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MATH 105 and 106 are designed for students whose background and algebraic skills are inadequate for the fast pace of MATH 111. In addition to covering the usual material of beginning calculus, these courses will have an extensive review of algebra and trigonometry. There will be a special emphasis on solving word problems.

MATH 105 starts with a quick review of algebraic manipulations, inequalities, absolute values and straight lines. Then the basic ideas of calculus--limits, derivatives, and integrals--are introduced, but only in the context of polynomial and rational functions. As various applications are studied, the algebraic techniques involved will be reviewed in more detail. When covering related rates and maximum-minimum problems, time will be spent learning how to approach, analyze and solve word problems. Four class meetings per week, one of which is a two-hour group-work day.

Note: While MATH 105 and 106 are sufficient for any course with a MATH 111 requisite, MATH 105 alone is not. However, students who plan to take MATH 121 should consider taking MATH 105 and then MATH 111, rather than MATH 106. Students cannot register for both MATH 105 and CHEM 151 in the same semester.

Fall semester. Professor TBA.

MATH 106 is a continuation of MATH 105. Trigonometric, logarithmic and exponential functions will be studied from the point of view of both algebra and calculus. The applications encountered in MATH 105 will reappear in problems involving these new functions. The basic ideas and theorems of calculus will be reviewed in detail, with more attention being paid to rigor. Four class meetings per week, one of which is a two-hour group-work day.

Requisite: MATH 105. Spring semester. Professor TBA.

Basic concepts of limits, derivatives, anti-derivatives; applications, including max/min problems and related rates; the definite integral, simple applications; trigonometric functions; logarithms and exponential functions. Four class hours per week.

Limited to 35 students per section. Fall and spring semesters. In the fall semester, the intensive section (Section 01) is open only to students listed as eligible on the Mathematics placement list. The intensive section replaces one weekly class hour with a 90-to-120-minute group work day. Professors TBA.

A continuation of MATH 111. Inverse trigonometric and hyperbolic functions; methods of integration, both exact and approximate; applications of integration to volume and arc length; improper integrals; l’Hôpital’s rule; infinite series, power series and the Taylor development; and polar coordinates. Four class hours per week.

Requisite: A grade of C or better in MATH 111 or consent of the Department. Limited to 35 students per section. Fall and spring semesters. Professor TBA.

(Offered as STAT 135 and MATH 135.) Introduction to Statistics via Modeling is an introductory statistics course that uses modeling as a unifying framework for much of statistics. The course provides a basic foundation in statistics with a major emphasis on constructing models from data. Students learn important concepts of statistics by mastering powerful and relatively advanced statistical techniques using computational tools. Topics include descriptive and inferential statistics, probability (including conditional probabilities and Bayes' rule), multiple regression and an introduction to causal inference. This is a more mathematically rigorous version of STAT 111, formerly MATH 130. (Students may not receive credit for both STAT 111 and MATH 135.) Four class hours per week (two will be held in the computer lab).

Requisite: MATH 111. Limited to 24 students. Fall and spring semesters. Lecturer Wang.

Mathematical modeling is the process of translating a real world problem into a mathematical expression, analyzing it using mathematical tools and numerical simulations, and then interpreting the results in the context of the original problem. Discussion of basic modeling principles and case studies will be followed by several projects from areas such as environmental studies and biology (e.g., air pollution, ground water flow, populations of interacting species, social networks). This course has no requisites; projects will be tailored to each student’s level of mathematical preparation. Four class hours per week, with occasional in-class computer labs.

Limited to 24 students. Spring semester. Professor TBA.

Elementary vector calculus; introduction to partial derivatives; multiple integrals in two and three dimensions; line integrals in the plane; Green’s theorem; the Taylor development and extrema of functions of several variables; implicit function theorems; Jacobians. Four class hours per week.

Requisite: A grade of C or better in MATH 121 or the consent of the instructor. Limited to 35 students per section. Fall and spring semesters. Professors TBA.

This course is an introduction to some topics in mathematics that do not require the calculus. The topics covered include logic, elementary set theory, functions, relations and equivalence relations, mathematical induction, counting principles, and graph theory. Additional topics may vary from year to year. This course serves as an introduction to mathematical thought and pays particular attention to helping students learn how to write proofs. Four class hours per week.

Limited to 25 students fall semester. No limit for spring semester. Fall and spring semesters. Professors TBA.

MATH 225 is a mathematical treatment of fractal geometry, a field of mathematics partly developed by Benoit Mandelbrot (1924-2010) that continues to be actively researched in the present day. Fractal geometry is a mathematical examination of the concepts of self-similarity, fractals, and chaos, and their applications to the modeling of natural phenomena. In particular, we will develop the iterated function system (IFS) method for describing fractals, examine Julia sets, Mandelbrot sets, and study the concept of fractal dimension, among other things. Through the teaching of these concepts, MATH 225 will also lend itself to familiarizing students with some of the formalisms and rigor of mathematical proofs.

Requisite: MATH 211 or consent of the instructor. Limited to 35 students. Omitted 2015-16. Professor Folsom.

(Offered as MATH 240 and BIOL 240.) With new experimental techniques leading to large biological data sets of increased quality, the ability to analyze biological systems using mathematical modeling approaches has become an integral part of modern biology. This course aims to provide students interested in the interface between biology and mathematics with an integrated understanding of some of the mathematical and computational techniques used in this field. The mathematical approaches we will use to study biological systems will include discrete and continuous dynamical models as well as probability models and parameter estimation algorithms.

Requisite: MATH 211 and BIOL 181 or 191, or permission of the instructor. Limited to 24 students. Omitted 2015-16.

An introduction to the theory of rational integers; divisibility, the unique factorization theorem; congruences, quadratic residues. Selections from the following topics: cryptology; Diophantine equations; asymptotic prime number estimates; continued fractions; algebraic integers. Four class hours per week. Offered in alternate years.

Requisite: MATH 121 or consent of the instructor. Spring semester. Professor TBA.

About 2300 years ago, Euclid introduced the axiomatic method to mathematics in his geometry textbook, the *Elements*. In this book, Euclid deduced the theorems of geometry from a small number of simple axioms about points, lines, and circles. Among his axioms is the parallel axiom, which asserts that if we are given a line and a point not on the line, then there is a unique line through the given point that is parallel to the given line.

Over 2000 years after Euclid, mathematicians discovered that by replacing Euclid's parallel axiom with its negation, we can develop a different kind of geometry in which we still have geometric objects like triangles and circles, but many of the theorems and formulas are different. For example, the sum of the angles of a triangle will always be less than 180 degrees, and this sum will determine the area of the triangle.

In this course we will study both Euclidean and non-Euclidean geometry. We will also consider the fascinating history of how non-Euclidean geometry was discovered. Four class hours per week.

Requisite: Mathematics 121. Fall semester. Professor Velleman

The study of differential equations is an important part of mathematics that involves many topics, both theoretical and practical. The course will cover first- and second-order ordinary differential equations, basic theorems concerning existence and uniqueness of solutions and continuous dependence on parameters, long-term behavior of solutions and approximate solutions. The focus of the course will be on connecting the theoretical aspects of differential equations with real-world applications from physics, biology, chemistry, and engineering. Four class hours per week.

Requisite: MATH 211 or consent of the instructor. Spring semester. Professor TBA.

The study of vector spaces over the real and complex numbers, introducing the concepts of subspace, linear independence, basis, and dimension; systems of linear equations and their solution by Gaussian elimination; matrix operations; linear transformations and their representations by matrices; eigenvalues and eigenvectors; and inner product spaces. Special attention will be paid to the theoretical development of the subject. Four class meetings per week.

Requisite: MATH 121 or consent of the instructor. This course and MATH 272 may not both be taken for credit. Fall and Spring semester.

The study of vector spaces over the real and complex numbers, introducing the concepts of subspace, linear independence, basis, and dimension; systems of linear equations and their solution by Gaussian elimination; matrix operations; linear transformations and their representations by matrices; eigenvalues and eigenvectors; and inner product spaces. Additional topics include ill-conditioned systems of equations, the LU decomposition, covariance matrices, least squares, and the singular value decomposition. Recommended for Economics majors who wish to learn linear algebra. Four class hours per week, with occasional in-class computer labs.

Requisite: MATH 121 or consent of the instructor. This course and MATH 271 may not both be taken for credit. Spring semester.

A graph is a collection of points with edges drawn between them. Graph theory was first introduced by Leonhard Euler in his solution to the Königsberg bridge problem in 1736. Since then, graph theory has become an active area of study in mathematics due both to its wide array of real life applications in biology, chemistry, social sciences and computer networking, and to its interactions with other branches of mathematics. The course will start with an overview of the fundamental concepts and general results in graph theory, followed by explorations of a variety of topics in graph theory and their applications, including: connectivity, planar graphs, directed graphs, greedy algorithms, matchings, vertex and edge colorings. The course will end with the introduction of a more advanced topic. Four class hours per week.

Requisite: MATH 271 or 272 or permission of the instructor. Recommended: MATH 220 or other prior experience with mathematical proofs. Limited to 30 students. Spring semester. Professor Sosa.

The first half of the course covers continuous and discrete Fourier transforms (including convolution and Plancherel’s formula), Fourier series (including convergence and the fast Fourier transform algorithm), and applications like heat conduction along a rod and signal processing. The second half of the course is devoted to wavelets: Haar bases, the discrete Haar transform in 1 and 2 dimensions with application to image analysis, multiresolution analysis, filters, and wavelet-based image compression like JPEG2000. Three class hours per week plus a weekly one-hour computer laboratory.

Requisite: MATH 211 and 271 or 272. Fall semester. Professor TBA.

An introduction to analytic functions; complex numbers, derivatives, conformal mappings, integrals. Cauchy’s theorem; power series, singularities, Laurent series, analytic continuation; Riemann surfaces; special functions. Four class hours per week.

Requisite: MATH 211 and prior experience with mathematical proofs, or consent of the instructor. Fall semester. Professor TBA.

A brief consideration of properties of sets, mappings, and the system of integers, followed by an introduction to the theory of groups and rings including the principal theorems on homomorphisms and the related quotient structures; integral domains, fields, polynomial rings. Four class hours per week.

Requisite: MATH 271 or 272 or consent of the instructor. Students with a grade of B+ or lower in linear algebra are encouraged to take another 200-level course with proofs before taking MATH 350. Limited to 25 students fall semester. Professor TBA. No limit for spring semester. Professor TBA.

Completeness of the real numbers; topology of n-space including the Bolzano-Weierstrass and Heine-Borel theorems; sequences, properties of functions continuous on sets; infinite series, uniform convergence. The course may also study the Gamma function, Stirling’s formula, or Fourier series. Four class hours per week.

Requisite: MATH 211 and prior experience with mathematical proofs (MATH 271 or 272 recommended), or consent of the instructor. Students with a grade of B+ or lower in linear algebra are encouraged to take another 200-level course with proofs before taking MATH 355. Limited of 25 students fall semester. Professor TBA. No limit for spring semester. Professor TBA.

(Offered as STAT 360 and MATH 360.) This course explores the nature of probability and its use in modeling real world phenomena. There are two explicit complementary goals: to explore probability theory and its use in applied settings, and to learn parallel analytic and empirical problem solving skills. The course begins with the development of an intuitive feel for probabilistic thinking, based on the simple yet subtle idea of counting. It then evolves toward the rigorous study of discrete and continuous probability spaces, independence, conditional probability, expectation, and variance. Distributions covered include the binomial, hypergeometric, Poisson, normal, Gamma, Beta, multinomial, and bivariate normal. Other topics include generating functions, order statistics, and limit theorems. Four class hours per week.

Requisite: MATH 121 or consent of the instructor. Limited to 24 students. Fall semester. Professor Horton.

A stochastic process is a collection of random variables used to model the evolution of a system over time. Unlike deterministic systems, stochastic processes involve an element of randomness or uncertainty. Examples include stock market fluctuations, audio signals, EEG recordings, and random movement such as Brownian motion and random walks. Topics will include Markov chains, martingales, Brownian motion, and stochastic integration, including Ito’s formula. Four class hours per week, with weekly in-class computer labs. \

Requisite: MATH 360 or consent of instructor. Limited to 24 students. Spring semester. Professor TBA.

(Offered as STAT 370 and MATH 370.) This course examines the theory behind common statistical inference procedures including estimation and hypothesis testing. Beginning with exposure to Bayesian inference, the course will cover Maximum Likelihood Estimators, sufficient statistics, sampling distributions, joint distributions, confidence intervals, hypothesis testing and test selection, non-parametric procedures, and linear models. Four class hours per week.

Requisite: STAT 111 or STAT 135 and STAT 360, or consent of the instructor. Spring semester. Professor Horton.

Most mathematicians consider set theory to be the foundation of mathematics, because everything that is studied in mathematics can be defined in terms of the concepts of set theory, and all the theorems of mathematics can be proven from the axioms of set theory. This course will begin with the axiomatization of set theory that was developed by Ernst Zermelo and Abraham Fraenkel in the early part of the twentieth century. We will then see how all of the number systems used in mathematics are defined in set theory, and how the fundamental properties of these number systems can be proven from the Zermelo-Fraenkel axioms. Other topics will include the axiom of choice, infinite cardinal and ordinal numbers, and models of set theory. Four class hours per week.

Requisite: MATH 220, 271, 272, or 355, or consent of the instructor. Fall semester. Professor TBA.

Mathematicians confirm their answers to mathematical questions by writing proofs. But what, exactly, is a proof? This course begins with a precise definition specifying what counts as a mathematical proof. This definition makes it possible to carry out a mathematical study of what can be accomplished by means of deductive reasoning and, perhaps more interestingly, what cannot be accomplished. Topics will include the propositional and predicate calculi, completeness, compactness, and decidability. At the end of the course we will study Gödel’s famous Incompleteness Theorem, which shows that there are statements about the positive integers that are true but impossible to prove. Four class hours per week. Offered in alternate years.

Requisite: MATH 220, 271, 272, or 355, or consent of the instructor. Omitted 2015-16.

Fall and spring semesters. The Department.

The quadratic formula shows us that the roots of a quadratic polynomial possess a certain symmetry. Galois Theory is the study of the corresponding symmetry for higher degree polynomials. We will develop this theory starting from a basic knowledge of groups, rings and fields. One of our main goals will be to prove that there is no general version of the quadratic formula for a polynomial of degree five or more. Along the way, we will also show that a circular cake can be divided into 17 (but not 7) equal slices using only a straight-edged knife.

Requisite: MATH 350 or consent of the instructor. Spring semester. Professor Ching.

The topic will vary from year to year. The topic for 2014 was computational algebraic geometry.

The study of geometric objects by means of their defining equations dates back to the introduction of coordinates by Descartes in 1637.

This course will introduce algorithmic methods for manipulating and understanding algebraic equations and will develop a dictionary between algebra and geometry. We will also explore the structure of ideals in polynomial rings and the resulting quotient rings. The course will end with student presentations on applications of algebraic geometry to robotics, geometric theorem proving, invariant theory, graph theory, and sudoku. Three class hours per week plus a weekly one-hour computer lab.

Requisite: MATH 350. Limited to 16 students. Omitted 2015-16.

An introduction to Lebesgue measure and integration; topology of the real numbers, inner and outer measures and measurable set; the approximation of continuous and measurable functions; the Lebesgue integral and associated convergence theorems; the Fundamental Theorem of Calculus. Four class hours per week.

Requisite: MATH 355. Spring semester. Professor TBA.

An introduction to general topology; the topology of Euclidean, metric and abstract spaces, with emphasis on such notions as continuous mappings, compactness, connectedness, completeness, separable spaces, separation axioms, and metrizable spaces. Additional topics may be selected to illustrate applications of topology in analysis or to introduce the student briefly to algebraic topology. Four class hours per week. Offered in alternate years.

Requisite: MATH 355. Spring semester. Professor TBA.

Open to seniors with the consent of the Department. Fall semester. The Department.

This course is an introduction to applied statistical methods useful for the analysis of data from all fields. Brief coverage of data summary and graphical techniques will be followed by elementary probability, sampling distributions, the central limit theorem and statistical inference. Inference procedures include confidence intervals and hypothesis testing for both means and proportions, the chi-square test, simple linear regression, and a brief introduction to analysis of variance (ANOVA). Four class hours per week (two will be held in the computer lab). Labs are not interchangeable between sections due to course content.

Limited to 24 students per section. Fall semester: Professor Kim. Spring semester: Professors Kim.

(Offered as STAT 135 and MATH 135.) Introduction to Statistics via Modeling is an introductory statistics course that uses modeling as a unifying framework for much of statistics. The course provides a basic foundation in statistics with a major emphasis on constructing models from data. Students learn important concepts of statistics by mastering powerful and relatively advanced statistical techniques using computational tools. Topics include descriptive and inferential statistics, probability (including conditional probabilities and Bayes' rule), multiple regression and an introduction to causal inference. This is a more mathematically rigorous version of STAT 111, formerly MATH 130. (Students may not receive credit for both STAT 111 and MATH 135.) Four class hours per week (two will be held in the computer lab).

Requisite: MATH 111. Limited to 24 students. Fall and spring semesters. Lecturer Wang.

Bayesian statistics is founded upon the idea that our beliefs about the world are constantly revised with the incorporation of new information. This course provides a principled introduction to Bayesian statistics. We begin with the basic building blocks of Bayesian inference: the likelihood, prior, and posterior distributions. We will then show how to simulate from the posterior distribution using the Markov chain Monte Carlo (MCMC) method. Single and multivariate models will be considered as well as hierarchical models, such as Bayesian linear regression, and other more advanced topics. The course will emphasize problem solving and data analysis via statistical software. Four class hours per week.

Requisite MATH 111 and STAT 111/135 or permission of the instructor. Limited to 20 students. Spring semester. Professor Wang.

This course is an introduction to nonparametric and distribution-free statistical procedures and techniques. These methods rely heavily on counting and ranking techniques and will be explored through both theoretical and applied perspectives. One- and two-sample procedures will provide students with alternatives to traditional parametric procedures, such as the t-test. We will also investigate correlation, regression, and one-way analysis of variance techniques in a nonparametric setting. A variety of other topics may be explored in the nonparametric setting including resampling techniques (for example, bootstrapping), categorical data and contingency tables, density estimation, and the two-way layout. The course will emphasize data analysis (with appropriate use of statistical software) and the intuitive nature of nonparametric statistics. Four class hours per week.

Requisite: STAT 111 or STAT 135 or equivalent. Omitted 2015-16.

This course is an intermediate applied statistics course that builds on the statistical data analysis methods introduced in STAT 111 or STAT 135. Students will learn how to pose a statistical question, perform appropriate statistical analysis of the data, and properly interpret and communicate their results. Emphasis will be placed on the use of statistical software, data manipulation, model fitting, and assessment. Topics covered will include ethics, experimental design, parametric and nonparametric methods, resampling approaches, analysis of variance models, multiple regression, model selection, and logistic regression. Four class hours per week.

Requisite: STAT 111 or 135 or consent of the instructor. Limited to 24 students. Fall semester: Professor Wang. Spring semester: Professor Kim.

This course is an intermediate applied statistics course that builds on the statistical concepts introduced in STAT 111 or STAT 135 and data analysis methods introduced in 200-level statistics courses. It will focus on the analysis and mapping of environmental and social data in a spatial context, including continuous process data and point process data. Other topics include descriptive and inferential techniques used in quantitative geographic analysis, parametric and nonparametric analyses, model assessment, and visualization. Students will build computing skills and use R for data display, modeling, and communication. Two class meetings per week, 80 minutes each.

Requisite: Previous 200-level statistics coursework, or any 2 courses in statistics, or permission of the instructor. Limited to 24 students. Fall semester. Visiting Professor Kim.

(Offered as STAT 360 and MATH 360.) This course explores the nature of probability and its use in modeling real world phenomena. There are two explicit complementary goals: to explore probability theory and its use in applied settings, and to learn parallel analytic and empirical problem solving skills. The course begins with the development of an intuitive feel for probabilistic thinking, based on the simple yet subtle idea of counting. It then evolves toward the rigorous study of discrete and continuous probability spaces, independence, conditional probability, expectation, and variance. Distributions covered include the binomial, hypergeometric, Poisson, normal, Gamma, Beta, multinomial, and bivariate normal. Other topics include generating functions, order statistics, and limit theorems. Four class hours per week.

Requisite: MATH 121 or consent of the instructor. Limited to 24 students. Fall semester. Professor Horton.

(Offered as STAT 370 and MATH 370.) This course examines the theory behind common statistical inference procedures including estimation and hypothesis testing. Beginning with exposure to Bayesian inference, the course will cover Maximum Likelihood Estimators, sufficient statistics, sampling distributions, joint distributions, confidence intervals, hypothesis testing and test selection, non-parametric procedures, and linear models. Four class hours per week.

Requisite: STAT 111 or STAT 135 and STAT 360, or consent of the instructor. Spring semester. Professor Horton.

Our world is awash in data. To allow decisions to be made based on evidence, there is a need for statisticians to be able to make sense of the data around us and communicate their findings. In this course, students will be exposed to advanced statistical methods and will undertake the analysis and interpretation of complex and real-world datasets that go beyond textbook problems. Course topics will vary from year to year depending on the instructor and selected case studies. Topics may include visualization techniques to summarize and display high dimensional data, advanced topics in design and linear regression, selected topics in data mining, nonparametric analysis, and analysis of network data. Through a series of case studies, students will develop the capacity to think and compute with data, undertake and assess analyses, and effectively communicate their results using written and oral presentation.

Requisite: STAT 230 (formerly MATH 230), STAT 370 (formerly MATH 430 and STAT 430) and the computer requirement; or consent of the instructor. Limited to 20 students. Fall semester. Professor Wang.