Mathematics (B.A. or B.S.)

Courses

MATH 106b, The Shape of Space  Ian Adelstein

This course provides an introduction to mathematical thinking through ideas in geometry and graph theory. Traditional lecture, worksheets, discussion, group work, and classroom activities all contribute to a dynamic learning experience. The course follows a historical narrative, starting from antiquity, to understand the foundations of mathematical thought. An axiomatic approach to geometry affords students the opportunity to construct proofs of classical theorems. The basics of graph theory are introduced in order to explore real world problems such as map coloring and bridge crossing. The ancient Greek method of exhaustion previews a discussion of the integral, and from here we explore the beautiful relationship between the geometry and topology of graphs, polyhedra, and surfaces. Throughout the course students build their mathematical and geometric intuition through problem solving and exercises in geometric imagining. Enrollment is limited to students who have not previously taken a course numbered at or above MATH 110.  QR
MW 2:30pm-3:45pm

* MATH 107a, Mathematics in the Real World  Brett Smith

The use of mathematics to address real-world problems. Applications of exponential functions to compound interest and population growth; geometric series in mortgage payments, amortization of loans, present value of money, and drug doses and blood levels; basic probability, Bayes's rule, and false positives in drug testing; elements of logic. No knowledge of calculus required. Enrollment limited to students who have not previously taken a high school or college calculus course.  QR
TTh 2:30pm-3:45pm

MATH 108b, Estimation and Error  Sudesh Kalyanswamy

A problem-based investigation of basic mathematical principles and techniques that help make sense of the world. Estimation, order of magnitude, approximation and error, counting, units, scaling, measurement, variation, simple modeling. Applications to demographics, geology, ecology, finance, and other fields. Emphasis on both the practical and the philosophical implications of the mathematics. No knowledge of calculus required. Enrollment limited to students who have not previously taken a high school or college calculus course.  QR
TTh 2:30pm-3:45pm

* MATH 110a, Introduction to Functions and Calculus I  Staff

Comprehensive review of precalculus, limits, differentiation and the evaluation of definite integrals, with applications. Precalculus and calculus topics are integrated. Emphasis on conceptual understanding and problem solving. Successful completion of MATH 110 and 111 is equivalent to MATH 112. No prior acquaintance with calculus is assumed; some knowledge of algebra and precalculus mathematics is helpful.  QR
HTBA

* MATH 111b, Introduction to Functions and Calculus II  Staff

Continuation of MATH 110. Comprehensive review of precalculus, limits, differentiation and evaluation of definite integrals, with applications. Precalculus and calculus topics are integrated. Emphasis on conceptual understanding and problem solving. Successful completion of both MATH 110 and 111 is equivalent to MATH 112. Prerequisite: MATH 110.  QR
HTBA

* MATH 112a or b, Calculus of Functions of One Variable I  Staff

Limits and their properties. Definitions and some techniques of differentiation and the evaluation of definite integrals, with applications. Use of the software package Mathematica to illustrate concepts. No prior acquaintance with calculus or computing assumed. May not be taken after MATH 110 or 111.  QR
HTBA

* MATH 115a or b, Calculus of Functions of One Variable II  Staff

A continuation of MATH 112. Applications of integration, with some formal techniques and numerical methods. Improper integrals, approximation of functions by polynomials, infinite series. Exercises involve the software package Mathematica. After MATH 112 or equivalent; open to freshmen with some preparation in calculus. May not be taken after MATH 116.  QR
HTBA

* MATH 116a, Mathematical Models in the Biosciences I: Calculus Techniques  John Hall

Introduction to topics in mathematical modeling that are applicable to biological systems. Discrete and continuous models of population, neural, and cardiac dynamics. Stability of fixed points and limit cycles of differential equations. Applications include Norton's chemotherapy scheduling and stochastic models of tumor suppressor gene networks. After MATH 112 or equivalent. May not be taken after MATH 115.  QR
TTh 9am-10:15am

* MATH 118a or b, Introduction to Functions of Several Variables  Staff

A combination of linear algebra and differential calculus of several variables. Matrix representation of linear equations, Gauss elimination, vector spaces, independence, basis and dimension, projections, least squares approximation, and orthogonality. Three-dimensional geometry, functions of two and three variables, level curves and surfaces, partial derivatives, maxima and minima, and optimization. Intended for students in the social sciences, especially Economics. May not be taken after MATH 120 or 222. Prerequisite: MATH 112.  QR
HTBA

* MATH 120a or b, Calculus of Functions of Several Variables  Staff

Analytic geometry in three dimensions, using vectors. Real-valued functions of two and three variables, partial derivatives, gradient and directional derivatives, level curves and surfaces, maxima and minima. Parametrized curves in space, motion in space, line integrals; applications. Multiple integrals, with applications. Divergence and curl. The theorems of Green, Stokes, and Gauss. After MATH 115, or with permission of instructor. May not be taken after MATH 121.  QR
HTBA

* MATH 121b, Mathematical Models in the Biosciences II: Advanced Techniques  John Hall

A continuation of MATH 116, focusing on epidemiological models, mathematical foundations of virus and antiviral dynamics, ion channel models and cardiac arrhythmias, and evolutionary models of disease.
After MATH 116, or with permission of instructor.  QR
TTh 9am-10:15am

MATH 160b / AMTH 160b / S&DS 160b, The Structure of Networks  Ronald Coifman

Network structures and network dynamics described through examples and applications ranging from marketing to epidemics and the world climate. Study of social and biological networks as well as networks in the humanities. Mathematical graphs provide a simple common language to describe the variety of networks and their properties.  QR
TTh 11:35am-12:50pm

MATH 222a or b / AMTH 222a or b, Linear Algebra with Applications  Staff

Matrix representation of linear equations. Gauss elimination. Vector spaces. Linear independence, basis, and dimension. Orthogonality, projection, least squares approximation; orthogonalization and orthogonal bases. Extension to function spaces. Determinants. Eigenvalues and eigenvectors. Diagonalization. Difference equations and matrix differential equations. Symmetric and Hermitian matrices. Orthogonal and unitary transformations; similarity transformations. After MATH 115 or equivalent. May not be taken after MATH 225.  QR
HTBA

MATH 225a or b, Linear Algebra and Matrix Theory  Staff

An introduction to the theory of vector spaces, matrix theory and linear transformations, determinants, eigenvalues, and quadratic forms. Some relations to calculus and geometry are included. After or concurrently with MATH 120. May not be taken after MATH 222.  QR
HTBA

* MATH 230a, Vector Calculus and Linear Algebra I  Patrick Devlin

A careful study of the calculus of functions of several variables, combined with linear algebra.  QR
HTBA

* MATH 231b, Vector Calculus and Linear Algebra II  Patrick Devlin

Continuation of MATH 230. Application of linear algebra to differential calculus. Inverse and implicit function theorems; the idea of a manifold; integration of differential forms; general Stokes' theorem.  QR
TTh 11:35am-12:50pm

* MATH 235b, Reflection Groups  Caglar Uyanik

Concepts of linear algebra are used to explore the algebraic and geometric properties of groups generated by reflections. Examples from reflection groups introduce elements of group theory, Lie algebras, and representation theory. Reflections in a real Euclidean space, groups generated by reflections, crystallographic groups, and Coxeter groups. Preference to sophomores majoring in mathematics or the sciences. Prerequisite: MATH 222 or 225.  QR
TTh 1pm-2:15pm

MATH 240b, Advanced Linear Algebra  Stefan Steinerberger

The course is designed to continue discussing various aspects of linear algebra starting at eigenvalues. Materials covered include generalized eigenvalues, the Jordan block decomposition, the Moore-Penrose pseudoinverse, singular values, and the basics of perturbation theory. Other material may be discussed at the instructor's discretion. After MATH 225 or MATH 230/231.
MWF 10:30am-11:20am

MATH 241a / S&DS 241a, Probability Theory  Winston Lin

Introduction to probability theory. Topics include probability spaces, random variables, expectations and probabilities, conditional probability, independence, discrete and continuous distributions, central limit theorem, Markov chains, and probabilistic modeling. After or concurrently with MATH 120 or equivalent.  QR
MW 9am-10:15am

MATH 242b / S&DS 242b, Theory of Statistics  Andrew Barron

Study of the principles of statistical analysis. Topics include maximum likelihood, sampling distributions, estimation, confidence intervals, tests of significance, regression, analysis of variance, and the method of least squares. Some statistical computing. After S&DS 241 and concurrently with or after MATH 222 or 225, or equivalents.  QR
MWF 9:25am-10:15am

MATH 244a or b / AMTH 244a or b, Discrete Mathematics  Staff

Basic concepts and results in discrete mathematics: graphs, trees, connectivity, Ramsey theorem, enumeration, binomial coefficients, Stirling numbers. Properties of finite set systems. Recommended preparation: MATH 115 or equivalent.  QR
HTBA

MATH 246a or b, Ordinary Differential Equations  Staff

First-order equations, second-order equations, linear systems with constant coefficients. Numerical solution methods. Geometric and algebraic properties of differential equations. After MATH 120 or equivalent; after or concurrently with MATH 222 or 225 or equivalent.  QR
HTBA

MATH 250a or b, Vector Analysis  Staff

Calculus of functions of several variables, using vector and matrix methods. The derivative as a linear mapping. Inverse and implicit function theorems. Transformation of multiple integrals. Line and surface integrals of vector fields. Curl and divergence. Differential forms. Theorems of Green and Gauss; general Stokes' theorem. After MATH 120, and 222 or 225 or equivalent.  QR
HTBA

MATH 251b / EENG 434b / S&DS 351b, Stochastic Processes  Amin Karbasi

Introduction to the study of random processes including linear prediction and Kalman filtering, Poison counting process and renewal processes, Markov chains, branching processes, birth-death processes, Markov random fields, martingales, and random walks. Applications chosen from communications, networking, image reconstruction, Bayesian statistics, finance, probabilistic analysis of algorithms, and genetics and evolution. Prerequisite: S&DS 241 or equivalent.  QR
MW 1pm-2:15pm

MATH 270a, Set Theory  Gregg Zuckerman

Algebra of sets; finite, countable, and uncountable sets. Cardinal numbers and cardinal arithmetic. Order types and ordinal numbers. The axiom of choice and the well-ordering theorem. After MATH 120 or equivalent.  QR
MW 1pm-2:15pm

MATH 300b, Topics in Analysis  Staff

An introduction to analysis, with topics chosen from infinite series, the theory of metric spaces, and fixed-point theorems with applications. Students who have taken MATH 230, 231 should take MATH 301 instead of this course. After MATH 250 or with permission of instructor.  QR
MW 11:35am-12:50pm

* MATH 301a, Introduction to Analysis  Peter Jones

Foundations of real analysis, including metric spaces and point set topology, infinite series, and function spaces. After MATH 230, 231 or equivalent.  QR
TTh 1pm-2:15pm

MATH 305b, Real Analysis  Hee Oh

The Lebesgue integral, Fourier series, applications to differential equations. After MATH 301 or with permission of instructor.  QR
MW 1pm-2:15pm

MATH 310a, Introduction to Complex Analysis  Franco Vargas Pallete

An introduction to the theory and applications of functions of a complex variable. Differentiability of complex functions. Complex integration and Cauchy's theorem. Series expansions. Calculus of residues. Conformal mapping. After MATH 231 or 250 or equivalent.  QR
TTh 11:35am-12:50pm

* MATH 315b, Intermediate Complex Analysis  Franco Vargas Pallete

Continuation of MATH 310. Topics may include argument principle, Rouché's theorem, Hurwitz theorem, Runge's theorem, analytic continuation, Schwarz reflection principle, Jensen's formula, infinite products, Weierstrass theorem. Functions of finite order, Hadamard's theorem, meromorphic functions. Mittag-Leffler's theorem, subharmonic functions. After MATH 310.  QR  RP
TTh 2:30pm-3:45pm

* MATH 320a, Measure Theory and Integration  Arie Levit

Construction and limit theorems for measures and integrals on general spaces; product measures; Lp spaces; integral representation of linear functionals. After MATH 305 or equivalent.  QR  RP
TTh 2:30pm-3:45pm

* MATH 325b, Introduction to Functional Analysis  Jeremy Hoskins

Hilbert, normed, and Banach spaces; geometry of Hilbert space, Riesz-Fischer theorem; dual space; Hahn-Banach theorem; Riesz representation theorems; linear operators; Baire category theorem; uniform boundedness, open mapping, and closed graph theorems. After MATH 320.  QR  RP
MW 1pm-2:15pm

MATH 330b / S&DS 400b, Advanced Probability  Sekhar Tatikonda

Measure theoretic probability, conditioning, laws of large numbers, convergence in distribution, characteristic functions, central limit theorems, martingales. Some knowledge of real analysis assumed.  QR
TTh 2:30pm-3:45pm

* MATH 345a, Modern Combinatorics  Mathias Schacht

Recent developments and important questions in combinatorics. Relations to other areas of mathematics such as analysis, probability, and number theory. Topics include probabilistic method, random graphs, random matrices, pseudorandomness in graph theory and number theory, Szemeredi's theorem and lemma, and Green-Tao's theorem. Prerequisite: MATH 244.  QR
TTh 9am-10:15am

MATH 350a, Introduction to Abstract Algebra  Marketa Havlickova

Group theory, structure of Abelian groups, and applications to number theory. Symmetric groups and linear groups including orthogonal and unitary groups; properties of Euclidean and Hermitian spaces. Some examples of group representations. Modules over Euclidean rings, Jordan and rational canonical forms of a linear transformation. After MATH 225, 231, or 222, with additional experience writing mathematical proofs.  QR
MWF 10:30am-11:20am

* MATH 354b, Number Theory  Rong Zhou

Prime numbers; quadratic reciprocity law, Gauss sums; finite fields, equations over finite fields; zeta functions. After MATH 350.  QR
MW 11:35am-12:50pm

MATH 360b, Introduction to Lie Groups  PhilSang Yoo

Lie groups as the embodiment of the idea of continuous symmetry. The exponential map on matrices and applications; spectral theory; examples and structure of Lie groups and Lie algebras; connections with geometry and physics. After MATH 350, 231, or 250. MATH 300 or 301 recommended.  QR
TTh 11:35am-12:50pm

MATH 370b, Fields and Galois Theory  Richard Kenyon

Rings, with emphasis on integral domains and polynomial rings. The theory of fields and Galois theory, including finite fields, solvability of equations by radicals, and the fundamental theorem of algebra. Quadratic forms. After MATH 350.  QR
TTh 9am-10:15am

MATH 380a, Modern Algebra I  Staff

A survey of algebraic constructions and theories at a sophisticated level. Topics include categorical language, free groups and other free objects in categories, general theory of rings and modules, artinian rings, and introduction to homological algebra. After MATH 350 and 370.  QR  RP
TTh 1pm-2:15pm