Teaching

I hadn't planned on going into mathematics when I started as an undergraduate, but after dabbling in some upper-level math courses I found the subject too beautiful to pass by. Perhaps because it took me a while to discover the aesthetic of mathematics for myself, one of my favorite parts of my job is being able to share an appreciation of this subject with students — especially students who have never enjoyed mathematics.

For research projects I've explored with undergraduates, please see my research page.

Current courses

Elements of Analysis I, Fall 2018

Math 302, Elements of Analysis I, is a course which explores the formality behind much of the content students learn through multivariable calculus. In this way the class is the opposite of many proof based math courses, where typically a new class of objects or unfamiliar quality is introduced, and the goal is to explore its ramifications (often by showing how the new idea generalizes, synthesizes or otherwise sheds light on previously understood content). In 302, by contrast, students typically have an established intuition for the topics we consider, and the challenge is to replace this intuitive understanding with an axiomatic one. In short, we will mostly be focused on resolving the "why" questions of calculus.

The class begins by detailing the axioms we need to describe the real numbers, then exploring their ramifications. We'll ultimately list a handful of axioms which uniquely characterize the real numbers, and we'll spend a considerable amount of time thinking about how these axioms distinguish the real numbers from other familiar sets of numbers (like the rational numbers). With this information in hand, we'll be able to start "doing analysis" in $\mathbb{R}^n$. This will require us to formalize notions like "limit" and "continuity," but will also require us to think carefully about "open" and "closed" sets. Our objective will be to establish the right abstract formulation for some concepts so that it is easy to prove more powerful theorems. As an example, instead of showing that all polynomials are continuous directly, we'll instead argue that the functions $f(x) = 1$ and $g(x) = x$ are continuous, and then we'll prove that the sum, difference and product of continuous functions is again functions. Taken together, these results will tell us that all polynomials are continuous, without requiring a direct ``$\varepsilon-\delta$ proof" of this fact.

The conceptual highlights of the course include (but are not necessarily limited to): the least upper bound property, the uncountability of the real numbers, basic point set topology, limits (of functions and sequences), continuity, connectedness, open covers, compactness, differentiation and integration.

The course webpage is here.

Advanced Linear Algebra, Fall 2018

The purpose of Math 322, Advanced Linear Algebra, is to delve deeper into the structure of vector spaces and their transformations. The abstraction and mathematical maturity developed in abstract algebra allow students to approach the content of this course with a new set of tools. The course progresses through Halmos' Finite dimensional vector spaces, with topics covering three main areas: vector spaces (including basic qualities of vectors, invariants of vector spaces, and canonical vector spaces constructions such as dual spaces, quotient spaces, and linear forms), transformations (including the formal construction of determinants and a review of canonical forms), and orthogonality (from the basics of inner product spaces to the spectral theorems). If time allows, the course will finish with a discussion of modules, a construction which generalizes vector spaces by replacing the scalar field with a scalar ring.

Aside from the content of the course, another distinguishing feature of this section of Math 322 is that lectures will primarily be delivered by students in the course.

The course webpage is here.

Previous courses

Calculus II, Summer 2018

This course picks up where a typical introductory Calculus class ends. After reviewing some of the basic definitions (especially integration), we'll begin developing more "advanced" techniques for computing anti-derivatives. Afterward we'll see some applications of integrals, particularly to geometric quantities like volume and length. We'll also explore how these ideas can be translated into new contexts, both in terms of how we express curves and how we express points in the plane. The second half of the course breaks from the focus on integration to consider sequences and series. The ultimate goal of this portion of the course is to determine what it means to add an infinite number of terms; though this begins by considering an infinite sum of real numbers, the real strength of this theory comes when considering infinite sums of functions.

The course webpage is here.

Abstract Algebra, Spring 2018

When a person thinks of algebra, they typically think of a process used to solve polynomial equations. Modern algebra, the subject of Math 305, is a discipline developed to answer precisely when formulas for solving equations exist. We all know the quadratic formula for solving degree 2 polynomial equations, and there are even cubic and quartic equations as well. One of the most surprising theorems in algebra was discovered in the early 1800s when it was proven that the general quintic equation has roots which cannot be expressed using addition, subtraction, multiplication, division and root extraction for rational numbers; somehow these polynomials are too complex to allow for a ``quintic formula."

The machinery necessary to understand this result includes several classes of mathematical objects which are abstractions of some familiar mathematical friends (most notably, the collection of integers, which is often written $\mathbb{Z}$). This class will be your introduction to these basic objects. We will spend the majority of our class time discussing groups, which is a set of elements which can be combined under a binary operation. Closer to the end of the semester we'll also consider algebraic sets which have even more structure than groups. Throughout the semester we will occasionally discuss how ideas from modern algebra appear in in everyday life, though most of our time together will be spent fully embracing the abstract nature of the discipline.

The course webpage is here.

Elementary Number Theory, Spring 2018

Number theory is the study of integers, primarily their structure under the operations of multiplication and addition. Though this seems a humble beginning, it is surprising how quickly one can ask mathematical questions about integers which are exceedingly difficult to resolve. Indeed, there are numerous problems which are thousands of years old that have yet to be resolved, as well as many other ancient problems which have only been resolved due to very sophisticated mathematics. It is this contrast between simplicity and complexity which forms the aesthetic of many number theoretic problems. In this class we'll develop some basic tools which allow us to begin analyzing the structures which govern the integers.

The course webpage is here.

Linear Algebra, Fall 2017

Math 206, Linear Algebra, begins by investigating techniques for solving systems of linear equations. The methods we develop suggest deeper structures which are responsible for the phenomena we see, and much of the class is spent contemplating these driving forces abstractly. The benefit of this abstract approach is that it produces machinery which is useful in a wide variety of contexts, not simply when something ``looks like" a system of linear equations. We will spend much of the semester analyzing structures inside ``real $n$-dimensional Euclidean space" (your old friend $\mathbb{R}^n$ from multivariable calculus); later in the semester we will see that we can develop a more abstract notion of an $n$-dimensional space which has many of the attractive qualities of $\mathbb{R}^n$. One of the punchlines in linear algebra is that any $n$-dimensional space (over $\mathbb{R}$) is ``the same as" $\mathbb{R}^n$ (a statement we'll work to make more precise as the semester unfolds).

For many students this class will serve as an introduction to abstract mathematics, so in addition to the linear algebra knowledge you'll accumulate throughout the term, you'll also be developing the meta-skills of reading, writing and creating mathematical proofs. If time permits, we will also consider some ``real-world" applications of linear algebra.

The course webpage is here.

Elements of Analysis I, Fall 2017

Math 302, Elements of Analysis I, is a course which explores the formality behind much of the content students learn through multivariable calculus. In this way the class is the opposite of many proof based math courses, where typically a new class of objects or unfamiliar quality is introduced, and the goal is to explore its ramifications (often by showing how the new idea generalizes, synthesizes or otherwise sheds light on previously understood content). In 302, by contrast, students typically have an established intuition for the topics we consider, and the challenge is to replace this intuitive understanding with an axiomatic one. In short, we will mostly be focused on resolving the "why" questions of calculus.

The class begins by detailing the axioms we need to describe the real numbers, then exploring their ramifications. We'll ultimately list a handful of axioms which uniquely characterize the real numbers, and we'll spend a considerable amount of time thinking about how these axioms distinguish the real numbers from other familiar sets of numbers (like the rational numbers). With this information in hand, we'll be able to start "doing analysis" in $\mathbb{R}^n$. This will require us to formalize notions like "limit" and "continuity," but will also require us to think carefully about "open" and "closed" sets. Our objective will be to establish the right abstract formulation for some concepts so that it is easy to prove more powerful theorems. As an example, instead of showing that all polynomials are continuous directly, we'll instead argue that the functions $f(x) = 1$ and $g(x) = x$ are continuous, and then we'll prove that the sum, difference and product of continuous functions is again functions. Taken together, these results will tell us that all polynomials are continuous, without requiring a direct ``$\varepsilon-\delta$ proof" of this fact.

The conceptual highlights of the course include (but are not necessarily limited to): the least upper bound property, the uncountability of the real numbers, basic point set topology, limits (of functions and sequences), continuity, connectedness, open covers, compactness, differentiation and integration.

The course webpage is here.

Calculus II, Summer 2017

This course picks up where a typical introductory Calculus class ends. After reviewing some of the basic definitions (especially integration), we'll begin developing more "advanced" techniques for computing anti-derivatives. Afterward we'll see some applications of integrals, particularly to geometric quantities like volume and length. We'll also explore how these ideas can be translated into new contexts, both in terms of how we express curves and how we express points in the plane. The second half of the course breaks from the focus on integration to consider sequences and series. The ultimate goal of this portion of the course is to determine what it means to add an infinite number of terms; though this begins by considering an infinite sum of real numbers, the real strength of this theory comes when considering infinite sums of functions.

The course webpage is here.

Advanced Number Theory, Spring 2017

Number theory has evolved through several stages in the past two millennia. Notions of primality and divisibility are indeed quite classical, and the ancients even knew a great deal about some relatively sophisticated ideas: the infinitude of primes, perfect numbers, etc. Innovation sat stagnant for about 15 centuries until, around 1625, Fermat acquired a copy of Diophantus and started pursuing the subject in earnest. Even this work might have fallen to the wayside had Euler not continued Fermat's work about 100 years later. Since that time, many mathematical greats have made significant contributions to the subject and discovered surprising connections between number theory and almost all other branches of mathematics. Perhaps the most striking feature of modern (i.e., post-Fermat) number theory is the divide between the simplicity of theorem statements and the complexity of techniques used to prove them.

In this class, we'll focus on answering the following question: for an integer $n$, how can we characterize those primes $p$ for which there exist integers $x$ and $y$ with $p = x^2+ny^2$. As a follow-up to 305, we'll be able to use the power of abstract algebra to answer this question, and we'll see that this seemingly innocuous problem requires some sophisticated machinery to unravel. One of interesting trends will uncover is the increased complexity of machinery necessary to resolve this problem as the value of $n$ increases.

The course webpage is here.

Probability and Statistics, Spring 2017

As the course title suggests, the content of Math 220 is divided into two related components. First, we learn the basics of probability theory. Roughly speaking, probability is the science of determining the likelihood of a certain event. This includes some fundamental concepts (e.g., conditional probability, independent events) and theorems (e.g., Bayes's Theorem), but also requires us to develop the ability to carefully enumerate possibilities. We will explore some common discrete and continuous probability distributions, and see how they are related to the so-called normal distribution via the Central Limit Theorem.

After discussing probability, we will turn our attention to statistics. Whereas the former asks us to determine the likelihood of an outcome based on an assumed model of a given random process, statistics asks us to determine a reasonable model for a random process based on observed outcomes of that process. For instance, if we roll a pair of fair dice, probability theory tells us that it's six times more likely that we would see a total of 7 on the dice than that we'd see a total of 12. If, on the other hand, we're given two dice and roll them 100 times, and we find that we get a total of 12 for 50 of those rolls, then we can reasonably infer that the dice aren't fair (and given more information, we could describe more explicitly the bias in the dice). We will cover a handful of statistical topics: sampling, estimation, confidence intervals, and hypothesis testing.

The course webpage is here.

Multivariable Calculus, Fall 2016

Math 205 (Multivariable Calculus) is a first attempt to understand what derivatives and integrals mean for a function whose domain or codomain isn't $\mathbb{R}$. In particular, this class will focus on functions $f:\mathbb{R}^n\to \mathbb{R}^m$, particularly when $n,m \leq 3$. Fortunately much of the differential and integral calculus that you remember has natural extensions in this setting. Our goal will be to complete chapters 11 and 13-17 from the 6th edition of Stewart's Calculus.

The course webpage is here.

Calculus II, Summer 2016

This course picks up where a typical introductory Calculus class ends. After reviewing some of the basic definitions (especially integration), we'll begin developing more "advanced" techniques for computing anti-derivatives. Afterward we'll see some applications of integrals, particularly to geometric quantities like volume and length. We'll also explore how these ideas can be translated into new contexts, both in terms of how we express curves and how we express points in the plane. The second half of the course breaks from the focus on integration to consider sequences and series. The ultimate goal of this portion of the course is to determine what it means to add an infinite number of terms; though this begins by considering an infinite sum of real numbers, the real strength of this theory comes when considering infinite sums of functions.

The course webpage is here.

Multivariable Calculus, Spring 2016

Math 205 (Multivariable Calculus) is a first attempt to understand what derivatives and integrals mean for a function whose domain or codomain isn't $\mathbb{R}$. In particular, this class will focus on functions $f:\mathbb{R}^n\to \mathbb{R}^m$, particularly when $n,m \leq 3$. Fortunately much of the differential and integral calculus that you remember has natural extensions in this setting. Our goal will be to complete chapters 11 and 13-17 from the 6th edition of Stewart's Calculus.

The course webpage is here.

Linear Algebra, Spring 2016

Math 206, Linear Algebra, begins by investigating techniques for solving systems of linear equations. The methods we develop suggest deeper structures which are responsible for the phenomena we see, and much of the class is spent contemplating these driving forces abstractly. The benefit of this abstract approach is that it produces machinery which is useful in a wide variety of contexts, not simply when something ``looks like" a system of linear equations. We will spend much of the semester analyzing structures inside ``real $n$-dimensional Euclidean space" (your old friend $\mathbb{R}^n$ from multivariable calculus); later in the semester we will see that we can develop a more abstract notion of an $n$-dimensional space which has many of the attractive qualities of $\mathbb{R}^n$. One of the punchlines in linear algebra is that any $n$-dimensional space (over $\mathbb{R}$) is ``the same as" $\mathbb{R}^n$ (a statement we'll work to make more precise as the semester unfolds).

For many students this class will serve as an introduction to abstract mathematics, so in addition to the linear algebra knowledge you'll accumulate throughout the term, you'll also be developing the meta-skills of reading, writing and creating mathematical proofs. If time permits, we will also consider some ``real-world" applications of linear algebra.

The course webpage is here.

Abstract Algebra, Fall 2015

When a person thinks of algebra, they typically think of a process used to solve polynomial equations. Modern algebra, the subject of Math 305, is a discipline developed to answer precisely when formulas for solving equations exist. We all know the quadratic formula for solving degree 2 polynomial equations, and there are even cubic and quartic equations as well. One of the most surprising theorems in algebra was discovered in the early 1800s when it was proven that the general quintic equation has roots which cannot be expressed using addition, subtraction, multiplication, division and root extraction for rational numbers; somehow these polynomials are too complex to allow for a ``quintic formula."

The machinery necessary to understand this result includes several classes of mathematical objects which are abstractions of some familiar mathematical friends (most notably, the collection of integers, which is often written $\mathbb{Z}$). This class will be your introduction to these basic objects. We will spend the majority of our class time discussing groups, which is a set of elements which can be combined under a binary operation. Closer to the end of the semester we'll also consider algebraic sets which have even more structure than groups. Throughout the semester we will occasionally discuss how ideas from modern algebra appear in in everyday life, though most of our time together will be spent fully embracing the abstract nature of the discipline.

The course webpage is here.

Probability and Statistics, Fall 2015

As the course title suggests, the content of Math 220 is divided into two related components. First, we learn the basics of probability theory. Roughly speaking, probability is the science of determining the likelihood of a certain event. This includes some fundamental concepts (e.g., conditional probability, independent events) and theorems (e.g., Bayes's Theorem), but also requires us to develop the ability to carefully enumerate possibilities. We will explore some common discrete and continuous probability distributions, and see how they are related to the so-called normal distribution via the Central Limit Theorem.

After discussing probability, we will turn our attention to statistics. Whereas the former asks us to determine the likelihood of an outcome based on an assumed model of a given random process, statistics asks us to determine a reasonable model for a random process based on observed outcomes of that process. For instance, if we roll a pair of fair dice, probability theory tells us that it's six times more likely that we would see a total of 7 on the dice than that we'd see a total of 12. If, on the other hand, we're given two dice and roll them 100 times, and we find that we get a total of 12 for 50 of those rolls, then we can reasonably infer that the dice aren't fair (and given more information, we could describe more explicitly the bias in the dice). We will cover a handful of statistical topics: sampling, estimation, confidence intervals, and hypothesis testing.

The course webpage is here.

Calculus II, Summer 2015

This course picks up where a typical introductory Calculus class ends. After reviewing some of the basic definitions (especially integration), we'll begin developing more "advanced" techniques for computing anti-derivatives. Afterward we'll see some applications of integrals, particularly to geometric quantities like volume and length. We'll also explore how these ideas can be translated into new contexts, both in terms of how we express curves and how we express points in the plane. The second half of the course breaks from the focus on integration to consider sequences and series. The ultimate goal of this portion of the course is to determine what it means to add an infinite number of terms; though this begins by considering an infinite sum of real numbers, the real strength of this theory comes when considering infinite sums of functions.

The course webpage is here.

Abstract Algebra, Spring 2015

When a person thinks of algebra, they typically think of a process used to solve polynomial equations. Modern algebra, the subject of Math 305, is a discipline developed to answer precisely when formulas for solving equations exist. We all know the quadratic formula for solving degree 2 polynomial equations, and there are even cubic and quartic equations as well. One of the most surprising theorems in algebra was discovered in the early 1800s when it was proven that the general quintic equation has roots which cannot be expressed using addition, subtraction, multiplication, division and root extraction for rational numbers; somehow these polynomials are too complex to allow for a ``quintic formula."

The machinery necessary to understand this result includes several classes of mathematical objects which are abstractions of some familiar mathematical friends (most notably, the collection of integers, which is often written $\mathbb{Z}$). This class will be your introduction to these basic objects. We will spend the majority of our class time discussing groups, which is a set of elements which can be combined under a binary operation. Closer to the end of the semester we'll also consider algebraic sets which have even more structure than groups. Throughout the semester we will occasionally discuss how ideas from modern algebra appear in in everyday life, though most of our time together will be spent fully embracing the abstract nature of the discipline.

The course webpage is here.

Advanced Number Theory, Spring 2015

Number theory has evolved through several stages in the past two millennia. Notions of primality and divisibility are indeed quite classical, and the ancients even knew a great deal about some relatively sophisticated ideas: the infinitude of primes, perfect numbers, etc. Innovation sat stagnant for about 15 centuries until, around 1625, Fermat acquired a copy of Diophantus and started pursuing the subject in earnest. Even this work might have fallen to the wayside had Euler not continued Fermat's work about 100 years later. Since that time, many mathematical greats have made significant contributions to the subject and discovered surprising connections between number theory and almost all other branches of mathematics. Perhaps the most striking feature of modern (i.e., post-Fermat) number theory is the divide between the simplicity of theorem statements and the complexity of techniques used to prove them.

In this class, we'll focus on answering the following question: for an integer $n$, how can we characterize those primes $p$ for which there exist integers $x$ and $y$ with $p = x^2+ny^2$. As a follow-up to 305, we'll be able to use the power of abstract algebra to answer this question, and we'll see that this seemingly innocuous problem requires some sophisticated machinery to unravel. One of interesting trends will uncover is the increased complexity of machinery necessary to resolve this problem as the value of $n$ increases.

The course webpage is here.

Linear Algebra, Fall 2014

Linear algebra begins by considering techniques for computing solutions to systems of linear equations. The methods we develop suggest that there are deeper structures which are responsible for the behavior we witness, and much of the class is spent in contemplating these driving principles abstractly. The benefit of this abstract approach is that it produces machinery which is useful in a wide variety of contexts, not simply when something ``looks like" a system of linear equations.

For many students this class will serve as an introduction to abstract mathematics, so in addition to the linear algebra knowledge you'll accumulate throughout the term, you'll also be developing the meta-skills of reading, writing and creating mathematical proofs. When time permits, we will also consider some ``real-world" applications of linear algebra.

The course webpage is here.

Multivariable Calculus, Fall 2014

Math 205 (Multivariable Calculus) is a first attempt to understand what derivatives and integrals mean for a function whose domain or codomain isn't $\mathbb{R}$. In particular, this class will focus on functions $f:\mathbb{R}^n\to \mathbb{R}^m$, particularly when $n,m \leq 3$. Fortunately much of the differential and integral calculus that you remember has natural extensions in this setting. Our goal will be to complete chapters 11 and 13-17 from the 6th edition of Stewart's Calculus.

The course webpage is here.

Calculus II, Summer 2014

This course picks up where a typical introductory Calculus class ends. After reviewing some of the basic definitions (especially integration), we'll begin developing more "advanced" techniques for computing anti-derivatives. Afterward we'll see some applications of integrals, particularly to geometric quantities like volume and length. The second half of the course breaks from the focus on integration to consider sequences and series. The ultimate goal of this portion of the course is to determine what it means to add an infinite number of terms; though this begins by considering an infinite sum of real numbers, the real strength of this theory comes when considering infinite sums of functions.

The course webpage is here.

Calculus II, Summer 2013

This course picks up where a typical introductory Calculus class ends. After reviewing some of the basic definitions (especially integration), we'll begin developing more "advanced" techniques for computing anti-derivatives. Afterward we'll see some applications of integrals, particularly to geometric quantities like volume and length. The second half of the course breaks from the focus on integration to consider sequences and series. The ultimate goal of this portion of the course is to determine what it means to add an infinite number of terms; though this begins by considering an infinite sum of real numbers, the real strength of this theory comes when considering infinite sums of functions.

The course webpage is here.

Elementary Number Theory, Wellesley College, Spring 2013

Number theory is the study of integers, primarily their structure under the operations of multiplication and addition. Though this seems a humble beginning, it is surprising how quickly one can ask mathematical questions about integers which are exceedingly difficult to resolve. Indeed, there are numerous problems which are thousands of years old that have yet to be resolved, as well as many other ancient problems which have only been resolved due to very sophisticated mathematics. It is this contrast between simplicity and complexity which forms the aesthetic of many number theoretic problems. In this class we'll develop some basic tools which allow us to begin analyzing the structures which govern the integers.

The course webpage is here.

Numerical Analysis, Wellesley College, Spring 2013

In a typical mathematics course, a student learns a combination of theory and computation, with the latter providing concrete examples of how the former is exhibited ``in the real world." Of course, ``the real world" in a mathematics class is the realm of pure mathematics, where real numbers have infinite decimal expansions and there are a continuum of numbers that are as close to zero as you like. In the real world that we experience on a day-to-day basis, however, concepts which rely on infinite precision have a harder time being implemented in a practical way; one simply only has so much memory that one can allocate to storing the decimal digits of , for instance. The typical answer to this problem is to allow ourselves to approximate real values in our computations, with the tacit assumption that these approximated values will be sufficient for any ``real" problem we might face. Moreover, if there's some situation where one needs additional precision when doing a certain computation, the assumption is that if one begins with a higher precision approximation to the number in question, then computations with this better approximation should themselves be more precise.

In many ways, numerical analysis is the class in which one investigates whether these assumptions are true. It is a class that is steeped both in practical application (almost by its nature), but also intimately connected to deeply theoretical --- and often philosophical --- considerations. Perhaps the most consistent question we will aim to answer in this class is: how can I approximate a particular mathematical computation, and how well do I understand the error in this approximation? In answering this question we'll consider how computers store and process real numbers, how algebraically equivalent expressions for two quantities can yield dramatically different computed results, and how to effectively compress information to retain only the most important information.

The course webpage is here.

Complex Analysis, Wellesley College, Fall 2012

The goal of this class is to learn what we can about functions of a single complex variable. In many ways this will mean attempting to translate familiar notions from calculus into the complex setting, and we'll see that there are certain surprises in the study of complex functions which one might not have predicted from studying real calculus. A constant theme in the course will be connecting the geometric understanding of the complex numbers with the algebraic operations we perform on them. By the end of the course we will understand the classical results on derivatives and path integrals for complex functions, and we will (hopefully) discuss one of the most important complex functions in all of mathematics: the Riemann zeta function.

The course webpage is here.

Abstract Algebra, Wellesley College, Fall 2012

When a person thinks of algebra, they typically think of a process used to solve polynomial equations. Modern algebra, the subject of Math 305, is a discipline developed to answer precisely when formulas for solving equations exist. We all know the quadratic formula for solving degree 2 polynomial equations, and there are even cubic and quartic equations as well. One of the most surprising theorems in algebra was discovered in the early 1800s when it was proven that the general quintic equation has roots which cannot be expressed using addition, subtraction, multiplication, division and root extraction for rational numbers; somehow these polynomials are too complex to allow for a ``quintic formula."

The machinery necessary to understand this result includes several classes of mathematical objects which are abstractions of some familiar mathematical friends (most notably, the collection of integers, which is often written $\mathbb{Z}$). This class will be your introduction to these basic objects. We will spend the majority of our class time discussing groups, which is a set of elements which can be combined under a binary operation. Closer to the end of the semester we'll also consider algebraic sets which have even more structure than groups. Throughout the semester we will occasionally discuss how ideas from modern algebra appear in in everyday life, though most of our time together will be spent fully embracing the abstract nature of the discipline.

The course webpage is here.

Mathematical Thought and Proof, Wellesley College, Spring 2012

This course is designed to show math enthusiasts what life is like after calculus. The heart of the course is in learning proof, the defining quality of mathematics and the source of its timeless truth. Students will spend the semester learning about the basic mathematical objects and proof techniques that will carry them through the rest of the undergraduate major (and, if interested, beyond). A secondary goal in the class is to show students some of the incredibly interesting mathematical results from the past several hundred years. For instance, a portion of the class will be devoted to discussing why there are certain infinite sets which are quantifiably larger than other infinite sets. In fact, there are infinitely many sizes of infinity!

The course webpage is here.

Abstract Algebra, Wellesley College, Spring 2012

When a person thinks of algebra, they typically think of a process used to solve polynomial equations. Modern algebra, the subject of Math 305, is a discipline developed to answer precisely when formulas for solving equations exist. We all know the quadratic formula for solving degree 2 polynomial equations, and there are even cubic and quartic equations as well. One of the most surprising theorems in algebra was discovered in the early 1800s when it was proven that the general quintic equation has roots which cannot be expressed using addition, subtraction, multiplication, division and root extraction for rational numbers; somehow these polynomials are too complex to allow for a ``quintic formula."

The machinery necessary to understand this result includes several classes of mathematical objects which are abstractions of some familiar mathematical friends (most notably, the collection of integers, which is often written $\mathbb{Z}$). This class will be your introduction to these basic objects. We will spend the majority of our class time discussing groups, which is a set of elements which can be combined under a binary operation. Closer to the end of the semester we'll also consider algebraic sets which have even more structure than groups. Throughout the semester we will occasionally discuss how ideas from modern algebra appear in in everyday life, though most of our time together will be spent fully embracing the abstract nature of the discipline.

The course webpage is here.

Calculus II, Section 6, Wellesley College, Fall 2011

This course picks up where a typical introductory Calculus class ends. After reviewing some of the basic definitions (especially integration), we'll begin developing more "advanced" techniques for computing anti-derivatives. Afterward we'll see some applications of integrals, particularly to geometric quantities like volume and length. The second half of the course breaks from the focus on integration to consider sequences and series. The ultimate goal of this portion of the course is to determine what it means to add an infinite number of terms; though this begins by considering an infinite sum of real numbers, the real strength of this theory comes when considering infinite sums of functions.

The course webpage is here.

Complex Analysis, Wellesley College, Fall 2011

The goal of this class is to learn what we can about functions of a single complex variable. In many ways this will mean attempting to translate familiar notions from calculus into the complex setting, and we'll see that there are certain surprises in the study of complex functions which one might not have predicted from studying real calculus. A constant theme in the course will be connecting the geometric understanding of the complex numbers with the algebraic operations we perform on them. By the end of the course we will understand the classical results on derivatives and path integrals for complex functions, and we will (hopefully) discuss one of the most important complex functions in all of mathematics: the Riemann zeta function.

The course webpage is here.

Multivariable Calculus, Section 3, Wellesley College, Spring 2011

Math 205 (Multivariable Calculus) is a first attempt to understand what derivatives and integrals mean for a function whose domain or codomain isn't $\mathbb{R}$. In particular, this class will focus on functions $f:\mathbb{R}^n\to \mathbb{R}^m$, particularly when $n,m \leq 3$. Fortunately much of the differential and integral calculus that you remember has natural extensions in this setting. Our goal will be to complete chapters 11 and 13-17 from the 6th edition of Stewart's Calculus.

The course content was managed through Wellesley's Sakai server here.

Number Theory, Section 1, Wellesley College, Spring 2011

Math 223 (Number Theory) is the study of integers under the operations of multiplication and addition. Though this seems a humble beginning, it is surprising how quickly one can ask mathematical questions about integers which are exceedingly difficult to resolve. Indeed, there are numerous problems which are thousands of years old that have yet to be resolved, as well as many other ancient problems which have only been resolved due to very sophisticated mathematics. It is this contrast between simplicity and complexity which forms the aesthetic of many number theoretic problems. In this class we'll develop some basic tools which allow us to begin analyzing the structures which govern the integers.

The course content was managed through Wellesley's Sakai server here.

Calculus II, Sections 1&2, Wellesley College, Fall 2010

This course picks up where a typical introductory Calculus class ends. After reviewing some of the basic definitions (especially integration), we'll begin developing more "advanced" techniques for computing anti-derivatives. Afterward we'll see some applications of integrals, particularly to geometric quantities like volume and length. The second half of the course breaks from the focus on integration to consider sequences and series. The ultimate goal of this portion of the course is to determine what it means to add an infinite number of terms; though this begins by considering an infinite sum of real numbers, the real strength of this theory comes when considering infinite sums of functions.

The course webpage is here.

Abstract Linear Algebra, Section M13, UIUC, Spring 2010

This is an introductory linear algebra course which places more focus on proof than a typical matrix algebra class. Aside from the theoretical bent, we stick to many of the traditional topics in a linear algebra class. We'll also hit a few highlights not typically covered in an introductory linear algebra class, including the singular value decomposition, stochastic matrices, the Cayley-Hamilton theorem and Jordan Canonical forms.

The course webpage is here.

Applied Linear Algebra (Calculus & Mathematica), Sections F83 & Z83, UIUC, Fall 2009

This is a Calculus & Mathematica version of the typical 415 course offered by the department. The class focuses on the singular value decomposition of a matrix and what it tells us about the geometric action of matrices. Since the SVD encodes much of classical matrix theory, this really is the `holy grail' of linear algebra.

Students follow the Mathematica notebooks which allow them to explore linear algebra and its applications through the lens of SVD. Topics range from interpretting SVD factors, to basic applications of the SVD, to understanding why the SVD theory works.

Course content is managed through the ClassComm system, so there is no course webpage.

Elementary Number Theory, Section X13, UIUC, Spring 2009

This is a basic introductory course to number theory with a focus on `elementary' methods. After covering basic topics like divisibility, primality, and the Fundamental Theorem of Arithmetic, we'll then introduce modular arithmetic, study basic arithmetic functions and their properties, learn about quadratic reciprocity, and investigate primitive roots for various moduli. If time permits, we'll also cover some of the current applications of elementary number theory, particularly the RSA cryptosystem.

This course includes a variety of methodologies to promote active and creative student engagement in the class. For instance, the primary course management tool is its own Wiki, located at http://math453spring2009.wikidot.com. Throughout the term students are generating and participating in discussions of course content through the Wiki's own forum. Students will be responsible for preparing their own Wiki pages and in-class presentations on number theoretic topics of their choice; these will occur at the end of the term.

Calculus II, Section BL1, UIUC, Spring 2009

This is a second course in calculus, beginning with techniques in integration, moving through infinite series, and concluding with parametric equations and coordinate changes. The course meets en masse for lecture twice per week and breaks into six smaller discussion sections twice per week. As the instructor for this section, I am in charge of managing all details for the course, from giving the lectures to coordinating the dicussion sections to writing tests, quizzes and homework assignments.

The course webpage is here.

Elementary Number Theory, Section D13, UIUC, Fall 2008

This is a basic introductory course to number theory with a focus on `elementary' methods. After covering basic topics like divisibility, primality, and the Fundamental Theorem of Arithmetic, we'll then introduce modular arithmetic, study basic arithmetic functions and their properties, learn about quadratic reciprocity, and investigate primitive roots for various moduli. If time permits, we'll also cover some of the current applications of elementary number theory, particularly the RSA cryptosystem.

This course includes a variety of methodologies to promote active and creative student engagement in the class. For instance, the primary course management tool is its own Wiki, located at http://math453fall2008.wikidot.com. Throughout the term students are generating and participating in discussions of course content through the Wiki's own forum. Students will be responsible for preparing their own Wiki pages and in-class presentations on number theoretic topics of their choice; these will occur at the end of the term.

Applied Linear Algebra (Calculus & Mathematica), Sections E83 & Z83, UIUC, Spring 2007

This is a Calculus & Mathematica version of the typical 415 course offered by the department. The class focuses on the singular value decomposition of a matrix and what it tells us about the geometric action of matrices. Since the SVD encodes much of classical matrix theory, this really is the `holy grail' of linear algebra.

Students follow the Mathematica notebooks which allow them to explore linear algebra and its applications through the lens of SVD. Topics range from interpretting SVD factors, to basic applications of the SVD, to understanding why the SVD theory works.

Course content is managed through the ClassComm system, so there is no course webpage. A syllabus is avaiable here.

Introductory Matrix Theory, Sections P1 & Q2, UIUC, Fall 2007

This course is basic introduction to the ideas of linear algebra, with an emphasis on real examples. The course necessarily covers a great deal of theory, and so serves the double-purpose of being something of an introduction to abstract mathematical thought. The course is designed by the Math Department at UIUC, and we follow their syllabus.

The course webpage is here.

Introductory Matrix Theory (Calculus & Mathematica), Section T8, UIUC, Fall 2007

This is a Calculus & Mathematica version of the typical 225 course offered by the department. Instead of the usual `systems of equations' approach to linear algebra, the class focuses on the singular value decomposition of a matrix and what it tells us about the geometric action of matrices. Since the SVD encodes much of classical matrix theory, this really is the `holy grail' of linear algebra.

Students follow Mathematica notebooks which allow them to explore linear algebra and its applications through the lens of SVD. Topics range from interpretting SVD factors to basic applications of the SVD.

Course content is managed through the ClassComm system, so there is no course webpage. A syllabus is avaiable here.

Math 51 Stanford University, Winter 2007 and Fall 2006

As the uber-TA for Math 51 in the Winter of 2007, I designed and maintained the course webpage for this edition of Math 51 (450+ enrollment). Additional responsibilities included writing and posting homework and exam solutions as well as various administrative tasks.

As the teaching assistant for the Accelerated Calculus for Engineers (ACE) section of Math 51 in the Fall of 2006, I led an extended discussion section twice a week for students enrolled in the Math 51 ACE. This page contains the few handouts I gave out to the class or solutions I posted for interested students.

Stanford Engineering Summer Academy -- Mathematics Module Stanford University, Summer 2006

The Stanford Summer Engineering Academy (SSEA) is designed to help attract and maintain `a diverse student body to the School of Engineering' and is a `rigorous introduction to [Stanford's] engineering, math, and physical sciences programs.' The mathematics module prepares students for either Math 51 (linear algebra and multivariable calculus) or Math 41/42 (single variable calculus) by introducing material they will see during the regular term. I designed and taught the module as well as coordinated discussion sections with course assistants.

The course webpage is here.

Matrix Theory and its Applications Stanford University, Summer 2006

As the name suggests, this course is an introduction to linear algebra with a special emphasis on presenting applications. I designed and taught the course and used Mathematica as a tool for visualizing concepts or presenting real-world applications. You can find these Mathematica calculations on the course webpage under Mathematica Examples.

The course webpage is here.

Etale cohomology reading group Stanford University, Winter and Spring 2006.

This was a collection of grad students interested in learning a little more about Etale cohomology. It was informal, but we put up a webpage to keep track of what we covered. Use material at your own risk!

Introduction to Calculus Stanford University, Winter 2006 and 2005

This class is the first in a three part series which introduces students to single variable calculus; the emphasis in this part of the series is differentiation and some of its applications. I designed and taught the course.

The course webpage for the 2006 version of the class is here, and the webpage for the 2005 version of the class is here.

MatLab exercises

Rob Easton and I wrote some MatLab exercises for the multivariable calculus/linear algebra series at Stanford. Each of these gives a `real world' application of the ideas which were covered in the classes.

± Assignment 1: Math 51
± Assignment 2: Math 51 (An alternate version with modified final exercise)
± Assignment 3: Math 51 (An alternate version with modified bonus problem)
± Assignment 1: Math 52
± Assignment 2: Math 52
± Assignment 3: Math 52