Introduction to differential and integral calculus for functions of one variable. The heart of calculus is the study of rates of change. Differential calculus concerns the process of finding the rate at which a quantity is changing (the derivative). Integral calculus reverses this process. Information is given about the derivative, and the process of integration finds the "integral," which measures accumulated change. This course aims to develop a thorough understanding of the concepts of differentiation and integration, and covers techniques and applications of differentiation and integration of algebraic, trigonometric, logarithmic, and exponential functions. MATH 115 is an introductory course designed for students who have not seen calculus before.

The course begins with applications and techniques of integration. It probes notions of limit and convergence and adds techniques for finding limits. Half of the course covers infinite sequences and series, where the basic question is, What meaning can we attach to a sum with infinitely many terms and why might we care? The course can help students improve their ability to reason abstractly and also teaches important computational techniques. Topics include integration techniques, l'Hôpital's rule, improper integrals, geometric and other applications of integration, infinite series, power series, and Taylor series. MATH 116 is the appropriate first course for many students who have had AB calculus in high school.

The course begins with applications and techniques of integration. It probes notions of limit and convergence and adds techniques for finding limits. Half of the course covers infinite sequences and series, where the basic question is, What meaning can we attach to a sum with infinitely many terms and why might we care? The course can help students improve their ability to reason abstractly and also teaches important computational techniques. Topics include integration techniques, l'Hôpital's rule, improper integrals, geometric and other applications of integration, infinite series, power series, and Taylor series. MATH 116 is the appropriate first course for many students who have had AB calculus in high school.

Most real-world systems that one may want to model, whether in the natural or in the social sciences, have many interdependent parameters. To apply calculus to these systems, we need to extend the ideas and techniques of single-variable Calculus to functions of more than one variable. Topics include vectors, matrices, determinants, polar, cylindrical, and spherical coordinates, curves, partial derivatives, gradients and directional derivatives, Lagrange multipliers, multiple integrals, vector calculus: line integrals, surface integrals, divergence, curl, Green's Theorem, Divergence Theorem, and Stokes’ Theorem.

Linear algebra is one of the most beautiful subjects in the undergraduate mathematics curriculum. It is also one of the most important with many possible applications. In this course, students learn computational techniques that have widespread applications in the natural and social sciences as well as in industry, finance, and management. There is also a focus on learning how to understand and write mathematical proofs and an emphasis on improving mathematical style and sophistication. Topics include vector spaces, subspaces, linear independence, bases, dimension, inner products, linear transformations, matrix representations, range and null spaces, inverses, and eigenvalues.

Linear algebra is one of the most beautiful subjects in the undergraduate mathematics curriculum. It is also one of the most important with many possible applications. In this course, students learn computational techniques that have widespread applications in the natural and social sciences as well as in industry, finance, and management. There is also a focus on learning how to understand and write mathematical proofs and an emphasis on improving mathematical style and sophistication. Topics include vector spaces, subspaces, linear independence, bases, dimension, inner products, linear transformations, matrix representations, range and null spaces, inverses, and eigenvalues.

This course is designed to examine the degree to which a function can be determined by an algebraic relationship it has with its derivative(s) --- a so-called ordinary differential equation (ODE). For instance, can one completely catalog all functions which equal their own derivative? In service of developing techniques for solving certain classes of differential equations, some fundamental notions from linear algebra and complex numbers are presented.  Differential equation topics include modeling with and solving first- and second-order ODEs, separable ODEs, and a discussion of higher order and non-linear ODEs; linear algebra topics include solving systems via elementary row operations, bases, dimension, determinants, column space, and eigenvalues/vectors.

Number theory is the study of the most basic mathematical objects: the natural numbers (1, 2, 3, etc.). It begins by investigating simple patterns: for instance, which numbers can be written as sums of two squares? Do the primes go on forever? How can we be sure? The patterns and structures that emerge from studying the properties of numbers are so elegant, complex, and important that number theory has been called "the Queen of Mathematics." Once studied only for its intrinsic beauty, number theory has practical applications in cryptography and computer science. Topics include the Euclidean algorithm, modular arithmetic, Fermat's and Euler's Theorems, public-key cryptography, quadratic reciprocity. MATH 223 has a focus on learning to understand and write mathematical proofs; it can serve as valuable preparation for MATH 305.

Combinatorics is the art of counting possibilities: for instance, how many different ways are there to distribute 20 apples to 10 kids? Graph theory is the study of connected networks of objects. Both have important applications to many areas of mathematics and computer science. The course will be taught emphasizing creative problem-solving as well as methods of proof, such as proof by contradiction and induction. Topics include: selections and arrangements, generating functions, recurrence relations, graph coloring, Hamiltonian and Eulerian circuits, and trees.

This course is a continuation of MATH 302 with a treatment of real analysis focusing on the theory of integration. We will see why Riemann integration is supplanted by Lebesgue integration, and how different metrics on function space yield different topological structures. Topics in this course will include the convergence of series of functions, summation tests, the Monotone Convergence Theorem, the Dominated Convergence Theorem, and Fatou's Lemma. If there is time, we will also study the theory of Fourier transforms.

Graph Theory has origins both in recreational mathematics problems (i.e., puzzles and games) and as a tool to solve practical problems in many areas of society. Topics covered will include trees and distance, connectivity and paths, network flow, graph coloring, directed graphs, and tournaments. In addition, students will gain a sense of what it means to do research in graph theory.

Topic for Spring 2025: Introduction to Cryptography

The goal of this course is to use modern public-key cryptography as a vehicle for learning various important concepts in advanced mathematics. Topics will include Diffie-Hellman key exchange, RSA cryptosystem, NTRU cryptosystem, elliptic curve cryptography, discrete logs, DES and AES, digital signatures, hash functions, error correcting codes and quantum cryptography. To understand these ideas, we will need to study ring theory, probability, number theory over a finite field, elliptic curves, linear algebra of lattices, and  NP-Completeness. There will also be a computational component to the course - encryption, decryption, and attacks on cryptosystems using a computer.

Introduction to differential and integral calculus for functions of one variable. The heart of calculus is the study of rates of change. Differential calculus concerns the process of finding the rate at which a quantity is changing (the derivative). Integral calculus reverses this process. Information is given about the derivative, and the process of integration finds the "integral," which measures accumulated change. This course aims to develop a thorough understanding of the concepts of differentiation and integration, and covers techniques and applications of differentiation and integration of algebraic, trigonometric, logarithmic, and exponential functions. MATH 115 is an introductory course designed for students who have not seen calculus before.

Introduction to differential and integral calculus for functions of one variable. The heart of calculus is the study of rates of change. Differential calculus concerns the process of finding the rate at which a quantity is changing (the derivative). Integral calculus reverses this process. Information is given about the derivative, and the process of integration finds the "integral," which measures accumulated change. This course aims to develop a thorough understanding of the concepts of differentiation and integration, and covers techniques and applications of differentiation and integration of algebraic, trigonometric, logarithmic, and exponential functions. MATH 115 is an introductory course designed for students who have not seen calculus before.

The course begins with applications and techniques of integration. It probes notions of limit and convergence and adds techniques for finding limits. Half of the course covers infinite sequences and series, where the basic question is, What meaning can we attach to a sum with infinitely many terms and why might we care? The course can help students improve their ability to reason abstractly and also teaches important computational techniques. Topics include integration techniques, l'Hôpital's rule, improper integrals, geometric and other applications of integration, infinite series, power series, and Taylor series. MATH 116 is the appropriate first course for many students who have had AB calculus in high school.

The course begins with applications and techniques of integration. It probes notions of limit and convergence and adds techniques for finding limits. Half of the course covers infinite sequences and series, where the basic question is, What meaning can we attach to a sum with infinitely many terms and why might we care? The course can help students improve their ability to reason abstractly and also teaches important computational techniques. Topics include integration techniques, l'Hôpital's rule, improper integrals, geometric and other applications of integration, infinite series, power series, and Taylor series. MATH 116 is the appropriate first course for many students who have had AB calculus in high school.

The course begins with applications and techniques of integration. It probes notions of limit and convergence and adds techniques for finding limits. Half of the course covers infinite sequences and series, where the basic question is, What meaning can we attach to a sum with infinitely many terms and why might we care? The course can help students improve their ability to reason abstractly and also teaches important computational techniques. Topics include integration techniques, l'Hôpital's rule, improper integrals, geometric and other applications of integration, infinite series, power series, and Taylor series. MATH 116 is the appropriate first course for many students who have had AB calculus in high school.

How can a candidate in a political race win the majority of votes yet lose the election? How can two competing candidates interpret the same statistic as being in their favor? How can the geometry of the voting district disenfranchise entire groups of voters? Can we quantify the power the President of the United States has? In this course, we will look at the mathematics behind these and related questions that arise in politics. We will study topics such as fairness, voting paradoxes, social choice, game theory, apportionment, gerrymandering, and data interpretation. The goal of the class will be to illustrate the importance of rigorous reasoning in various social and political processes while providing an introduction to some fascinating mathematics.

Most real-world systems that one may want to model, whether in the natural or in the social sciences, have many interdependent parameters. To apply calculus to these systems, we need to extend the ideas and techniques of single-variable Calculus to functions of more than one variable. Topics include vectors, matrices, determinants, polar, cylindrical, and spherical coordinates, curves, partial derivatives, gradients and directional derivatives, Lagrange multipliers, multiple integrals, vector calculus: line integrals, surface integrals, divergence, curl, Green's Theorem, Divergence Theorem, and Stokes’ Theorem.

Most real-world systems that one may want to model, whether in the natural or in the social sciences, have many interdependent parameters. To apply calculus to these systems, we need to extend the ideas and techniques of single-variable Calculus to functions of more than one variable. Topics include vectors, matrices, determinants, polar, cylindrical, and spherical coordinates, curves, partial derivatives, gradients and directional derivatives, Lagrange multipliers, multiple integrals, vector calculus: line integrals, surface integrals, divergence, curl, Green's Theorem, Divergence Theorem, and Stokes’ Theorem.

Linear algebra is one of the most beautiful subjects in the undergraduate mathematics curriculum. It is also one of the most important with many possible applications. In this course, students learn computational techniques that have widespread applications in the natural and social sciences as well as in industry, finance, and management. There is also a focus on learning how to understand and write mathematical proofs and an emphasis on improving mathematical style and sophistication. Topics include vector spaces, subspaces, linear independence, bases, dimension, inner products, linear transformations, matrix representations, range and null spaces, inverses, and eigenvalues.

This course is designed to examine the degree to which a function can be determined by an algebraic relationship it has with its derivative(s) --- a so-called ordinary differential equation (ODE). For instance, can one completely catalog all functions which equal their own derivative? In service of developing techniques for solving certain classes of differential equations, some fundamental notions from linear algebra and complex numbers are presented.  Differential equation topics include modeling with and solving first- and second-order ODEs, separable ODEs, and a discussion of higher order and non-linear ODEs; linear algebra topics include solving systems via elementary row operations, bases, dimension, determinants, column space, and eigenvalues/vectors.

Combinatorics is the art of counting possibilities: for instance, how many different ways are there to distribute 20 apples to 10 kids? Graph theory is the study of connected networks of objects. Both have important applications to many areas of mathematics and computer science. The course will be taught emphasizing creative problem-solving as well as methods of proof, such as proof by contradiction and induction. Topics include: selections and arrangements, generating functions, recurrence relations, graph coloring, Hamiltonian and Eulerian circuits, and trees.

Real analysis is the study of the rigorous theory of the real numbers, Euclidean space, and calculus. The goal is to thoroughly understand the familiar concepts of continuity, limits, and sequences. Topics include compactness, completeness, and connectedness; continuous functions; differentiation and integration; limits and sequences; and interchange of limit operations as time permits.

This course covers some basic notions of point-set topology, such as topological spaces, metric spaces, connectedness and compactness, Heine-Borel Theorem, quotient spaces, topological groups, groups acting on spaces, homotopy equivalences, separation axioms, Euler characteristic, and classification of surfaces. Additional topics include the study of the fundamental group (time permitting).

Linear algebra at this more advanced level is a basic tool in many areas of mathematics and other fields. The course begins by revisiting some linear algebra concepts from MATH 206 in a more sophisticated way, making use of the mathematical maturity picked up in MATH 305. Such topics include vector spaces, linear independence, bases, and dimensions, linear transformations, and inner product spaces. Then we will turn to new notions, including dual spaces, reflexivity, annihilators, direct sums and quotients, tensor products, multilinear forms, and modules. One of the main goals of the course is the derivation of canonical forms, including triangular form and Jordan canonical forms. These are methods of analyzing matrices that are more general and powerful than diagonalization (studied in MATH 206). We will also discuss the spectral theorem, the best example of successful diagonalization, and its applications.

How can a candidate in a political race win the majority of votes yet lose the election? How can two competing candidates interpret the same statistic as being in their favor? How can the geometry of the voting district disenfranchise entire groups of voters? Can we quantify the power the President of the United States has? In this course, we will look at the mathematics behind these and related questions that arise in politics. We will study topics such as fairness, voting paradoxes, social choice, game theory, apportionment, gerrymandering, and data interpretation. The goal of the class will be to illustrate the importance of rigorous reasoning in various social and political processes while providing an introduction to some fascinating mathematics.

This course is designed to examine the degree to which a function can be determined by an algebraic relationship it has with its derivative(s) --- a so-called ordinary differential equation (ODE). For instance, can one completely catalog all functions which equal their own derivative? In service of developing techniques for solving certain classes of differential equations, some fundamental notions from linear algebra and complex numbers are presented.  Differential equation topics include modeling with and solving first- and second-order ODEs, separable ODEs, and a discussion of higher order and non-linear ODEs; linear algebra topics include solving systems via elementary row operations, bases, dimension, determinants, column space, and eigenvalues/vectors.

Introduction to differential and integral calculus for functions of one variable. The heart of calculus is the study of rates of change. Differential calculus concerns the process of finding the rate at which a quantity is changing (the derivative). Integral calculus reverses this process. Information is given about the derivative, and the process of integration finds the "integral," which measures accumulated change. This course aims to develop a thorough understanding of the concepts of differentiation and integration, and covers techniques and applications of differentiation and integration of algebraic, trigonometric, logarithmic, and exponential functions. MATH 115 is an introductory course designed for students who have not seen calculus before.

The course begins with applications and techniques of integration. It probes notions of limit and convergence and adds techniques for finding limits. Half of the course covers infinite sequences and series, where the basic question is, What meaning can we attach to a sum with infinitely many terms and why might we care? The course can help students improve their ability to reason abstractly and also teaches important computational techniques. Topics include integration techniques, l'Hôpital's rule, improper integrals, geometric and other applications of integration, infinite series, power series, and Taylor series. MATH 116 is the appropriate first course for many students who have had AB calculus in high school.

[Registration as of 1/22/2024] The Workday waitlist for this section has been changed to an instructor-maintained waitlist. To add yourself to the waitlist send email to the instructor.

The course begins with applications and techniques of integration. It probes notions of limit and convergence and adds techniques for finding limits. Half of the course covers infinite sequences and series, where the basic question is, What meaning can we attach to a sum with infinitely many terms and why might we care? The course can help students improve their ability to reason abstractly and also teaches important computational techniques. Topics include integration techniques, l'Hôpital's rule, improper integrals, geometric and other applications of integration, infinite series, power series, and Taylor series. MATH 116 is the appropriate first course for many students who have had AB calculus in high school.

Most real-world systems that one may want to model, whether in the natural or in the social sciences, have many interdependent parameters. To apply calculus to these systems, we need to extend the ideas and techniques of single-variable Calculus to functions of more than one variable. Topics include vectors, matrices, determinants, polar, cylindrical, and spherical coordinates, curves, partial derivatives, gradients and directional derivatives, Lagrange multipliers, multiple integrals, vector calculus: line integrals, surface integrals, divergence, curl, Green's Theorem, Divergence Theorem, and Stokes’ Theorem.

Most real-world systems that one may want to model, whether in the natural or in the social sciences, have many interdependent parameters. To apply calculus to these systems, we need to extend the ideas and techniques of single-variable Calculus to functions of more than one variable. Topics include vectors, matrices, determinants, polar, cylindrical, and spherical coordinates, curves, partial derivatives, gradients and directional derivatives, Lagrange multipliers, multiple integrals, vector calculus: line integrals, surface integrals, divergence, curl, Green's Theorem, Divergence Theorem, and Stokes’ Theorem.

Linear algebra is one of the most beautiful subjects in the undergraduate mathematics curriculum. It is also one of the most important with many possible applications. In this course, students learn computational techniques that have widespread applications in the natural and social sciences as well as in industry, finance, and management. There is also a focus on learning how to understand and write mathematical proofs and an emphasis on improving mathematical style and sophistication. Topics include vector spaces, subspaces, linear independence, bases, dimension, inner products, linear transformations, matrix representations, range and null spaces, inverses, and eigenvalues.

This course is designed to examine the degree to which a function can be determined by an algebraic relationship it has with its derivative(s) --- a so-called ordinary differential equation (ODE). For instance, can one completely catalog all functions which equal their own derivative? In service of developing techniques for solving certain classes of differential equations, some fundamental notions from linear algebra and complex numbers are presented.  Differential equation topics include modeling with and solving first- and second-order ODEs, separable ODEs, and a discussion of higher order and non-linear ODEs; linear algebra topics include solving systems via elementary row operations, bases, dimension, determinants, column space, and eigenvalues/vectors.

Number theory is the study of the most basic mathematical objects: the natural numbers (1, 2, 3, etc.). It begins by investigating simple patterns: for instance, which numbers can be written as sums of two squares? Do the primes go on forever? How can we be sure? The patterns and structures that emerge from studying the properties of numbers are so elegant, complex, and important that number theory has been called "the Queen of Mathematics." Once studied only for its intrinsic beauty, number theory has practical applications in cryptography and computer science. Topics include the Euclidean algorithm, modular arithmetic, Fermat's and Euler's Theorems, public-key cryptography, quadratic reciprocity. MATH 223 has a focus on learning to understand and write mathematical proofs; it can serve as valuable preparation for MATH 305.

[Registration as of 1/22/2024] The Workday waitlist for this section has been changed to an instructor-maintained waitlist. To add yourself to the waitlist send email to the instructor.

Combinatorics is the art of counting possibilities: for instance, how many different ways are there to distribute 20 apples to 10 kids? Graph theory is the study of connected networks of objects. Both have important applications to many areas of mathematics and computer science. The course will be taught emphasizing creative problem-solving as well as methods of proof, such as proof by contradiction and induction. Topics include: selections and arrangements, generating functions, recurrence relations, graph coloring, Hamiltonian and Eulerian circuits, and trees.

Topic for 2023-2024: PDEs and Geometric Analysis

This course is an introduction to Geometric Analysis through the study of partial differential equations (PDEs).  The focus is on studying the properties of solutions.  Topics covered include: the Laplace and Heat equation, maximum principles, gradient flows, and the role of non-smoothness in PDEs.  In the second half of the course we study geometric PDEs such as the minimal surface equation and curve shortening flow in the plane.

Topic for Spring 24: Galois Theory

This course offers a continued study of the algebraic structures introduced in MATH 305, culminating in the Fundamental Theorem of Galois Theory, a beautiful result that depicts the circle of ideas surrounding field extensions, polynomial rings, and automorphism groups. Applications of Galois theory include the unsolvability of the quintic by radicals and geometric impossibility proofs, such as the trisection of angles and duplication of cubes. Cyclotomic extensions and Sylow theory may be included in the syllabus.

[Registration as of 1/22/2024] The Workday waitlist for this section has been changed to an instructor-maintained waitlist. To add yourself to the waitlist send email to the instructor.

Topic for Spring 2024: Knot Theory

Topology deals with the properties of an object that no amount of bending, twisting, stretching, or shrinking can change (unlike geometry, which deals with the rigid properties of objects, such as length and angles).  Take a piece of string, tie a knot in it, and glue the ends together. The result is a knot. Knot theory is a branch of topology that deals with knots and links in three-dimensional space. Given a knot, how do you decide if it can be untangled? Given two knots, how do you decide if one can be made to look like the other, using only bending, twisting, stretching, or shrinking? The study of knots is over 100 years old, and some of the most exciting results have occurred in the last ten years. Knot theory has evolved from an area in "pure" mathematics to include applications in molecular biology, chemistry, fluid dynamics, and quantum mechanics. This course is an introduction to the theory of knots. Among other topics, we will cover methods of knot tabulation, surfaces applied to knots, polynomials associated to knots, and applications of knot theory.