory ECON 5 Advanced Macroeconomics: Monetary, Fiscal, and Stabilization Policies ECON 465 Market Design: Theory and Applications ECON 470 Bargaining Theory and Applications ECON 640 Econometrics II ECON 650 Financial Econometrics ECON 660 Big Data ECON 750 Investments II ECON 759 Data, Statistics, Finance ECON 80 Economics and Psychology ECON 80 Behavioral Economics ECON 850 Theory of Economic Growth ECON 860 The Theory of General Equilibrium ECON 870 Game Theory and Applications to Economics One 000-level course from the "data methods" group: 4 ECON 305 Economics of Education: Research ECON 30 Labor Economics ECON 360 Health Economics ECON 40 Urban Economics ECON 50 Economic Development ECON 50 The Economic Analysis of Institutions ECON 530 Health, Hunger and the Household in Developing Countries ECON 640 Econometrics II ECON 650 Financial Econometrics ECON 759 Data, Statistics, Finance ECON 765 Finance, Regulation, and the Economy: Research One additional 000-level economics course Total Credits 3 No course may be used to simultaneously satisfy (a) and (b). APMA 0330 and APMA 0340 may be substituted with advisor approval. 3 Or ECON 0 with permission. 4 No course may be used to simultaneously satisfy the "mathematical economics" and the "data methods" requirements. Standard program for the Sc.B. degree (Advanced Economics track): Prerequisites: MATH 000 Introductory Calculus, Part II MATH 050 Linear Algebra Course Requirements: Applied Mathematics Requirements (a) APMA 0350 & APMA 0360 Applied Ordinary Differential Equations and Applied Partial Differential Equations I Select one of the following: APMA 060 Introduction to Scientific Computing (preferred) 4 Applied Mathematics

5 Applied Mathematics 5 CSCI 0040 Introduction to Scientific Computing and Problem Solving (preferred) CSCI 050 Introduction to Object-Oriented Programming and Computer Science CSCI 070 Computer Science: An Integrated Introduction Select one of the following: APMA 00 Operations Research: Probabilistic Models APMA 0 Operations Research: Deterministic Models APMA 650 Statistical Inference I or APMA 655 Statistical Inference I (b) Select two of the following: APMA 00 Operations Research: Probabilistic Models APMA 0 Operations Research: Deterministic Models APMA 660 Statistical Inference II APMA 670 Statistical Analysis of Time Series APMA 680 Nonparametric Statistics APMA 690 Computational Probability and Statistics APMA 700 The Mathematics of Insurance APMA 740 Recent Applications of Probability and Statistics MATH 00 Analysis: Functions of One Variable Economics Requirements: ECON 30 Intermediate Microeconomics (Mathematical) 3 ECON 0 Intermediate Macroeconomics ECON 630 Econometrics I Three 000-level courses from the "mathematical-economics" group: 4 3 ECON 70 Welfare Economics and Social Choice Theory ECON 5 Advanced Macroeconomics: Monetary, Fiscal, and Stabilization Policies ECON 465 Market Design: Theory and Applications ECON 470 Bargaining Theory and Applications ECON 640 Econometrics II ECON 650 Financial Econometrics ECON 660 Big Data ECON 750 Investments II ECON 759 Data, Statistics, Finance ECON 80 Economics and Psychology ECON 80 Behavioral Economics ECON 850 Theory of Economic Growth ECON 860 The Theory of General Equilibrium ECON 870 Game Theory and Applications to Economics One 000-level course from the "data methods" group: 4 ECON 305 Economics of Education: Research ECON 30 Labor Economics ECON 360 Health Economics ECON 40 Urban Economics ECON 50 Economic Development ECON 50 The Economic Analysis of Institutions ECON 530 Health, Hunger and the Household in Developing Countries ECON 640 Econometrics II ECON 650 Financial Econometrics ECON 759 Data, Statistics, Finance ECON 765 Finance, Regulation, and the Economy: Research Two additional 000-level economics courses Total Credits 6 No course may be used to simultaneously satisfy (a) and (b). APMA 0330 and APMA 0340 may be substituted with advisor approval. 3 Or ECON 0 with permission. 4 No course may be used to simultaneously satisfy the "mathematical economics" and the "data methods" requirements. Standard program for the A.B. degree (Mathematical Finance track): Prerequisites: MATH 000 Introductory Calculus, Part II MATH 050 Linear Algebra Course Requirements: Applied Mathematics Requirements (a) APMA 0350 & APMA 0360 Applied Ordinary Differential Equations and Applied Partial Differential Equations I Select one of the following: APMA 060 Introduction to Scientific Computing (preferred) CSCI 0040 Introduction to Scientific Computing and Problem Solving (preferred) CSCI 050 Introduction to Object-Oriented Programming and Computer Science CSCI 070 Computer Science: An Integrated Introduction APMA 00 Operations Research: Probabilistic Models APMA 650 Statistical Inference I or APMA 655 Statistical Inference I (b) Select one of the following: APMA 80 Introduction to Numerical Solution of Differential Equations APMA 330 Applied Partial Differential Equations II APMA 660 Statistical Inference II APMA 670 Statistical Analysis of Time Series APMA 680 Nonparametric Statistics APMA 690 Computational Probability and Statistics APMA 700 The Mathematics of Insurance APMA 70 Monte Carlo Simulation with Applications to Finance (preferred) APMA 740 Recent Applications of Probability and Statistics MATH 00 Analysis: Functions of One Variable Economics Requirements: ECON 30 Intermediate Microeconomics (Mathematical) 3 ECON 0 Intermediate Macroeconomics ECON 630 Econometrics I Select two 000-level courses from the "financial economics" group: ECON 650 Financial Econometrics ECON 70 Investments I ECON 70 Corporate Finance ECON 730 Venture Capital, Private Equity, and Entrepreneurship Applied Mathematics 5

6 6 Applied Mathematics ECON 750 Investments II ECON 759 Data, Statistics, Finance ECON 760 Financial Institutions ECON 765 Finance, Regulation, and the Economy: Research ECON 770 Fixed Income Securities ECON 780 Corporate Strategy ECON 790 Corporate Governance and Management Select one 000-level course from the "mathematical economics" group: ECON 70 Welfare Economics and Social Choice Theory ECON 5 Advanced Macroeconomics: Monetary, Fiscal, and Stabilization Policies ECON 465 Market Design: Theory and Applications ECON 470 Bargaining Theory and Applications ECON 640 Econometrics II ECON 650 Financial Econometrics ECON 660 Big Data ECON 750 Investments II ECON 759 Data, Statistics, Finance ECON 80 Economics and Psychology ECON 80 Behavioral Economics ECON 850 Theory of Economic Growth ECON 860 The Theory of General Equilibrium ECON 870 Game Theory and Applications to Economics Select one 000-level course from the "data methods" group: ECON 305 Economics of Education: Research ECON 30 Labor Economics ECON 360 Health Economics ECON 40 Urban Economics ECON 50 Economic Development ECON 50 The Economic Analysis of Institutions ECON 530 Health, Hunger and the Household in Developing Countries ECON 69 Applied Research Methods for Economists ECON 640 Econometrics II ECON 650 Financial Econometrics ECON 759 Data, Statistics, Finance ECON 765 Finance, Regulation, and the Economy: Research Total Credits 3 APMA 0330 and APMA 0340 may be substituted with advisor approval. No course may be used to simultaneously satisfy the "financial economics," the "mathematical economics," or the "data methods" requirements. 3 Or ECON 0 with permission. Standard program for the Sc.B. degree (Mathematical Finance track): Prerequisites: MATH 000 Introductory Calculus, Part II MATH 050 Linear Algebra Course Requirements: Applied Mathematics requirements: (a) APMA 0350 & APMA 0360 Applied Ordinary Differential Equations and Applied Partial Differential Equations I Select one of the following: APMA 060 Introduction to Scientific Computing (preferred) CSCI 0040 Introduction to Scientific Computing and Problem Solving (preferred) CSCI 050 Introduction to Object-Oriented Programming and Computer Science CSCI 070 Computer Science: An Integrated Introduction APMA 00 Operations Research: Probabilistic Models APMA 650 Statistical Inference I or APMA 655 Statistical Inference I (b) Select two of the following: APMA 80 Introduction to Numerical Solution of Differential Equations APMA 330 Applied Partial Differential Equations II APMA 660 Statistical Inference II APMA 670 Statistical Analysis of Time Series APMA 680 Nonparametric Statistics APMA 690 Computational Probability and Statistics APMA 700 The Mathematics of Insurance APMA 70 Monte Carlo Simulation with Applications to Finance (preferred) APMA 740 Recent Applications of Probability and Statistics MATH 00 Analysis: Functions of One Variable Economics Requirements: ECON 30 Intermediate Microeconomics (Mathematical) 3 ECON 0 Intermediate Macroeconomics ECON 630 Econometrics I Select three 000-level courses from the "financial economics" group: 3 ECON 650 Financial Econometrics ECON 70 Investments I ECON 70 Corporate Finance ECON 730 Venture Capital, Private Equity, and Entrepreneurship ECON 750 Investments II ECON 759 Data, Statistics, Finance ECON 760 Financial Institutions ECON 765 Finance, Regulation, and the Economy: Research ECON 770 Fixed Income Securities ECON 780 Corporate Strategy ECON 790 Corporate Governance and Management Select two 000-level courses from the "mathematical economics" group: ECON 70 Welfare Economics and Social Choice Theory ECON 5 Advanced Macroeconomics: Monetary, Fiscal, and Stabilization Policies ECON 465 Market Design: Theory and Applications ECON 470 Bargaining Theory and Applications ECON 640 Econometrics II ECON 650 Financial Econometrics ECON 660 Big Data 6 Applied Mathematics

7 Applied Mathematics 7 ECON 750 Investments II ECON 759 Data, Statistics, Finance ECON 80 Economics and Psychology ECON 80 Behavioral Economics ECON 850 Theory of Economic Growth ECON 860 The Theory of General Equilibrium ECON 870 Game Theory and Applications to Economics Select one 000-level course from the "data methods" group: ECON 305 Economics of Education: Research ECON 30 Labor Economics ECON 360 Health Economics ECON 40 Urban Economics ECON 50 Economic Development ECON 50 The Economic Analysis of Institutions ECON 530 Health, Hunger and the Household in Developing Countries ECON 640 Econometrics II ECON 650 Financial Econometrics ECON 759 Data, Statistics, Finance ECON 765 Finance, Regulation, and the Economy: Research Total Credits 6 APMA 0330 and APMA 0340 may be substituted with advisor approval. No course may be used to simultaneously satisfy the "financial economics," the "mathematical economics," or the "data methods" requirements. 3 Or ECON 0 with permission. Honors and Capstone Requirement Admission to candidacy for honors in the concentration is granted on the following basis: 3.7 GPA for Economics courses, and a 3.5 GPA overall. To graduate with honors, a student must write an honors thesis in the senior year following the procedures specified by the concentration (see Economics Department website). Beginning with the class of 06, students not writing an honors thesis must complete an alternative senior capstone project and obtain the approval of a faculty sponsor. Professional Track The requirements for the professional track include all those of the standard track, as well as the following: Students must complete two two-to-four month full-time professional experiences, doing work that is related to their concentration programs. Such work is normally done within an industrial organization, but may also be at a university under the supervision of a faculty member. On completion of each professional experience, the student must write and upload to ASK a reflective essay about the experience addressing the following prompts, to be approved by the student's concentration advisor: Which courses were put to use in your summer's work? Which topics, in particular, were important? In retrospect, which courses should you have taken before embarking on your summer experience? What are the topics from these courses that would have helped you over the summer if you had been more familiar with them? Are there topics you should have been familiar with in preparation for your summer experience, but are not taught at Brown? What are these topics? What did you learn from the experience that probably could not have been picked up from course work? Is the sort of work you did over the summer something you would like to continue doing once you graduate? Explain. Would you recommend your summer experience to other Brown students? Explain. Applied Mathematics Graduate Program The department of Applied Mathematics offers graduate programs leading to the Master of Science (Sc.M.) degree and the Doctor of Philosophy (Ph.D.) degree. For more information on admission and program requirements, please visit the following website: Courses APMA Introduction to Applied Complex Variables. Applications of complex analysis that do not require calculus as a prerequisite. Topics include algebra of complex numbers, plane geometry by means of complex coordinates, complex exponentials, and logarithms and their relation to trigonometry, polynomials, and roots of polynomials, conformal mappings, rational functions and their applications, finite Fourier series and the FFT, iterations and fractals. Uses MATLAB, which has easy and comprehensive complex variable capabilities. APMA Introduction to Mathematical Modeling. We will explore issues of mathematical modeling and analysis. Five to six self-contained topics will be discussed and developed. The course will include seminars in which modeling issues are discussed, lectures to provide mathematical background, and computational experiments. Required mathematical background is knowledge of one-variable calculus, and no prior computing experience will be assumed. FYS APMA 000. Elementary Probability for Applications. This course serves as an introduction to probability and stochastic processes with applications to practical problems. It will cover basic probability and stochastic processes such as basic concepts of probability and conditional probability, simple random walk, Markov chains, continuous distributions, Brownian motion and option pricing. Enrollment limited to 0 first year students. FYS APMA 00. What s the big deal with Data Science?. This seminar serves as a practical introduction to the interdisciplinary field of data science. Over the course of the semester, students will be exposed to the diversity of questions that data science can address by reading current scholarly works from leading researchers. Through handson labs and experiences, students will gain facility with computational and visualization techniques for uncovering meaning from large numerical and text-based data sets. Ultimately, students will gain fluency with data science vocabulary and ideas. There are no prerequisites for this course. FYS WRIT Fall APMA00 S0 603 TTh 0:30-:50(3) (K. Kinnaird) APMA 00. Mathematics of Finance. The current volatility in international financial markets makes it imparative for us to become competent in financial calculations early in our liberal arts and scientific career paths. This course is designed to prepare the student with those elements of mathematics of finance appropriate for the calculations necessary in financial transactions. APMA 060. Introduction to Scientific Computing. For students in any discipline that may involve numerical computations. Includes instruction for programming in MATLAB. Applications discussed include solution of linear equations (with vectors and matrices) and nonlinear equations (by bisection, iteration, and Newton's method), interpolation, and curve-fitting, difference equations, iterated maps, numerical differentiation and integration, and differential equations. Prerequisite: MATH 000 or its equivalent. Spr APMA060 S MWF 9:00-9:50(0) (G. Fu) Applied Mathematics 7

8 8 Applied Mathematics APMA 080. Modeling the World with Mathematics: An Introduction for Non-Mathematicians. Mathematics is the foundation of our technological society and most of its powerful ideas are quite accessible. This course will explain some of these using historical texts and Excel. Topics include the predictive power of 'differential equations' from the planets to epidemics, oscillations and music, chaotic systems, randomness and the atomic bomb. Prerequisite: some knowledge of calculus. APMA 000. Introduction to Modelling. This course provides an introduction to the mathematical modeling of selected biological, chemical, engineering, and physical processes. The goal is to illustrate the typical way in which applied mathematicians approach practical applications, from understanding the underlying problem, creating a model, analyzing the model using mathematical techniques, and interpreting the findings in terms of the original problem. Single-variable calculus is the only requirement; all other techniques from differential equations, linear algebra, and numerical methods, to probability and statistics will be introduced in class. Prerequisites: Math 000 or equivalent. APMA Methods of Applied Mathematics I, II. This course will cover mathematical techniques involving ordinary differential equations used in the analysis of physical, biological, and economic phenomena. The course emphasizes established methods and their applications rather than rigorous foundation. Topics include: first and second order differential equations, an introduction to numerical methods, series solutions, and Laplace transformations. Fall APMA0330 S0 606 MWF :00-:50() (V. Dobrushkin) Spr APMA0330 S MWF :00-:50(05) (S. Akopian) APMA Methods of Applied Mathematics I, II. Mathematical techniques involving differential equations used in the analysis of physical, biological and economic phenomena. Emphasis on the use of established methods, rather than rigorous foundations. I: First and second order differential equations. II: Applications of linear algebra to systems of equations; numerical methods; nonlinear problems and stability; introduction to partial differential equations; introduction to statistics. Prerequisite: MATH 000, 070, 080, 090, 000, or 0350, or advanced placement. Fall APMA0340 S MWF :00-:50() (Y. Guo) Spr APMA0340 S MWF :00-:50(05) (V. Dobrushkin) APMA Applied Ordinary Differential Equations. This course gives a comprehensive introduction to the qualitative and quantitative theory of ordinary differential equations and their applications. Specific topics covered in the course are applications of differential equations in biology, chemistry, economics, and physics; integrating factors and separable equations; techniques for solving linear systems of differential equations; numerical approaches to solving differential equations; phase-plane analysis of planar nonlinear systems; rigorous theoretical foundations of differential equations. Format: Six hours of lectures, and two hours of recitation. Prerequisites: MATH 000, MATH 070, MATH 080, MATH 090, MATH 000, MATH 0350 or advanced placement. MATH 050 (can be taken concurrently). Fall APMA0350 S MWF :00-:50(07) (S. Akopian) Spr APMA0350 S MWF :00-:50(06) (B. Kunsberg) APMA Applied Partial Differential Equations I. Covers the same material as APMA 0340, albeit of greater depth. Intended primarily for students who desire a rigorous development of the mathematical foundations of the methods used, for those students considering one of the applied mathematics concentrations, and for all students in the sciences who will be taking advanced courses in applied mathematics, mathematics, physics, engineering, etc. Three hours lecture and one hour recitation. Prerequisite: MATH 000, 070, 080, 090, 000, or 0350, or advanced placement. Fall APMA0360 S MWF :00-:50(06) (M. Maxey) Spr APMA0360 S0 478 TTh 0:30-:50(09) (J. Darbon) APMA 040. Mathematical Methods in the Brain Sciences. Basic mathematical methods commonly used in the neural and cognitive sciences. Topics include: introduction to probability and statistics, emphasizing hypothesis testing and modern nonparametric methods; introduction to differential equations and systems of differential equations, emphasizing qualitative behavior and simple phase-plane analysis. Examples from neuroscience, cognitive science, and other sciences. Prerequisite: MATH 000 or equivalent. APMA Essential Statistics. A first course in probability and statistics emphasizing statistical reasoning and basic concepts. Topics include visual and numerical summaries of data, representative and non-representative samples, elementary discrete probability theory, the normal distribution, sampling variability, elementary statistical inference, measures of association. Examples and applications from the popular press and the life, social and physical sciences. No prerequisites. Spr APMA0650 S0 479 TTh 9:00-0:0(0) (K. Kinnaird) APMA 070. Quantitative Models of Biological Systems. Quantitative dynamic models help understand problems in biology and there has been rapid progress in recent years. The course provides an introduction to the concepts and techniques, with applications to population dynamics, infectious diseases, enzyme kinetics, aspects of cellular biology. Additional topics covered will vary. Mathematical techniques will be discussed as they arise in the context of biological problems. Prerequisites: APMA 0330, 0340 or 0350, 0360, or written permission. Fall APMA070 S0 604 MWF :00-:50(6) (L. Bienenstock) APMA 080. Inference in Genomics and Molecular Biology. Sequencing of genomes has generated a massive quantity of fundamental biological data. Drawing traditional and Bayesian statistical inferences from these data, including; motif finding; hidden Markov models; other probabilistic models, significances in high dimensions; and functional genomics. Emphasis - application of probability theory to inferences on data sequence, the goal of enabling students to construct prob models. Statistical topics: Bayesian inferences, estimation, hypothesis testing and false discovery rates, statistical decision theory. Enroll in 080 for more in depth coverage of the class. Prerequisite: APMA 650, 655 or MATH 60 or CSCI 450; BIOL 000 recommended, programming skills required. Fall APMA080 S TTh 9:00-0:0(08) (C. Lawrence) APMA 70. Introduction to Computational Linear Algebra. Focuses on fundamental algorithms in computational linear algebra with relevance to all science concentrators. Basic linear algebra and matrix decompositions (Cholesky, LU, QR, etc.), round-off errors and numerical analysis of errors and convergence. Iterative methods and conjugate gradient techniques. Computation of eigenvalues and eigenvectors, and an introduction to least squares methods. Fall APMA70 S0 607 MWF 0:00-0:50(4) (G. Fu) APMA 80. Introduction to Numerical Solution of Differential Equations. Fundamental numerical techniques for solving ordinary and partial differential equations. Overview of techniques for approximation and integration of functions. Development of mult-istep and multi-stage methods, error analysis, step-size control for ordinary differential equations. Solution of two-point boundary value problems, introduction to methods for solving linear partial differential equations. Students will be required to use Matlab (or other computer languages) to implement the mathematical algorithms under consideration: experience with a programming language is therefore strongly recommended. Prerequisites: APMA 0330, 0340 or 0350, Applied Mathematics

9 Applied Mathematics 9 APMA 90. Finite Volume Method for CFD: A Survey. This course will provide students with an overview of the subjects necessary to perform robust simulations of computational fluid dynamics (CFD) problems. After an initial overview of the finite volume method and fluid mechanics, students will use the finite volume library OpenFOAM to explore the different components that make up a modern CFD code (discretization, linear algebra, timestepping, boundary conditions, splitting schemes, and multiphysics) and learn how to navigate a production scale software library. APMA 00. Operations Research: Probabilistic Models. Basic probabilistic problems and methods in operations research and management science. Methods of problem formulation and solution. Markov chains, birth-death processes, stochastic service and queueing systems, the theory of sequential decisions under uncertainty, dynamic programming. Applications. Prerequisite: APMA 650, 655 or MATH 60, or equivalent. Spr APMA00 S TTh :00-:0(0) (K. Ramanan) APMA 0. Operations Research: Deterministic Models. An introduction to the basic mathematical ideas and computational methods of optimizing allocation of effort or resources, with or without constraints. Linear programming, network models, dynamic programming, and integer programming. Fall APMA0 S TTh 0:30-:50(3) (B. Rozovsky) APMA 50. Advanced Engineering Mechanics (ENGN 370). Interested students must register for ENGN 370. APMA 60. Introduction to the Mechanics of Solids and Fluids. An introduction to the dynamics of fluid flow and deforming elastic solids for students in the physical or mathematical sciences. Topics in fluid mechanics include statics, simple viscous flows, inviscid flows, potential flow, linear water waves, and acoustics. Topics in solid mechanics include elastic/plastic deformation, strain and stress, simple elastostatics, and elastic waves with reference to seismology. Offered in alternate years. APMA 330. Applied Partial Differential Equations II. Mathematical methods based on functions of a complex variable. Fournier series and its applications to the solution of one-dimensional heat conduction equations and vibrating strings. Series solution and special functions. Vibrating membrance. Sturm-Liouville problem and eigenfunction expansions. Fournier transform and wave propagations. Fall APMA330 S0 604 MWF :00-:50(06) (D. Sanz-Alonso) APMA 340. Methods of Applied Mathematics III, IV. See Methods Of Applied Mathematics III, IV (APMA 330) for course description. APMA 360. Topics in Chaotic Dynamics. This course gives an overview of the theory and applications of dynamical systems modeled by differential equations and maps. We will discuss changes of the dynamics when parameters are varied, investigate periodic and homoclinic solutions that arise in applications, and study the impact of additional structures such as time reversibility and conserved quantities on the dynamics. We will also study systems with complicated "chaotic" dynamics that possess attracting sets which do not have an integer dimension. Applications to chemical reactions, climate, epidemiology, and phase transitions will be discussed. This course can be used as a senior seminar. WRIT Spr APMA360 S MWF :00-:50(07) (J. Mallet-Paret) APMA 650. Statistical Inference I. APMA 650 is an integrated first course in mathematical statistics. The first half of APMA 650 covers probability and the last half is statistics, integrated with its probabilistic foundation. Specific topics include probability spaces, discrete and continuous random variables, methods for parameter estimation, confidence intervals, and hypothesis testing. Prerequisite: One year of university-level calculus. At Brown, this corresponds to MATH 000, MATH 070, MATH 080, MATH 090, MATH 000, or MATH A score of 4 or 5 on the AP Calculus BC exam is also sufficient. Fall APMA650 S TTh :00-:0(0) (B. Kunsberg) Spr APMA650 S MWF :00-:50(04) (D. Sanz-Alonso) APMA 655. Statistical Inference I. Students may opt to enroll in 655 for more in depth coverage of APMA 650. Enrollment in 655 will include an optional recitation section and required additional individual work. Applied Math concentrators are encouraged to take 655. Prerequisite (for either version): MATH 000, 070, 080, 090, 000, or Fall APMA655 S0 604 MWF :00-:50(6) (H. Wang) APMA 660. Statistical Inference II. APMA 660 is designed as a sequel to APMA 650 to form one of the alternative tracks for an integrated year's course in mathematical statistics. The main topic is linear models in statistics. Specific topics include likelihood-ratio tests, nonparametric tests, introduction to statistical computing, matrix approach to simple-linear and multiple regression, analysis of variance, and design of experiments. Prerequisite: APMA 650, 655 or equivalent, basic linear algebra. Spr APMA660 S TTh :30-3:50() (B. Gidas) APMA 670. Statistical Analysis of Time Series. Time series analysis is an important branch of mathematical statistics with many applications to signal processing, econometrics, geology, etc. The course emphasizes methods for analysis in the frequency domain, in particular, estimation of the spectrum of time-series, but time domain methods are also covered. Prerequisites: elementary probability and statistics on the level of APMA APMA 680. Nonparametric Statistics. A systematic treatment of distribution-free alternatives to classical statistical tests. These nonparametric tests make minimum assumptions about distributions governing the generation of observations, yet are of nearly equal power to the classical alternatives. Prerequisite: APMA 650 or equivalent. Offered in alternate years. APMA 690. Computational Probability and Statistics. Examination of probability theory and mathematical statistics from the perspective of computing. Topics selected from random number generation, Monte Carlo methods, limit theorems, stochastic dependence, Bayesian networks, dimensionality reduction. Prerequistes: A calculusbased course in probability or statistics (e.g. APMA650 or MATH60) is required, and some programming experience is strongly recommended. Prerequisite: MATH 000, 070, 080, 090, 000, or 0350, or equivalent placement. Fall APMA690 S0 608 MWF :00-:50(07) (N. Garcia Trillos) APMA 700. The Mathematics of Insurance. The course consists of two parts: the first treats life contingencies, i.e. the construction of models for individual life insurance contracts. The second treats the Collective Theory of Risk, which constructs mathematical models for the insurance company and its portfolio of policies as a whole. Suitable also for students proceeding to the Institute of Actuaries examinations. Prerequisites: Probability Theory to the level of APMA 650 or MATH 60. APMA 70. Information Theory. Information theory is the study of the fundamental limits of information transmission and storage. This course, intended primarily for advanced undergraduates and beginning graduate students, offers a broad introduction to information theory and its applications: Entropy and information, lossless data compression, communication in the presence of noise, channel capacity, channel coding, source-channel separation, lossy data compression. Prerequisite: one course in probability. Fall APMA70 S MWF 9:00-9:50(0) (G. Menon) APMA 70. Monte Carlo Simulation with Applications to Finance. The course will cover the basics of Monte Carlo and its applications to financial engineering: generating random variables and simulating stochastic processes; analysis of simulated data; variance reduction techniques; binomial trees and option pricing; Black-Scholes formula; portfolio optimization; interest rate models. The course will use MATLAB as the standard simulation tool. Prerequisites: APMA 650 or MATH 60 Applied Mathematics 9