Amherst College Mathematics and Statistics for 2018-19

Mathematics

105 Calculus with Algebra

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 Pflueger.2017-18: Offered in Fall 2017 Other years: Offered in Fall 2009, Fall 2010, Fall 2011, Fall 2012, Fall 2013, Fall 2014, Fall 2015, Fall 2016, Fall 2018

106 Calculus with Elementary Functions

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.2017-18: Offered in Spring 2018 Other years: Offered in Spring 2008, Spring 2009, Spring 2010, Spring 2011, Spring 2012, Spring 2013, Spring 2014, Spring 2015, Spring 2016, Spring 2017

111 Introduction to the Calculus

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 30 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. Fall and spring semesters. The Department.2017-18: Offered in Fall 2017 and Spring 2018 Other years: Offered in Fall 2009, Spring 2010, Fall 2010, Spring 2011, Fall 2011, Spring 2012, Fall 2012, Spring 2013, Fall 2013, Spring 2014, Fall 2014, Spring 2015, Fall 2015, Spring 2016, Fall 2016, Spring 2017, Fall 2018

121 Intermediate Calculus

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 placement into MATH 121 or consent of the Department. Limited to 30 students per section. Fall and spring semesters. The Department. 2017-18: Offered in Fall 2017 and Spring 2018 Other years: Offered in Fall 2009, Spring 2010, Fall 2010, Spring 2011, Fall 2011, Spring 2012, Fall 2012, Spring 2013, Fall 2013, Spring 2014, Fall 2014, Spring 2015, Fall 2015, Spring 2016, Fall 2016, Spring 2017, Fall 2018

135 Introduction to Statistics via Modeling

(Offered as STAT 135 and MATH 135) Introduction to Statistics via Modeling is an introductory statistics course that uses modeling as a unifying framework. 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, visualization, probability, study design, ANOVA, and multiple regression. Students who have taken a semester of calculus (MATH 111 or higher, or equivalent placement) or who are majoring or planning to major in mathematics and/or statistics should take this course instead of STAT 111/111E. (Students who have taken STAT 111/111E or PSYC 122 may not also receive credit for STAT/MATH 135. Students who have taken ECON 360/361 will be admitted only with permission of instructor.)Requisite: MATH 111. Limited to 24 students. Fall and spring semesters. Fall semester Professors Bailey, Correia and Horton. Spring semester Professors Bailey, Correia, Liao and Visiting Assistant Professor Matheson.2017-18: Offered in Fall 2017 and Spring 2018 Other years: Offered in Spring 2014, Spring 2015, Fall 2015, Spring 2016, Fall 2016, Spring 2017, Fall 2018

140 Mathematical Modeling

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 including biology and the social sciences (e.g., flocking and schooling behavior, disease spread in populations, generation of artificial societies). 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. Fall semester. Professor Leise.2017-18: Offered in Fall 2017 Other years: Offered in Fall 2008, Fall 2012, Fall 2014, Spring 2016

150 Voting and Elections: A Mathematical Perspective

The outcomes of many elections, whether to elect the next United States president or to rank college football teams, can displease many of the voters. How can perfectly fair elections produce results that nobody likes? We will analyze different voting systems, including majority rule, plurality rule, Borda count, and approval voting, and assess a voter’s power to influence the election under each system, for example, by calculating the Banzhaf power index. We will prove Arrow’s Theorem and discuss its implications. After exploring the pitfalls of various voting systems through both theoretical analysis and case studies, we will try to answer some pressing questions: Which voting system best reflects the will of the voters? Which is least susceptible to manipulation? What properties should we seek in a voting system, and how can we best attain them?Limited to 24 students. Fall semester. Professor Leise. 2017-18: Not offered Other years: Offered in Fall 2016, Fall 2018

205 ≠ (Inequality)

(Offered as MATH 205, BLST 209 and HIST 209 [US]) This course will look at issues surrounding inequality in K-12 math education. Mathematics has a reputation for being something that either you can do or you can’t: the so-called "geniuses" know all the answers already, whereas for everyone else it is a constant struggle. In addition, math and other STEM fields have traditionally been discouraged as career paths for women and for students from underrepresented groups. At Amherst today, students from those groups are still in the minority in math classes. We’ll ask why this is, whether it can and should be changed, and if so, how.
Our discussions will be guided by some of the following questions: To what extent is math ability an innate talent that you are either born with or not? How and why is variation in accomplishment in mathematics related to race, gender and socio-economic class? What mathematics should we teach in schools and how should those teachers be prepared? What is "math phobia," how does it develop and how can it be treated? How do attitudes towards math in the general public affect student learning?
Limited to 25 students. Admission with consent of the instructor. Spring semester. Professors Ching and Moss.2017-18: Offered in Spring 2018 Other years: Offered in Spring 2017

211 Multivariable Calculus

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 placement into MATH 211 or consent of the Department. Limited to 30 students per section. Fall and spring semesters. The Department.2017-18: Offered in Fall 2017 and Spring 2018 Other years: Offered in Fall 2009, Spring 2010, Fall 2010, Spring 2011, Fall 2011, Spring 2012, Fall 2012, Spring 2013, Fall 2013, Spring 2014, Fall 2014, Spring 2015, Fall 2015, Spring 2016, Fall 2016, Spring 2017, Fall 2018

220 Mathematical Reasoning and Proof

This course serves as an introduction to mathematical reasoning and pays particular attention to helping students learn how to write proofs. The topics covered may include logic, elementary set theory, functions, relations and equivalence relations, mathematical induction, sequences, and quantifiers. Additional topics may vary from semester to semester. . Four class hours per week. Limited to 25 students. Fall semester: Professor TBA. Spring semester: Professor TBA.2017-18: Offered in Fall 2017 and Spring 2018 Other years: Offered in Spring 2010, Spring 2011, Spring 2012, Spring 2013, Spring 2014, Spring 2015, Fall 2015, Spring 2016, Fall 2016, Spring 2017, Fall 2018

225 Fractal Geometry

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 2017-18.2017-18: Not offered Other years: Offered in Fall 2013, Fall 2014, Fall 2016

250 Number Theory

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. Limited to 25 students. Spring semester. Professor TBA.2017-18: Offered in Spring 2018 Other years: Offered in Fall 2010, Fall 2012, Spring 2014, Spring 2015, Spring 2016, Spring 2017

255 Geometry

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 Sosa Castillo.2017-18: Offered in Fall 2017 Other years: Offered in Fall 2011, Fall 2015

260 Differential Equations

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. Limited to 25 students. Spring semester. Professor TBA.2017-18: Offered in Spring 2018 Other years: Offered in Spring 2009, Spring 2011, Spring 2013, Spring 2014, Spring 2015, Spring 2016, Spring 2017

271 Linear Algebra

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. MATH 271 will feature both proofs and applications, with special attention 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. Limited to 25 students. Fall semester Professor Daniels. Spring semesters Professor TBA.2017-18: Offered in Fall 2017 and Spring 2018 Other years: Offered in Fall 2007, Spring 2008, Fall 2008, Spring 2009, Fall 2011, Spring 2012, Fall 2012, Spring 2013, Fall 2013, Spring 2014, Fall 2014, Spring 2015, Fall 2015, Spring 2016, Fall 2016, Spring 2017, Fall 2018

272 Linear Algebra with Applications

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. MATH 272 will feature both proofs and applications, with special attention paid to applied topics such as least squares and singular value decomposition.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. Limited to 25 students. Fall semester: Professor TBA. Spring semester: Professor TBA. 2017-18: Offered in Fall 2017 and Spring 2018 Other years: Offered in Fall 2007, Spring 2008, Fall 2008, Spring 2009, Fall 2009, Spring 2010, Fall 2010, Spring 2011, Fall 2011, Spring 2012, Fall 2012, Spring 2013, Fall 2013, Spring 2014, Fall 2014, Spring 2015, Fall 2015, Spring 2016, Fall 2016, Spring 2017, Fall 2018

280 Graph Theory

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 instructor. MATH 220 or other prior experience with mathematical proofs is recommended. Limited to 30 students. Spring semester. Visiting Professor Sosa Castillo.2017-18: Offered in Spring 2018 Other years: Offered in Spring 2016

281 Combinatorics

This course emphasizes enumerative combinatorics, a classical subject in mathematics related to the theory of counting. Problems in this area often pertain to finding the number of possible arrangements of a set of objects under some particular constraints. This course incorporates a wide set of problems involving enumerative combinatorics, as well as theory and applications. Topics include the sum and product rules, combinations and permutations, binomial and multinomial coefficients, the principle of inclusion and exclusion, generating functions, recurrence relations, Catalan, Stirling, Bell and Eulerian numbers, partitions, tableaux, and stable marriage. Additional topics may vary.
Requisite: MATH 121, and MATH 220 or other prior experience with basic mathematical proof techniques (e.g., induction) by consent of instructor. Limited to 24 students. Fall semester. Professor Folsom.2017-18: Offered in Fall 2017

284 Numerical Analysis

This course will study numerical techniques for a variety of problems, such as finding roots of polynomials, interpolation, numerical integration, numerical solutions of differential equations, and matrix computations. We will study the underlying theory behind the algorithms, including error analysis, and the algorithms will be implemented using mathematical software to facilitate numerical experimentation.Requisite: MATH 211 and either 271 or 272, or permission of the instructor. Limited to 24 students. Fall semester. Professor Yacoubou Djima. 2017-18: Not offered Other years: Offered in Fall 2016, Fall 2018

294 Optimization

Optimization is a branch of applied mathematics focused on algorithms to determine maxima and minima of functions, often under constraints. Applications range from economics and finance to machine learning and information retrieval. This course will first develop advanced linear algebra tools, and then will study methods of convex optimization, including linear, quadratic, second-order cone, and semidefinite models. Several applications will be explored, and algorithms will be implemented using mathematical software to aid numerical experimentation.
Requisite: MATH 211 and 271 or 272, or permission of the instructor. Limited to 30 students. Omitted 2017-18. 2017-18: Not offered Other years: Offered in Spring 2017

310 Introduction to the Theory of Partitions

The theory of partitions is a fundamental branch of combinatorics and number theory pertaining to enumerative properties and patterns of the integers. With its mathematical origins tracing back to the seventeenth century, partition theory has evolved through contributions made by many influential mathematicians including Euler, Legendre, Hardy, Ramanujan, Selberg and Dyson, and continues to be an active area of study today. Topics include partition identities and bijections, Ferrers diagrams and Durfee squares, partition generating functions and q-series, the pentagonal number theorem, q-binomial numbers (Gaussian polynomials), and partition congruences.
Requisite: MATH 220 and 121, or other significant experience with proofs, or by consent of instructor. Limited to 24 students. Omitted 2017-18.
2017-18: Not offered Other years: Offered in Fall 2016

320 Wavelet and Fourier Analysis

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. Post-doctoral Fellow Yacoubou Djima. 2017-18: Offered in Fall 2017 Other years: Offered in Fall 2007, Fall 2009, Fall 2011, Fall 2013, Fall 2015

345 Functions of a Complex Variable

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.2017-18: Offered in Fall 2017 Other years: Offered in Fall 2009, Fall 2010, Fall 2011, Fall 2012, Fall 2013, Fall 2014, Fall 2015, Fall 2016, Fall 2018

350 Groups, Rings and Fields

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 211 and either 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 Pflueger. Spring semester Professor TBA.2017-18: Offered in Fall 2017 and Spring 2018 Other years: Offered in Spring 2010, Spring 2011, Spring 2012, Spring 2013, Spring 2014, Fall 2014, Spring 2015, Fall 2015, Spring 2016, Fall 2016, Spring 2017, Fall 2018

355 Introduction to Analysis

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 either 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 355. Limited to 25 students. Fall semester Professor Call. Spring semester: Professor TBA. 2017-18: Offered in Fall 2017 and Spring 2018 Other years: Offered in Spring 2010, Spring 2011, Spring 2012, Spring 2013, Fall 2013, Spring 2014, Fall 2014, Spring 2015, Fall 2015, Spring 2016, Fall 2016, Spring 2017, Fall 2018

359 An Introduction to the p-adic Numbers

The p-adic numbers were first introduced by Kurt Hensel near the end of the nineteenth century. Since their introduction they have become a central object in modern number theory, algebraic geometry, and algebraic topology. These numbers give a new set of fields (one for each prime p) that contain the rational numbers and behave in some ways like the real numbers. While these fields are similar to the real numbers in some respects, they also possess some unique and unexpected properties. The main objective of this course will be to rigorously construct the p-adic numbers and explore some of the algebraic, analytic, and topological properties that make them interesting.
Requisite: MATH 350 or consent of the instructor. Limited to 32 students. Omitted 2017-18.2017-18: Not offered Other years: Offered in Spring 2017

360 Probability

(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 Liao.2017-18: Offered in Fall 2017 Other years: Offered in Fall 2008, Fall 2010, Fall 2011, Fall 2012, Fall 2013, Fall 2014, Fall 2015, Fall 2016, Fall 2018

365 Stochastic Processes

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 Leise.2017-18: Offered in Spring 2018 Other years: Offered in Spring 2014, Spring 2016

370 Theoretical Statistics

(Offered as STAT 370 and MATH 370) This course examines the theory underlying common statistical procedures including visualization, exploratory analysis, estimation, hypothesis testing, modeling, and Bayesian inference. Topics include maximum likelihood estimators, sufficient statistics, confidence intervals, hypothesis testing and test selection, non-parametric procedures, and linear models.Requisite: STAT 111 or STAT 135 and STAT 360, or consent of the instructor. Limited to 25 students. Spring semester. Professor Horton.2017-18: Offered in Spring 2018 Other years: Offered in Spring 2009, Spring 2013, Spring 2015, Spring 2016, Spring 2017

375 Introduction to Representation Theory

Representation theory concerns the study of groups by expressing their elements as linear transformations of vector spaces. This approach gives us a more concrete way to think about groups via matrices, and it allows us to use tools from linear algebra to study them. Topics covered in this course include group actions, representations and modules, subrepresentations and homomorphisms, irreducibility, characters and character tables, and induced and restricted representations. We will also explore some applications to physics and other areas of mathematics.
Requisite: MATH 350 or consent of the instructor. Limited to 24 students. Priority to pre-registered students, math majors, and juniors and seniors. Omitted 2017-18.2017-18: Not offered Other years: Offered in Fall 2016

378 Differential Geometry of Curves and Surfaces

This course will explore the geometry of curves and surfaces in n-dimensional Euclidean space. For curves, the key concepts are curvature and torsion, while for surfaces, the key players are Gaussian curvature, geodesics, and the Gauss-Bonnet Theorem. Other topics covered may include (time permitting) the Four Vertex Theorem, map projections, the Hairy Ball Theorem, and minimal surfaces.
Requisites: MATH 211, MATH 271 or 272, and MATH 355 or consent of the instructor. Spring semester. Professor Cox.2017-18: Offered in Spring 2018

385 Mathematical Logic

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. Spring semester. Professor Ching.2017-18: Offered in Spring 2018 Other years: Offered in Fall 2008, Fall 2010, Spring 2015

405 Lie Algebras

Lie algebras originally arose as a way of studying certain continuous transformation groups called Lie groups. Lie algebras are simpler objects than Lie groups since they can be studied using tools from linear algebra, yet they still provide a lot of information about their associated Lie groups. This class serves as a first introduction to the theory of Lie algebras. We will examine the structure of finite dimensional Lie algebras, ideals and homomorphisms, nilpotent and solvable Lie algebras, Cartan subalgebras, semisimplicity, root systems, and the classification of semisimple Lie algebras. This classification is not only a fundamental result in Lie Theory, but is also an archetype of classifications that appear in other areas of math. More amazingly, this classification is embodied in simple combinatorial pictures called Dynkin diagrams, which underlie surprisingly disparate fields, such as geometric group theory, quiver representation theory, and string theory.
Requisite: MATH 350 or consent of the instructor. Limited to 18 students. Fall semester. Visiting Professor Naqvi.2017-18: Offered in Fall 2017

410 Galois Theory

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. Fall semester. Professor Cox.2017-18: Not offered Other years: Offered in Fall 2013, Spring 2016, Fall 2018

450 Measure Theory and Integration

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 R. Benedetto.2017-18: Offered in Spring 2018 Other years: Offered in Spring 2008, Spring 2010, Spring 2012, Spring 2014, Spring 2016

455 Topology

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. 2017-18: Not offered Other years: Offered in Spring 2009, Spring 2011, Spring 2013, Spring 2015, Spring 2017

460 Analytic Number Theory

MATH 460 is an introduction to Analytic Number Theory, a foundational subject in mathematics which dates back to the 1800s and is still a major research area today. The subject generally uses tools and techniques which are analytic in nature to solve problems primarily related to integers. Asymptotic and summation results and methods are of great significance in Analytic Number Theory. Two primary course objectives are to state and prove two major theorems: Dirichlet's Theorem on Primes in Arithmetic Progressions, and the Prime Number Theorem. In particular, we will study Selberg's "elementary" proof of the Prime Number Theorem, as well as an analytic proof. Additional topics may include: arithmetic functions, especially their averages, their asymptotics, and related summation formulae; Dirichlet convolutions; characters and Gauss sums; and an introduction to Dirichlet series, such as the Riemann zeta-function and L-functions. Further topics may vary.
Requisite: At least two among MATH 345, MATH 350, and MATH 355, with MATH 345 preferred; or by consent of instructor. Prior experience with number theory, such as MATH 250, may be helpful but is not required. Spring semester. Professor Folsom. 2017-18: Offered in Spring 2018

490, 390 Special Topics

Fall and spring semesters. The Department.2017-18: Offered in Fall 2017 and Spring 2018 Other years: Offered in Fall 2009, Fall 2010, Fall 2011, Spring 2012, Fall 2012, Spring 2013, Fall 2013, Spring 2014, Fall 2014, Spring 2015, Fall 2015, Spring 2016, Fall 2016, Spring 2017, Fall 2018

498, 499 Senior Departmental Honors

Open to seniors with the consent of the Department. Fall semester. The Department.2017-18: Offered in Fall 2017 Other years: Offered in Fall 2009, Fall 2010, Fall 2011, Fall 2012, Fall 2013, Fall 2014, Fall 2015, Fall 2016, Fall 2018

Statistics

111, 111E Introduction to Statistics

Introduction to Statistics provides a basic foundation in descriptive and inferential statistics, including constructing models from data. Students will learn to think critically about data, produce meaningful graphical and numerical summaries of data, apply basic probability models, and utilize statistical inference procedures using computational tools. Topics include basic descriptive and inferential statistics, visualization, study design, and multiple regression. Students who have taken a semester of calculus (MATH 111 or higher, or equivalent placement) or who are planning to major in statistics should take STAT 135/MATH 135 instead of this course. (Students who have taken STAT/MATH 135, PSYC 122, or ECON 360/361 may not also receive credit for STAT 111/111E, and STAT 111/111E does not count towards the major in Mathematics.)Limited to 24 students per section. Fall semester. Visiting Assistant Professor Matheson.2017-18: Offered in Fall 2017 and Spring 2018 Other years: Offered in Fall 2008, Spring 2009, Fall 2009, Spring 2010, Fall 2010, Spring 2011, Fall 2011, Spring 2012, Fall 2012, Spring 2013, Fall 2013, Spring 2014, Fall 2014, Spring 2015, Fall 2015, Spring 2016, Fall 2016, Spring 2017

135 Introduction to Statistics via Modeling

(Offered as STAT 135 and MATH 135) Introduction to Statistics via Modeling is an introductory statistics course that uses modeling as a unifying framework. 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, visualization, probability, study design, ANOVA, and multiple regression. Students who have taken a semester of calculus (MATH 111 or higher, or equivalent placement) or who are majoring or planning to major in mathematics and/or statistics should take this course instead of STAT 111/111E. (Students who have taken STAT 111/111E or PSYC 122 may not also receive credit for STAT/MATH 135. Students who have taken ECON 360/361 will be admitted only with permission of instructor.)Requisite: MATH 111. Limited to 24 students. Fall and spring semesters. Fall semester Professors Bailey, Correia and Horton. Spring semester Professors Bailey, Correia, Liao and Visiting Assistant Professor Matheson.2017-18: Offered in Fall 2017 and Spring 2018 Other years: Offered in Spring 2014, Spring 2015, Fall 2015, Spring 2016, Fall 2016, Spring 2017

220 Bayesian Modeling and Inference

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 instructor. Limited to 20 students. Omitted 2018-19.2017-18: Not offered Other years: Offered in Spring 2016

225 Nonparametric Statistics

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 and regression 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 one-way and two-way layouts for analysis of variance. The course will emphasize data analysis (with appropriate use of statistical software) and the intuitive nature of nonparametric statistics.Requisite: STAT 111 or STAT 135. Limited to 24 students. Omit 2018-19.2017-18: Offered in Spring 2018 Other years: Offered in Spring 2015

230 Intermediate Statistics

This course is an intermediate applied statistics course that builds on the statistical data analysis methods introduced in STAT 111/111E 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 wrangling, model fitting, and assessment. Topics covered will include ethics, experimental design, resampling approaches, analysis of variance models, multiple regression, model selection, and logistic regression.Requisite: STAT 111 or 135. Limited to 24 students (4 spots reserved for first-year students in fall). Fall and spring semesters. Fall semester: Professor Wagaman. Spring semester: Professors Wagaman and Correia.2017-18: Offered in Fall 2017 and Spring 2018 Other years: Offered in Spring 2008, Spring 2012, Fall 2013, Spring 2014, Fall 2016, Spring 2017

231 Data Science

Computational data analysis is an essential part of modern statistics and data science. This course provides a practical foundation for students to think with data by participating in the entire data analysis cycle. Students will generate statistical questions and then address them through data acquisition, cleaning, transforming, modeling, and interpretation. This course will introduce students to tools for data management and wrangling that are common in data science and will apply those tools to real-world applications. Students will undertake practical analyses of large, complex, and messy data sets leveraging modern computing tools.Requisite: STAT 111 or STAT 135 and COSC 111 or consent of the instructor. Limited to 24 students. Fall semester: Professor Horton. Spring semester: Professor Horton.2017-18: Offered in Fall 2017 and Spring 2018

265 Spatial Statistics

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. Omitted 2018-19. 2017-18: Not offered Other years: Offered in Fall 2015

360 Probability

(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 Liao.2017-18: Offered in Fall 2017 Other years: Offered in Fall 2008, Fall 2011, Fall 2012, Fall 2013, Fall 2014, Fall 2015, Fall 2016, Fall 2018

370 Theoretical Statistics

(Offered as STAT 370 and MATH 370) This course examines the theory underlying common statistical procedures including visualization, exploratory analysis, estimation, hypothesis testing, modeling, and Bayesian inference. Topics include maximum likelihood estimators, sufficient statistics, confidence intervals, hypothesis testing and test selection, non-parametric procedures, and linear models.Requisite: STAT 111 or STAT 135 and STAT 360, or consent of the instructor. Limited to 25 students. Spring semester. Professor Horton.2017-18: Offered in Spring 2018 Other years: Offered in Spring 2009, Spring 2013, Spring 2015, Spring 2016, Spring 2017

495 Advanced Data Analysis

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 but will include static and dynamic visualization techniques to summarize and display high dimensional data, advanced topics in design and linear regression, ethics, and selected topics in data mining. Other topics may vary but might include nonparametric analysis, spatial data, 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, STAT 370, and the computing requirement; or consent of the instructor. Recommended requisite: STAT 231. Limited to 20 students. Fall semester. Professor Wagaman. 2017-18: Offered in Fall 2017 Other years: Offered in Fall 2014, Fall 2015, Fall 2016