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 Cox.
Other years: Offered in Fall 2011, Fall 2012, Fall 2013, Fall 2014, Fall 2015, Fall 2017, Fall 2018, Fall 2019, Fall 2020, Fall 2021, Fall 2022, Fall 2023, Fall 2024MATH 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 Cox.
Other years: Offered in Spring 2012, Spring 2013, Spring 2014, Spring 2015, Spring 2016, Spring 2017, Spring 2018, Spring 2019, Spring 2020, Spring 2021, Spring 2022, Spring 2023, Spring 2025Basic 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.
Other years: Offered in Fall 2011, Spring 2012, Fall 2012, Spring 2013, Fall 2013, Spring 2014, Fall 2014, Spring 2015, Fall 2015, Spring 2016, Fall 2022, Spring 2023, Fall 2023, Fall 2024, Spring 2025A 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.
Other years: Offered in Fall 2011, Spring 2012, Fall 2012, Spring 2013, Fall 2013, Spring 2014, Fall 2014, Spring 2015, Fall 2015, Spring 2016, Fall 2022, Spring 2023, Fall 2023, Fall 2024, Spring 2025Mathematical 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. Fall semester. Professor Leise.
Other years: Offered in Fall 2012, Fall 2014, Spring 2016, Fall 2017, Spring 2019, Spring 2021, Spring 2022, Spring 2023, Spring 2025Elementary 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.
Other years: Offered in Fall 2011, Spring 2012, Fall 2012, Spring 2013, Fall 2013, Spring 2014, Fall 2014, Spring 2015, Fall 2015, Spring 2016, Fall 2022, Spring 2023, Fall 2023, Fall 2024, Spring 2025This 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.
Spring semester. Professor R. Benedetto.
Other years: Offered in Spring 2012, Spring 2013, Spring 2014, Spring 2015, Fall 2015, Spring 2016, Spring 2017, Fall 2022, Spring 2023, Fall 2023, Fall 2024, Spring 2025MATH 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.
Requesite: MATH 211 or consent of the instructor. Limited to 35 students. Fall semester. Professor Folsom.
Other years: Offered in Spring 2023, Spring 2025With 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 instructor. Limited to 24 students. Spring semester. Professor Dresch.
2023-24: Not offeredAn 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 Call.
Other years: Offered in Fall 2012, Spring 2014, Spring 2015, Spring 2016, Spring 2023, Spring 2025The 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 D'Ambroise.
Other years: Offered in Spring 2013, Spring 2014, Spring 2015, Spring 2016, Spring 2017, Spring 2018, Spring 2019, Spring 2020, Spring 2021, Spring 2022, Spring 2023, Fall 2023, Fall 2024, Spring 2025The 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 semester. Professors R.Benedetto and Dresch.
Other years: Offered in Fall 2011, Fall 2012, Fall 2013, Fall 2014, Fall 2015, Spring 2016, Spring 2017, Fall 2017, Fall 2022, Spring 2023, Fall 2023, Fall 2024, Spring 2025The 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. Professor D'Ambroise and Dresch.
Other years: Offered in Spring 2012, Spring 2013, Spring 2014, Spring 2015, Spring 2016, Spring 2017, Fall 2017, Spring 2018, Fall 2018, Spring 2019, Fall 2019, Spring 2020, Spring 2021, Fall 2021, Spring 2022, Fall 2022, Spring 2023, Fall 2023, Fall 2024, Spring 2025The 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. Omitted 2014-15.
2023-24: Not offeredAn 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. Fall semester. Professor D'Ambroise.
Other years: Offered in Fall 2011, Fall 2012, Fall 2013, Fall 2014, Fall 2015, Fall 2017, Fall 2018, Fall 2019, Fall 2020, Fall 2021, Fall 2022, Fall 2023, Fall 2024A 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. Fall semester. Professor Daniels. Spring semester. Professor Folsom.
Other years: Offered in Spring 2012, Spring 2013, Spring 2014, Fall 2014, Spring 2015, Fall 2015, Spring 2016, Spring 2017, Fall 2017, Fall 2022, Spring 2023, Fall 2023, Fall 2024, Spring 2025Completeness 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. Fall semester. Professor Velleman. Spring semester: Professor Ching.
Other years: Offered in Spring 2012, Spring 2013, Fall 2013, Spring 2014, Fall 2014, Spring 2015, Fall 2015, Spring 2016, Spring 2017, Fall 2017, Fall 2022, Spring 2023, Fall 2023, Fall 2024, Spring 2025A 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. Omitted 2014-15.
2023-24: Not offeredMost 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. Omitted 2014-15.
2023-24: Not offeredMathematicians 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 Velleman.
Other years: Offered in Spring 2015, Spring 2018, Fall 2019, Spring 2022Fall and spring semesters. The Department.
Other years: Offered in 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 2017, Spring 2018, Fall 2018, Spring 2019, Fall 2019, Spring 2020, Fall 2020, Spring 2021, Fall 2021, Spring 2022, Fall 2022, Spring 2023, Fall 2023, Fall 2024, Spring 2025The 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. Omitted 2014-15.
2023-24: Not offeredThe topic will vary from year to year. The topic for fall 2014 is 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. Fall semester. Professor Cox.
2023-24: Not offeredAn 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.
2023-24: Not offeredOpen to seniors with the consent of the Department. Fall semester. The Department.
Other years: Offered in Spring 2012, Spring 2013, Spring 2014, Spring 2015, Spring 2016, Spring 2017, Spring 2018, Spring 2019, Spring 2020, Spring 2021, Spring 2022, Spring 2023, Spring 2025