Math 225 introduces students to a few key ideas in discrete mathematics, including enumeration techniques, mathematical induction, generating functions, recurrence relations, and graph theory. For many students, this will also be one of the first classes they take that spends a considerable amount of class time discussing (and assigns a significant number of homework homework problems requiring) mathematical proof.

The course webpage is here.

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.

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.

Math 225 introduces students to a few key ideas in discrete mathematics, including enumeration techniques, mathematical induction, generating functions, recurrence relations, and graph theory. For many students, this will also be one of the first classes they take that spends a considerable amount of class time discussing (and assigns a significant number of homework homework problems requiring) mathematical proof.

The course webpage is here.

Most culture's mathematics begins with the introduction of the counting numbers and their structure under addition, with multiplication following soon thereafter. In western mathematics these two operations (together with the usual comparisons) were the language in which mathematics was written for thousands of years. The inverses of these operations were also considered, though ``improper fractions" were socially accepted long before negative numbers were thought to have real meaning. Though one might expect the acceptance of negative numbers to far predate that for complex numbers, the two were finally considered "natural'' around the same time when it was discovered that there are integer-coefficient cubic equations with real (positive) roots that nonetheless require square roots of negative numbers to be formulated (you might look up casus irreducibilis if you want to find out more about this). This was the first in a long line of discoveries that pointed to the fact that the nature of the complex numbers is important to understand, even if you think you only care about "real" phenomena.

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.

Roughly speaking, probability is the mathematics of determining the likelihood that a certain outcome occurs. Answering this question in a particular context requires us to understand --- with a lot of precision --- precisely what sorts of outcomes could occur, as well as some notion of how those outcomes are (for lack of a better term) doled out. For instance: if we roll a standard six-sided die, then the outcomes that are possible are the numbers $1$, $2$, $3$, $4$, $5$, and $6$; furthermore, each of these outcomes is equally likely. As a result, we can say that there's a 50/50 chance that we roll an odd number (or a one-third chance that we roll a multiple of $3$, etc.). On the other hand, if our die is "unfair" (say if there's additional weight that is situated on one face of the die), then we still have the same suite of possible outcomes, but they don't all occur with equal likelihood. This changes probability calculations from the standard die, despite the fact that much is the same in these two situations.

To aid in our investigation, we will develop a framework and vocabulary for understanding probabilistic questions. This includes some fundamental concepts (e.g., conditional probability, independent events) and theorems (e.g., Bayes's Theorem). In addition to this, we will also need to develop facility in a number of calculations. For example, we must develop the ability to carefully enumerate possibilities (and, believe it or not, counting is really hard), but we must also know how to add up infinite series and compute certain integrals. By the end of the semester, we will have explored many of the common discrete and continuous probability distributions, and we will see how they are related to the so-called normal distribution via the Central Limit Theorem.

The course webpage is here.

Math 116Z, Applied Calculus II, is a second course in calculus which aims highlight applications of the subject from the real world. The course begins where a typical introduction to calculus finishes: at the definition of the integral and its connections to so-called anti-derivatives. After refamiliarizing ourselves with these basic terms, we'll spend the first half of the course covering integrals in depth: first by developing techniques for evaluating difficult integrals, and then by discussing a variety of applications of the integral.

In the second half of the course we change gears, aiming to generalize the notion of tangent line that is introduced in differential calculus. We will quickly see that we can create tangent polynomials of arbitrarily large degree, and that higher-degree polynomials typically give a better fit to the original curve. The question arises whether these polynomials will "eventually" agree with the function; answering this question requires us to grapple with the question of infinite sums and how we can use them to build functions. This will result in a concept known as Taylor series. At the end of the course, we might briefly mention how these same ideas can be used to express functions as sums of sines and cosines, yielding so-called Fourier series. We might also cover some selected topics related to differential equations.

The course webpage is here.

QR140, Introduction to Quantitative Reasoning, is a course that works to support critical thinking skills that revolve around numbers and data. Given that our lives are awash numeric information, the content that we cover will be applicable in many facets of your life (including subsequent coursework at Wellesley, but also when living your ``real life"). The goal is for students to leave the course with increased familiarity and comfortability with interpretting and manipulating quantitative information: to be shrewd and skeptical when viewing the numeric arguments of others, and to be responsible and convincing when using quantitative data in their own arguments.

The course webpage is here.

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, and the canonical results from single-variable (real) differentiation and integration.

The course webpage is here.

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 some physical quantities like center of mass. 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.

As the name suggests, Math 115 (Calculus I) is the first course in calculus. The course begins by refamiliarizing ourselves with some of the basic properties of functions: reviewing the library of "basic" functions, creating new functions from old, associating a graphical depiction to a function, and defining the notion of an inverse function. The central concept of calculus --- the limit --- is then introduced and explored as a tool for answering the so-called "tangent line problem." In addition to exploring limits in a handful of contexts, we'll see how this notion of "derivative" can be used to create a new function from our original function, and we'll explore how these two functions are related to each other. We then spend a number of days developing some standard rules for differentiation, after which we apply the derivative to answer a handful of questions (e.g., where does a function take on its maximum and minimum values?). We finish the class with a few days of integration theory; this begins with a geometric question quite different from the "tangent line problem," but ultimately we'll see that it has extremely close connections to the derivative operator.

The course webpage is here.

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.

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.

One of the surprises in analysis is how useful notions like closed or open --- and especially compactness --- are in proving analytic statements. Indeed, after slogging through a few weeks of notions like these, one can typically move through the whole of single-variable differential calculus in about a week. The central place these concepts play in the development of such a fundamental mathematical discipline hints at how powerful these tools can be, and so it's natural to see where else these concepts might be useful. Math 307 (Topology) does precisely this, using the notion of openness as an axiomatic starting point for developing a theory that can sit entirely on its own. As with other abstractions performed in mathematics (e.g., eschewing $\mathbb{R}^n$ in favor of studying general $n$-dimensional vector spaces over $\mathbb{R}$), this axiomatic approach is both foreign and powerful: what does one lose when openness isn't defined in the way we're used to, and how do we both leverage and ignore our intuition in familiar topological settings?

We'll start with the central definition of the course --- topological space --- and develop a slew of familiar and unfamiliar examples. We'll see some concepts and behavior from analysis that hold in more general topological spaces, but we'll also see that topology has room for plenty of "pathological" examples that showcase just how nice $\mathbb{R}^n$ is (topologically speaking). We'll also see that there's a way to describe when two objects are "the same" topologically, and we'll examine the kinds of properties that are (and aren't!) preserved under this topological equivalence.

The course webpage is here.

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 some physical quantities like center of mass. 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.

As the name suggests, Math 115 (Calculus I) is the first course in calculus. The course begins by refamiliarizing ourselves with some of the basic properties of functions: reviewing the library of "basic" functions, creating new functions from old, associating a graphical depiction to a function, and defining the notion of an inverse function. The central concept of calculus --- the limit --- is then introduced and explored as a tool for answering the so-called "tangent line problem." In addition to exploring limits in a handful of contexts, we'll see how this notion of "derivative" can be used to create a new function from our original function, and we'll explore how these two functions are related to each other. We then spend a number of days developing some standard rules for differentiation, after which we apply the derivative to answer a handful of questions (e.g., where does a function take on its maximum and minimum values?). We finish the class with a few days of integration theory; this begins with a geometric question quite different from the "tangent line problem," but ultimately we'll see that it has extremely close connections to the derivative operator.

The course webpage is here.

In elementary probability theory, the fundamental object of interest is a collection of outcomes from a given experiment or observation (the so-called sample space $\Omega$). One then studies probability functions on this space: functions $\mathbb{P}:\wp(\Omega) \to [0,1]$ which associated a number between $0$ and $1$ to subsets of $\Omega$ according to some mild axioms, namely that $\mathbb{P}(\Omega)=1$ and that $\mathbb{P}$ is ``additive" on (countable) disjoint sets. Often this analysis is broken into two diametrically opposed cases, those in which the objects in question are discrete (in which a healthy dose of combinatorics is often what is needed to resolve questions), and those where they are instead continuous (in which case one uses the tools of calculus --- particularly integrals --- are the key integredient). The first pass at probability, however, often passes over subtleties which become more glaring as one's mathematical sophistication grows: is it feasible to attach a probability to all subsets of $\Omega$ in a sensible way? how do probability functions behave under limits? is the bifurcation between continuous and discrete accurate? and how does one resolve probability functions which are modeled by non-integrable functions? In this course we give a proper treatment of probability by starting with a general approach --- measure theory --- that unifies the discrete and continuous cases. We then review the key facets of probability theory (random variables, expectation, etc.) in this light, and use it to answer questions which the naive theory cannot easily resolve.

The course webpage is here.

When a person thinks of algebra, they typically think of a process used to solve polynomial equations. Modern algebra, the subject of Math 305, seems to break from this theme, instead focusing on axiomatic systems that encompass some familiar algebraic objects, but without much reference to solving polynomials. In this course, we show how the tools of modern algebra really do connect back to the backbone of algebra by developing Galois theory. In its purest form, Galois theory is an attempt to create a catalog of intermediate fields for a given field extension. More concretely, this means if we're given fields $E$ and $F$ with $F \subseteq E$, Galois theory seeks to find an object which parameterizes all those fields $L$ with $F \subseteq L \subseteq E$. On the surface, this problem doesn't seem to be related to solutions to polynomials, but we'll see throughout the course that field extensions are intimately connected to solutions to (irreducible) polynomials, and that finding "nice" solutions to those polynomials can be neatly described in terms of how "easily" one can "climb" from $F$ to $E$.

Aside from the topic of this course, it's worth noting that the pedagogical approach to this particular class is different than most: students will be responsible for the vast majority of the lecturing in the course.

The course webpage is here.

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.

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.

As the name suggests, Math 115 (Calculus I) is the first course in calculus. The course begins by refamiliarizing ourselves with some of the basic properties of functions: reviewing the library of "basic" functions, creating new functions from old, associating a graphical depiction to a function, and defining the notion of an inverse function. The central concept of calculus --- the limit --- is then introduced and explored as a tool for answering the so-called "tangent line problem." In addition to exploring limits in a handful of contexts, we'll see how this notion of "derivative" can be used to create a new function from our original function, and we'll explore how these two functions are related to each other. We then spend a number of days developing some standard rules for differentiation, after which we apply the derivative to answer a handful of questions (e.g., where does a function take on its maximum and minimum values?). We finish the class with a few days of integration theory; this begins with a geometric question quite different from the "tangent line problem," but ultimately we'll see that it has extremely close connections to the derivative operator.

The course webpage is here.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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 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.

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.

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.

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.

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.

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 ¹, 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.

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