You've done linear algebra, so what about solving equations that aren't linear? Algebraic geometry is the study of geometric objects coming from algebraic stuctures and using both geometry and algebra to understand the other better. Much like vector spaces are solutions to linear equations, we'll learn about how solutions to polynomial equations can be viewed geometrically and begin an exploration of affine varieties. We'll begin by getting a glimpse into what algebraic geometry is all about, and quickly realize that there's a lot of cool algebra that geometry elucidates. We'll encounter a little commutative algebra, some topology, and even some category theory! Time permitting, we may explore more advanced topics such as an introduction to curves.

Suggested References: Elementary Algebraic Geometry (Hulek, 2003)

Prerequisites: Math 31 or the equivalent

Representation theory takes the perspective that the right way to understand algebraic structures is to understand what they “do.” More precisely, a representation of a group is a vector space on which the group acts by linear transformations. This perspective has applications in physics and opens up a beautiful mathematical theory of its own. We will focus on the representation theory of finite groups and how it can be understood by looking at what are called “irreducible” representations, which act like atoms making up other representations.

Suggested References: TBD

Prerequisites: Math 31 or the equivalent

How do people make decisions? How do people in a group interact? How and why do friendships form and break? These are just a few questions that can be answered using mathematical models. This project could take many different paths, depending on interest. We could study networks, game theory, evolutionary dynamics, voting systems, coordination, etc. Before starting, we will discuss what topic and text would be best for you.

Suggested References: Newman's Networks, Nowak's Evolutionary Dynamics, Arrow's Social Choice and Individual Values, etc

Prerequisites: Linear algebra recommended for some topics

In a normal course on point-set topology one discusses compactness, connectedness, continuity, and perhaps metric spaces or manifolds if time permits. Thus many of the beautiful results of the 20th century are not shown. Some of the topics for ”advanced” point-set topology include the metrization theorems of Urysohn, Smirnov, Nagata, and Bing, the concept of paracompactness (which now plays a huge rule in the study of smooth manifolds and geometry), the study of locally compact Hausdorff spaces, topological groups, and perhaps most beautiful of them all, continuum theory and it’s amazing applications to the topology of the Euclidean plane. If there’s interest, one could even discuss the impossibility of some topological problems (instances of Godel’s incompleteness theorems realizing themselves in the real world).

Suggested References: Topology (Mukres, 2nd Edition, 2000), General Topology (Kelley, 1955), Topology Volume II (Kuratowski, 1968)

Prerequisites: Math 54 or the equivalent

Most undergraduates have seen the basics of set theory, the notion of union, intersection, Cartesian products, etc., and perhaps some useful theorems like De Morgan’s Laws and the distributive properties. But from there we often jump straight to higher subjects leaving the bulk of set theory behind. Many problems in the foundations of mathematics, such as Russell’s paradox, are resolved in ZFC and many constructions are able to be made explicit. Topics from ZFC could include the axioms such as regularity, pairing, choice, etc., and their uses, as well as constructions of the natural numbers, the integers, and the reals. In addition we could also tackle Zorn’s lemma and the well-ordering theorem, results that are used in abundance in analysis and algebra, and see how they’re equivalent to the axiom of choice.

Suggested References: Naïve Set Theory (Halmos), Set Theory: An Introduction to Independence Proofs (Kunnen), Set Theory (Jech, Third edition, 2006)

Prerequisites: Proof Based class

Perhaps the most beautiful form of mathematics since it is very pictorial and visual, the study of topological manifolds is the study of spaces which locally look like Euclidean space R n. Such a simple concept has crept into every area of mathematics in the 20th and 21st centuries, including algebra, analysis, algebraic geometry, physics, differential geometry, computer graphics, and more. One of the main problems in topological manifolds is proving if two are topologically equivalent, and classifying all manifolds of dimension n. This problem is not too difficult in dimension 2, nearly impossible in dimension 3, and provably impossible in dimension 4. We’ll study some of the tools (like homotopy, homology, and cohomology) that are used to tackle these problems, as well as learn some of the classic topological results like Brauer’s invariance of domain, his fixed-point theorem, and more.

Suggested References: Introduction to Topological Manifolds (Lee)

Prerequisites: Math 54 or the equivalent

A real number is a Cauchy sequence of rational numbers that get closer and closer together. But what happens when we change what it means for two rational numbers to be "close"? Enter the strange and beautiful world of p-adic analysis; a world where every triangle is isosceles, open sets are closed, and every point in a ball is its center. Topics in p-adic analysis include Ostrowski's Theorem, which classifies the absolute values on the rational numbers, Hensel's Lemma, a counterpart to Newton's method, p-adic integration, and p-adic interpolation of continuous function. We can learn arithmetic, algebra, and analysis in a p-adic univerrse, and we will find (with patience) that the theory here is in many ways simpler and more elegant than the real analysis that you grew up with.

Suggested References: p-adic Numbers, p-adic Analysis, and Zeta-Functions (Koblitz)

Prerequisites: Math 25 and Math 35 or the equivalent

Calculus is often viewed as the study of limits, or in other words, the study of how a function behaves when you get arbitrarily close to a particular point. But neither Newton nor Leibniz thought of calculus in this way; rather, they imagined the behavior of a function when you get infinitely close to a particular point. Nonstandard analysis formalizes this intution by enriching the real numbers with new "hyperreal" numbers that are infinitely large, or infinitely small, but in any case behave very much like the real numbers you've studied already. Explore how a little bit of logic and a different perspective clarifies and reformulates calculus in a way that is simple, intuitive, and powerful. Topics of study include the construction of the hyperreals, nonstandard definitions and nonstandard proofs of classic results in real analysis, and transfer principle, which lets us carry information from real analysis to nonstandard analysis and back again.

Suggested References: Lectures on the Hyperreals: An Introduction to Nonstandard Analysis (Goldblatt)

Prerequisites: Math 25 and Math 35 or the equivalent

The prime numbers 2, 3, 5, 7, 11, 13, ... are one of the most important sequences in mathematics. But what does the average prime number look like? Does the sum 1/2 + 1/3 + 1/5 + ... converge, or diverge? Are there infinitely many primes of the form p = 3n + 1? Questions like these undergird the rich and compelling discipline of analytic number theory. On average, how many factors does a natural number have? We can answer all of these questions together, as we apply techniques from basic algebra and complex analysis to get a deeper understanding of the natural numbers and how they behave.

Suggested References: Introduction to Analytic Number Theory (Apostol)

Prerequisites: Math 71 or the equivalent, and Math 43 or equivalent

A knot is an embedding of the circle into three-dimensions. It is just a tangled up circle sitting in space. Despite this simple description, it can be very hard to tell two such knots apart. Classically, mathematicians used easy to describe but hard to compute invariants of knots to tell them apart: crossing number, genus, unknotting number, slice genus, etc. Eventually, they progressed to associating polynomials to knots: the Alexander polynomial, Jones polynomial, HOMFLY polynomial, etc. The current state of knot theory is to associate even more complicated invariants, categorifications of these polynomials, to knots, to tell them apart, and in many cases, give us information about the more classical invariants. We can take a tour of these classical invariants, then through polynomials and onto categorifications. We will see lots of pictures and diagrams on the way.

Instead of considering pure math, we could also look for applications of knots, particularly to protein folding. There is a relatively new way of considering proteins in the language of knots, and we could learn about that and see what we can see. Whatever interests us the most.

Suggested References: The Knot Book (Adams), Knots and Links (Rolfsen), An Introduction to Knot Theory (Lickorish)

Prerequisites: Math 31 would probably be useful, and maybe Math 54, but don’t let those stand in the way of you learning interesting math.