Teaching
This course is concerned with the theoretical underpinnings of the set of real numbers as well as functions defined on that set. Throughout the course we will place emphasis on rigorous proof, developing intuition, as well as building a bank of (counter)examples of the various properties we will encounter.
We will begin by exploring the axiom of completeness of the real numbers, and follow this up by studying the concept of a sequence of real numbers, as well as the all-important definition of a limit of a sequence. Then we will turn our attention to some topological aspects of the real line, and use that language to help us understand limits of functions, continuity, and the Intermediate Value Theorem. We will then define the notion of a derivative, and prove the Mean Value Theorem, which is an extremely useful theorem in the study of differentiable functions. We will then study sequences and series of functions, as well as study new notions of convergence in this context. Finally, we will wrap up the semester looking at power series and Taylor series as an application of the previous topic.
In first semester Calculus, one studies functions of a single (real) variable, and then proceeds to study limits, continuity, derivatives (and their applications) and finally the definite integral and the Fundamental Theorem of Calculus. In Calculus II, one learns about u-substitution, and also about Taylor series which may be used to represent many of the functions that are met along the way.
It turns out that each of these topics has a direct parallel in higher dimensions. In this class, we will focus on the case of real-valued functions of two and three variables. Along the way, we will also encounter some new topics that are unique to the higher-dimensional setting.
The purpose of this course is twofold. First, we will develop the necessary tools to study linear systems of equations. Such equations are called linear since we don’t multiply the variables by each other – all we are allowed to do is multiply them by constants and compare them with constants. Despite their simplicity, such equations arise in a plethora of applications in science and engineering.
While studying such systems, one is naturally led to introduce the concept of vector space and linear transformation. An important example of linear transformations are functions defined in terms of differentiation. We call a system of equations involving some functions and one (or more) of their derivatives a system of differential equations. For example, the equation below is a differential equation: x′′ = −x, where x is a function of t, and x′′ means the second derivative of x with respect to t. So this equation says that the second derivative of x is equal to negative the value of x at every time t. Differential equations also arise in many applications in science and engineering, and their study is the second main part of the course.
Even though the actual solutions to differential equations are usually not linear, the sets of solutions share many important properties with solution sets to systems of linear equations. This is one reason why these topics are often taught together – ideas and examples on one side help illuminate the other.
This course develops the tools necessary to solve such systems of equations. Systems of linear equations are pervasive not just in mathematics and statistics, but also in many other areas of the natural and social sciences including economics and physics. Often times in such applications, the number of variables or equations can reach into the hundreds of thousands or even millions. One therefore needs a well-developed theory in order to answer questions about such systems; this is the goal of the course. We will cover a significant portion of the text, including linear transformations, subspaces, bases, determinants, eigenvalues and eigenvectors, orthogonal projections and the QR factorization.
In this course, we develop linear algebra from a more abstract point of view focusing on vector spaces over a field, and linear functions (maps) between them. This will allow for applications to a much broader class of objects than those covered in a first course, such as vectors and matrices with real entries.
We will see many of the same topics one covers in a traditional linear algebra course, but we will go into more depth, and focus much more on proofs rather than computations. Our approach follows the text and avoids any discussion or use of determinants until the very end. At the end of the course, we will develop determinants from the point of view of multilinear algebra, providing (in my opinion) the proper motivation for the definition of determinant.
In first semester Calculus, one studies functions of a single (real) variable, and then proceeds to study limits, continuity, derivatives (and their applications) and finally the definite integral and the Fundamental Theorem of Calculus. In Calculus II, one learns about u-substitution, and also about Taylor series which may be used to represent many of the functions that are met along the way.
It turns out that each of these topics has a direct parallel in higher dimensions. In this class, we will focus on the case of real-valued functions of two and three variables. Along the way, we will also encounter some new topics that are unique to the higher-dimensional setting.
In linear algebra, one studies the solution sets of systems of linear equations, and care is taken to develop algorithms for effective computation with these sets, as well as theory to explain and understand their properties. The primary algorithmic tools in linear algebra are of course matrix algebra, row reduction of matrices and determinants. For example, in most linear algebra courses one learns to answer the following questions using these tools:
Solve systems of linear equations.
Determine structure of the solutions to a linear system.
Determine when a system of linear equations has a solution.
Detect ‘redundant’ systems.
Pass between parametric and implicit descriptions of solution sets.
When the equations in a system are no longer linear, (as is the case in many applications from robotics, combinatorics, invariant theory, and integer programming), the usual properties of their solution set, as well as the algorithms used to answer the above questions, no longer apply. However, we would still like to be able to answer these questions.
In this course, we will develop the computational and theoretical tools necessary to provide algorithms to answer each of these questions (and other questions we discover along the way). This suite of tools, collectively called “Gröbner basis theory”, is essentially a generalization of the ideas of row reduction to the case of polynomial equations.
Once we provide an algorithmic answer to the above questions for solution sets of polynomial equations, one can apply this algorithm anywhere that systems of polynomial equations naturally arise. We will cover some of these applications in class, but I will leave many of them as project ideas.
Calculus is, loosely speaking, a set of mathematical tools which allows one to understand how a function changes locally, as well as how to use this information to answer geometric questions about the function.
We will begin with the definition of limit, and then move to the definition of derivative by means of the limit of the slopes of secant lines. We will then discuss how one may use the derivative to reason about the change of a function in applications, to sketch a graph of the function, and to use (anti)-derivatives to compute areas.
Calculus is the study of the ‘local’ behavior of functions, and how we may use this local information to infer global properties. This involves things like limits, derivatives and integrals. In the first semester of calculus, you were primarily focused on studying properties of limits and derivatives and were probably able to touch on integrals a bit at the end. In the first third of this course, we complete the introductory material on integrals, and proceed to study several different integration techniques that will help us apply the fundamental theorem of calculus in more contexts.
The second third is occupied by laying the groundwork for Taylor’s Theorem. This is a very important theorem that gives us a way to approximate a function by a polynomial function using its derivatives. It has applications in many other areas such as physics, economics, linear algebra and differential equations. The mathematics in this part is likely to be quite different than what you have seen before, so it is especially important to stay caught up on homework and reading during this material.
In the final third, we will study several applications of integration such as volume, work, and arc length, as well as cover some topics related to the calculus of parametric and polar curves.
Linear algebra is arguably the most useful and applicable course in any mathematics curriculum. In a first course (such as our MTH 121), much effort is spent developing language and terminology to aid in the study of matrices. So much in fact that often, only the major applications are covered such as eigenvalues and eigenvectors, orthogonality and projections.
Quite often, a second course in linear algebra (such as our MTH 324) studies vector spaces over a field in generality. Such a course requires abstract algebra (MTH 321), and as such its topics are generally much more abstract in nature. As such, many students interested in the applications of linear algebra (rather than the theory) do not take such a course.
This course seeks to find a middle ground. Here, we will assume nothing more than a standard first course in linear algebra, as well as a few things from some other courses you have no doubt taken, such as Calculus II. We will essentially pick right up where your original linear algebra class left off by reviewing the standard material for the first two or three weeks. Then, we will make our way through the following topics:
Linear algebra over the complex numbers (we cover this while reviewing previous semester material)
The Spectral Theorem
Singular value decomposition, square roots of matrices, and applications
Pseudoinverses
Quadratic Forms
Geršgorin’s Circle Theorem
Markov chains
The Perron-Frobenius Theorem and (sub)stochastic matrices 9. Analytic functions of matrices
Finite fields, and linear algebra in that context
An Introduction to coding theory
Each topic will be coupled with applications which show off its utility.
This semester we will continue our coverage of two important areas of abstract algebra: Galois (and field) Theory, and Representation Theory. Both of these topics concern groups (which was the primary topic of the first semester), but also use concepts such as rings and fields (which were covered briefly at the end of the first semester).
Galois Theory is the study of symmetries of roots (with respect to certain automorphisms) of a polynomial equation defined over a field. It provides a deep connection between the group of automorphisms of a field extension and the intermediate fields of that extension. It allows one to prove that the quintic equation does not have a general solution, to show that certain geometric constructions are (im)possible, and several other problems from mathematical antiquity. We will begin with a review of the ring theory we will need to study polynomials over a field, then discuss field extensions. Finally, we will introduce the Galois group of a (Galois) extension which will lead us to the Fundamental Theorem of Galois Theory. We will end with our main application, the insolvability of the general quintic.
Next comes the representation theory of groups; a complex representation of a group G is nothing more than a group homomorphism from G to the set of invertible n × n matrices over the complex numbers. It turns out that if G is a finite group, then every complex representation can be broken down into a direct product of smaller representations which come from a finite list (which depends on G). This result lies at the intersection of group theory and linear algebra and has a wide variety of applications in mathematics. Our approach to this theory will be via objects known as group characters, and we will compute the character table for several well-known groups such as cyclic groups, dihedral groups and symmetric groups. Some of our treatment will apply to fields other than the complex numbers as well.
For Galois and field theory, we will use will be Algebra: Chapter 0 by Aluffi, which uses a categorical approach to abstract algebra as a whole. I will do my best to bring everyone up to speed on the category theory we will require, but just know that we will not be needing it at the level of detail we covered it in the first semester. For representation theory, we will use Chapter 10 of Algebra by Artin. I will also pull from Chapter 1 of Sagan’s text on the Representation Theory of the symmetric group.
In first semester Calculus, one studies functions of a single (real) variable, and then proceeds to study limits, continuity, derivatives (and their applications) and finally the definite integral and the Fundamental Theorem of Calculus. In Calculus II, one learns about u-substitution, and also about Taylor series which may be used to represent many of the functions that are met along the way.
It turns out that each of these topics has a direct parallel in higher dimensions. In this class, we will focus on the case of real-valued functions of two and three variables. Along the way, we will also encounter some new topics that are unique to the higher-dimensional setting.
Abstract Algebra is the area of mathematics which studies collections of objects which satisfy certain axioms. The axioms one chooses to work with are often motivated by applications from such varied fields as theoretical physics, chemistry, integer programming, and robotics. While some of the results were known to mathematicians such as Euler, Lagrange, Fermat, Galois, and others, abstract algebra as a field unto itself did not emerge until the beginning of the 20th century.
One major difference in our approach to this material is that we will be weaving the notion of a category throughout our study. Loosely speaking, a category is a collection of ‘objects’, together with a collection of ’morphisms’ which give us a way to ‘compare’ objects. Examples of categories abound in mathematics, with perhaps the friendliest first encounter being the category of sets. For this reason, we will begin with a brief review of some fundamental ideas such as sets and functions, along with a small amount of category theory.
After our brief introduction to category theory, we will study groups. A group is a set of objects combined with an operation that allows us to combine two objects together to get a new object. A very important example of a group is the set of symmetries of a geometric object, such as a square, a cube, an icosahedron, or even a sphere. Another example is the set of bijections from a set to itself. There are many other examples and we will spend much of the semester exploring both the general theory of groups, as well as many examples.
Once we finish our material on groups, we will begin our discussion on rings, which provide a framework that is useful in areas such as number theory and algebraic geometry. This will also help us build the foundation for field theory and Galois theory that we will cover in the beginning of the second semester course.
Calculus is the study of the ‘local’ behavior of functions, and how we may use this local information to infer global properties. This involves things like limits, derivatives and integrals. In the first semester of calculus, you were primarily focused on studying properties of limits and derivatives and were probably able to touch on integrals a bit at the end. In the first third of this course, we complete the introductory material on integrals, and proceed to study several different integration techniques that will help us apply the fundamental theorem of calculus in more contexts.
The second third is occupied by laying the groundwork for Taylor’s Theorem. This is a very important theorem that gives us a way to approximate a function by a polynomial function using its derivatives. It has applications in many other areas such as physics, economics, linear algebra and differential equations. The mathematics in this part is likely to be quite different than what you have seen before, so it is especially important to stay caught up on homework and reading during this material.
In the final third, we will study several applications of integration such as volume, work, and arc length, as well as cover some topics related to the calculus of parametric and polar curves.
This course develops the tools necessary to solve such systems of equations. Systems of linear equations are pervasive not just in mathematics and statistics, but also in many other areas of the natural and social sciences including economics and physics. Often times in such applications, the number of variables or equations can reach into the hundreds of thousands or even millions. One therefore needs a well-developed theory in order to answer questions about such systems; this is the goal of the course. We will cover a significant portion of the text, including linear transformations, subspaces, bases, determinants, eigenvalues and eigenvectors, orthogonal projections and the QR factorization.
Calculus is the study of the ‘local’ behavior of functions, and how we may use this local information to infer global properties. This involves things like limits, derivatives and integrals. In the first semester of calculus, you were primarily focused on studying properties of limits and derivatives and were probably able to touch on integrals a bit at the end. In the first third of this course, we complete the introductory material on integrals, and proceed to study several different integration techniques that will help us apply the fundamental theorem of calculus in more contexts.
The second third is occupied by laying the groundwork for Taylor’s Theorem. This is a very important theorem that gives us a way to approximate a function by a polynomial function using its derivatives. It has applications in many other areas such as physics, economics, linear algebra and differential equations. The mathematics in this part is likely to be quite different than what you have seen before, so it is especially important to stay caught up on homework and reading during this material.
In the final third, we will study several applications of integration such as volume, work, and arc length, as well as cover some topics related to the calculus of parametric and polar curves.
In this course, we will attempt to cover the very basics of three important areas of abstract algebra. In many cases, we will only give the ideas of the proofs in order to provide an overview of the theory and not get bogged down in the details.
Introduction to Representation Theory of Groups - A complex representation of a group G is nothing more than a group homomorphism from G to the set of invertible n × n matrices over the complex numbers. It turns out that if G is a finite group, then every complex representation can be broken down into a direct product of smaller representations which come from a finite list (which depends on G). This result lies at the intersection of group theory and linear algebra and has a wide variety of applications in mathematics. Our approach to this theory will be via objects known as group characters, and we will compute the character table for several well-known groups such as cyclic groups, dihedral groups and symmetric groups.
Introduction to Field Theory and Galois Theory - A field is a number system where one can add, subtract, multiply and divide by nonzero elements. Examples of such objects are the rational numbers, the real numbers and the complex numbers. We will begin by studying extensions of fields, which parallel the construction of adjoining the square root of −1 to the real numbers to obtain the complex numbers. Once we have a good grasp of field theory, we will turn our attention to Galois theory which provides a deep connection between the structure of a field extension and the roots of a polynomial. This culminates in the statement that there does not exist a formula for the zeroes of a general polynomial of degree five or higher in terms of radicals.
Introduction to Algebraic Geometry - A primary object of study in linear algebra is the solution set of a system of linear equations. The solution set of such a system is relatively easy to describe using a translation of a linear subspace. When the system of equations are defined by higher degree polynomials, the solution sets are much more difficult to describe. Understanding such solution sets is the subject of Algebraic Geometry. It turns out that understanding the ring of functions defined on the solution set helps us to understand the solution set itself. We will use the computer system Macaulay2 to explore this connection between geometry and algebra.
Calculus is the study of the ‘local’ behavior of functions, and how we may use this local information to infer global properties. This involves things like limits, derivatives and integrals. In the first semester of calculus, you were primarily focused on studying properties of limits and derivatives and were probably able to touch on integrals a bit at the end. In the first third of this course, we complete the introductory material on integrals, and proceed to study several different integration techniques that will help us apply the fundamental theorem of calculus in more contexts.
The second third is occupied by laying the groundwork for Taylor’s Theorem. This is a very important theorem that gives us a way to approximate a function by a polynomial function using its derivatives. It has applications in many other areas such as physics, economics, linear algebra and differential equations. The mathematics in this part is likely to be quite different than what you have seen before, so it is especially important to stay caught up on homework and reading during this material.
In the final third, we will study several applications of integration such as volume, work, and arc length, as well as cover some topics related to the calculus of parametric and polar curves.
Abstract Algebra is the area of mathematics which studies collections of objects which satisfy certain axioms. The axioms one chooses to work with are often motivated by applications from such varied fields as theoretical physics, chemistry, integer programming, and robotics. While some of the results were known to mathematicians such as Gauss, Euler, Lagrange, Fermat, Galois, and others, abstract algebra as a field unto itself did not emerge until the beginning of the 20th century.
For the the first two-thirds of the semester, we will study groups. A group is a set of objects combined with an operation that allows us to combine two objects together to get a new object. A very important example of a group is the set of symmetries of a geometric object, such as a square, a cube, an icosahedron, or even a sphere. Another example is the set of bijective (that is, one-to-one and onto) functions from a set to itself. There are many other examples and we will spend much of the semester exploring both the general theory of groups, as well as many examples.
Once we finish our material on groups, we will also go on to discuss rings, which provide a framework that is useful in areas such as number theory and algebraic geometry. A ring is a set of objects that have two operations: addition and multiplication. Here, the algebraic manipulation will feel a bit more familiar, as it quite similar to the number systems you are used to in algebra and calculus.