Geometry and Relativity
(Offered as PHYS 102 and MATH 102) On January 27th, 1921, Albert Einstein gave a lecture titled “Geometry and Experience" at the Prussian Academy of Science. In this lecture he reflects on the interdependence of geometry and physics. To commemorate the centenary of such an inspiring event, this course will explore the natural connections between geometry (axioms, the notions of space and time, dimension and curvature) and relativity (the relativity principle, simultaneity, thought experiments). No background in physics or mathematics (besides basic high school algebra and trigonometry) will be assumed. The course is designed for students who do not intend to major in mathematics or physics. Omitted 2022-23. Professor Jagannathan.2023-24: Not offered
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 an 80-minute 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. The Department.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
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.
Requisite: MATH 105. Spring semester. Professor D. Benedetto.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
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.
Requisite: Math Placement into 111, or consent of the Department. 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 includes one additional weekly class meeting. 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 2022, Spring 2023, Fall 2023
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.
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.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
Introduction to Statistics via Modeling
(Offered as STAT 135 and MATH 135) This course 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, 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. (Students who have taken STAT 111, STAT136, or PSYC 122 may not also receive credit for STAT/MATH 135. Students who have taken ECON 360/361 will be admitted only with consent of the instructor.) No prior experience with statistical software is expected.
Student has completed or is in process of completing MATH 111 or has placement in MATH 121 or above, or has statistics placement of STAT135 or has consent of the instructor. Limited to 24 students per section. Fall and spring semesters. The Department.Other years: Offered in Spring 2014, Spring 2015, Fall 2015, Spring 2016, Fall 2022, Spring 2023, Fall 2023
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.
Limited to 24 students. Spring semester. Professor Moore.Other years: Offered in Fall 2012, Fall 2014, Spring 2016, Fall 2017, Spring 2019, Spring 2021, Spring 2022, Spring 2023
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. Omitted 2023-24.2023-24: Not offered
Elementary vector calculus; introduction to partial derivatives; multiple integrals in two and three dimensions; line integrals in the plane; Green’s theorem.
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.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
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.
Limited to 25 students. Spring and fall semesters. The Department.Other years: Offered in Spring 2012, Spring 2013, Spring 2014, Spring 2015, Fall 2015, Spring 2016, Spring 2017, Fall 2022, Spring 2023, Fall 2023
This course 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, study the concept of fractal dimension among other theoretical concepts, and examine Julia and Mandelbrot sets (time permitting). Through the teaching of these concepts, the course 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 25 students. Professor Folsom.Other years: Offered in Spring 2023
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.Other years: Offered in Fall 2012, Spring 2014, Spring 2015, Spring 2016, Spring 2023
Mathematics of Public-Key Cryptography
Many security problems arise when two computers must communicate on a channel with eavesdroppers or malicious attackers. Public-key cryptography applies ideas from number theory and abstract algebra to address these problems. This course concerns the mathematical theory and algorithms needed to construct the most commonly-used public-key ciphers and digital signature schemes, as well as the attacks that must be anticipated when designing such systems. Several topics from number theory, abstract algebra, and algorithms will be introduced, including discrete logarithms, integer factorization algorithms, and elliptic curves. Depending on time and student interest, we may cover some newer systems that are believed to be secure against attacks by quantum computers but not yet commonly implemented in practice. Students will write short programs to implement the systems and to break badly implemented systems. No prior programming experience is expected; basic aspects of programming in Python will be taught in class. Four class hours per week.
Requisite: Experience writing proofs, such as MATH 220/221 or 271/272, or consent of the instructor. Spring semester. Professor Pflueger.2023-24: Not offered
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.
Requisite: MATH 121. Fall semester.2023-24: Not offered
Combinatorial Geometry: Packings and Polytopes
This course is a survey of geometry in dimensions 2, 3, 4, and higher. We will consider questions such as: How do we know the angles of a triangle add up to 180 degrees? (Spoiler: usually they don't.) What are the different ways we could tile our kitchen floor? How many tennis balls fit in a bucket? How many regular polyhedra are there in four dimensions? And what shape is the universe? We will examine how mathematicians and scientists, from Euclid to Einstein, have tried to give definitive answers to these questions, and we will explore some of what is still unknown about how the mathematical world reflects what we see around us.
Requisite: MATH 211 or instructor permission. Limited to 25 students. Fall semester. Professor Ching.Other years: Offered in Fall 2023
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.
Requisite: MATH 211 or consent of the instructor. Limited to 25 students. Fall semester.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
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.
Requisite: MATH 211 or 220, or consent of the instructor. This course and MATH 272 may not both be taken for credit. Limited to 25 students. Fall and spring semesters. The Department.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
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. This course will feature both proofs and applications, with special attention paid to applied topics such as least squares and singular value decomposition.
Requisite: MATH 211 or 220, or consent of the instructor. This course and MATH 271 may not both be taken for credit. Limited to 25 students. Fall and Spring semester: The Department.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
A Second Course in Linear Algebra
This course is a continuation of the material in MATH 271 and 272, providing more insight into abstract vector spaces and operator theory. Topics may include least squares estimates, singular value decompositions, Jordan canonical forms, inner product spaces, linear functionals and duals, orthogonal polynomials, vector and matrix norms, the spectral theorem, eigenvalue inequalities, and error-correcting codes. Time permitting, applications to graph theory and discrete dynamical systems may be explored. Four class hours per week.
Requisites: MATH 271, MATH 272, or consent of the instructor. Spring semester. Limited to 25 students.2023-24: Not offered
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.
Requisite: MATH 271 or 272 or consent of the instructor. MATH 220 or other prior experience with mathematical proofs is recommended. Limited to 25 students. Fall semester. Professor Contreras.Other years: Offered in Fall 2022, Fall 2023
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. Additional topics may vary.
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 consent of the instructor. Limited to 24 students. Spring semester.
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. Possible topics include 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.Other years: Offered in Fall 2022
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 121 and 220, or other significant experience with proofs, or by consent of the instructor. Limited to 24 students. Omitted 2021-22.2023-24: Not offered
Computational Algebraic Geometry
The central object of study in algebraic geometry is a system of polynomial equations and its solution set. The theory has connections to many other areas in pure mathematics including number theory, representation theory, several complex variables, and combinatorics. Applied algebraic geometry is used in areas including mathematical biology, algebraic statistics, robotics, and computer vision. In this course, we will study algebraic geometry from a computational point of view, beginning with Buchberger's algorithm for computing Groebner bases, and working towards the ideal-variety correspondence. Topics may include projective algebraic geometry, toric varieties, and applications, depending on student interest.
Requisite: MATH 271 or 272, and experience with writing mathematical proofs (such as in MATH 220); or permission of the instructor.Limited to 25 students. Fall semester. The Department.
Pending Faculty Approval2023-24: Not offered
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. Omitted 2021-22.2023-24: Not offered
Calculus of Variations
What is the shape of a hanging chain? What shape of fixed perimeter encloses the most area? The calculus of variations answers questions such as these through maximizing or minimizing values of functionals over different input functions. Topics covered will include functional differentiation, the Euler-Lagrange equations, necessary and sufficient conditions for extrema, and direct minimization methods. Additionally, there will be applications to physics and optimization, including Hamiltonian mechanics and image processing.
Fall semester. Professor KraislerOther years: Offered in Fall 2023
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; special functions.
Requisite: MATH 211 and prior experience with mathematical proofs, or consent of the instructor. Limited to 25 students. Fall semester.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
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.
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.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
Introduction to Analysis
Completeness of the real numbers; topology of n-space including the Bolzano-Weierstrass and Heine-Borel theorems; sequences, properties of continuous functions on sets; infinite series, uniform convergence.
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. The Department.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
(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.
Student has completed or is in the process of completing MATH 121, or has MATH 211 placement, or has consent of the instructor. Limited to 24 students. Fall semester. Professor Horton.Other years: Offered in Fall 2011, Fall 2012, Fall 2013, Fall 2014, Fall 2015, Fall 2016, Fall 2017, Fall 2018, Fall 2019, Fall 2020, Fall 2021, Fall 2022, Fall 2023
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 the instructor. Limited to 24 students. Professor Leise.2023-24: Not offered
(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.
Student has completed or is in the process of completing: STAT 111 or MATH/STAT 135 or STAT136 or has STAT 230 placement, and STAT/MATH 360, or has consent of the instructor. Limited to 25 students. Spring semester. Professor Donges.Other years: Offered in Spring 2023
Real Analysis in Higher Dimensions
This course builds upon the material in MATH 355 (Introduction to Analysis) in order to rigorously develop basic tools for studying functions of more than one real variable. While the setting in MATH 355 is the real number line, the context for this course will be the n-dimensional Euclidean space. Many facets of analysis on this n-dimensional space will be explored including its topological properties as well as differentiation and Riemann integration in n-variables. The course will cover fundamental results such as the celebrated implicit and inverse function theorems. Time permitting, we may discuss additional topics such as analysis on metric spaces. Four class hours per week.
Requisite: MATH 355 and either MATH 271 or 272; or consent of the instructor. Limited to 25 students. Omitted 2021-22.2023-24: Not offered
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.2023-24: Not offered
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.Other years: Offered in Spring 2015, Spring 2018, Fall 2019, Spring 2022
Independent reading course.
Fall 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
Lie Groups and Lie Algebras
Lie groups and Lie algebras appear naturally in the study of symmetries of geometric objects. Lie algebras carry local information and can be studied using tools from linear algebra. Finite dimensional Lie groups can be constructed using techniques from calculus and group theory.
This course serves as a first introduction to Lie groups and Lie algebras. We will examine the structure of finite dimensional matrix Lie groups, the exponential and differentiation maps, as well as compact Lie groups. We will also study Lie algebras, ideals and homomorphisms, nilpotent and solvable Lie algebras, Cartan subalgebras, semisimplicity and 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. Four class meetings per week.
Requisite: MATH 350 or consent of the instructor. Fall semester.2023-24: Not offered
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. Prof. Daniels.2023-24: Not offered
Introduction to Commutative Algebra
Commutative algebra is known as the study of commutative rings and their ideals and modules. Besides being an important branch of algebra for its own sake, commutative algebra has strong ties to other areas, such as algebraic geometry and algebraic number theory, as it provides essential tools for them. This course is an introductory course in commutative algebra. We will explore more about rings (especially polynomial rings) and ideals, which are taught in Math 350. We will also introduce another important algebraic structure, namely modules over rings. Other fundamental topics include Noetherian rings, The Hilbert Basis Theorem, Gröbner bases, localization, primary decompositions, and tensor products.
Requisite: Math 350 or consent of the instructor. Limited to 24 students. Omitted 2023-24. Visiting Assistant Professor Gunturkun.2023-24: Not offered
The Arithmetic of Elliptic Curves
An elliptic curve is the set of zeros of a cubic polynomial in two variables. If the polynomial has rational coefficients, it is natural to ask for a description of those zeros whose coordinates are either integers or rational numbers. Our study of elliptic curves will focus on this fundamental problem and reveal a fascinating interplay between algebra, geometry, analysis and number theory. Topics discussed will include the geometry and group structure of elliptic curves, the Nagell-Lutz Theorem describing points of finite order, and the Mordell-Weil theorem on the finite generation of the group of rational points. Additional topics may include elliptic curve cryptography, Lenstra's algorithm using elliptic curves to factor large integers, the Thue-Siegel Theorem on the finiteness of the set of integer points, and the crucial role the theory of elliptic curves played in Wiles' proof of Fermat's Last Theorem. By bringing together techniques from a wide range of mathematical disciplines, we plan to illustrate the unity of mathematics and introduce active areas of research. Four class hours per week.
Requisite: MATH 350 or consent of the instructor. Omitted 2021-22.2023-24: Not offered
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.Other years: Offered in Spring 2012, Spring 2014, Spring 2016
This course will explore the geometry of curves, surfaces and higher dimensional geometric objects 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) two-dimensional Riemmanian geometry, differential forms and manifolds. MATH 378 and MATH 408 may not both be taken for credit.
Spring semester. Professor Contreras
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. Prof. Rasheed.2023-24: Not offered
Analytic Number Theory
This course 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. 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: MATH 355. Limited to 25 students. Fall semester. Professor Folsom.Other years: Offered in Fall 2023
Senior Departmental Honors
Open 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