Department of Mathematics

Category theory provides a structure and language in which to study all other types of mathematics. One has the category of sets, the category of groups, the category of abelian groups, etc. In studying general category theory one obtains new techniques for proving results that were otherwise out of reach or incredibly difficult. It allows us to study the relationship between seemingly different structures, such as geometric things and algebraic things. Sometimes, category theory inspires a new question whose answer has nothing to do with category theory in general but is of great use. In short, knowing a bit of category theory can help in almost every field of mathematics. 

Suggested Book(s):

1. An Introduction to Category Theory by Harold Simmons
2. More advanced texts as determined by student interest

Algebraic Varieties are the central object of study in Algebraic Geometry. A variety is the set of common zeroes of a collection of polynomials. The aim of this project is to understand Hilbert Nullstellensatz for Affine Varieties. We will start with a review of polynomial rings and field theory. After doing some commutative algebra including Hilbert Basis theory and Noether normalization, we will focus on Theory of Affine Varieties and Nullstellensatz. If time permits, we will do lots and lots of examples together and some generic properties of varieties. 

Prerequisites:

Some knowledge of rings and fields is essential, knowledge of pointset topology would be helpful but not essential. 

Suggested Book(s):

1. Introduction to commutative algebra by Atiyah and MacDonald.

Cohomology theories arise naturally in many contexts in algebra and are a important tool in all branches of algebra. Homological algebra is the branch of mathematics that studies (co)homology in a general algebraic setting.  The formalism of derived functors is an abstraction of all such theories. In this project, we will learn about derived functors and their instances like Ext, Tor and group cohomology. Depending on time and interest, we may also take a brief look at spectral sequences and their applications.

Homological algebra is a useful tool to understand many of the algebraic behaviors of abelian categories. Instead of learning such properties case by case, we will learn the description of those properties in the language of categories. The language of category theory will not only provide a new aspect of those algebraic properties in a general setting, but also give some specific new results. This course will be helpful for students who want to become a math graduate students working in topology, algebra, or number theory. This course will cover additive and abelian categories, derived functors, sheaves, and cohomology (including group cohomology, simplicial cohomology, and sheaf cohomology). 

Prerequisites:

Algebra 1 (Math 131a) or equivalent (Basic properties of groups and rings).

Suggested Book(s):

1.  Sur quelques points d’alg`ebre homologique by Alexander Grothendieck
2. Algebraic Topology by Allen Hatcher
3. An Introduction to Homological Algebra by Charles A. Weibel

The mathematical study of knots and links began in the 19th century when physicists proposed that knotted and linked vortices in the aether might provide an accurate model for atoms and molecules. Although the Michelson-Morley experiment in 1887 effectively disproved the existence of the aether and put paid to this physical interpretation of knot theory, the early 20th century and the birth of topology brought to light deep connections between knot theory and the study of three and four dimensional manifolds. Many questions about 3- and 4-manifolds can be reformulated into questions about the existence and properties of certain knots and links and these questions have far reaching implications in physics and other areas of mathematics. The last fifty years have seen an explosion of beautiful theorems about knots and links, several of which have earned Fields Medals for their authors, and knot theory is now a prominent, active area of research. Dale Rolfsen's book "Knots and Links" gives an exposition of the classical theory of knots and provides the background needed to understand the discoveries of the last fifty years. The aim of this reading program will be to learn the major theorems detailed in this book and to understand how knot theory interacts with the other areas of mathematics.

Suggested Book(s):

1. Knots and Links by Dan Rolfsen

In this reading program we will explore the magnificent world of partial differential equations. Starting with the classical equations: transport equation, harmonic equation, heat equation, and wave equation, we will explore the basic ideas of how people approach and perceive PDEs of different types. Later we shall focus on the theory of elliptic PDEs, which plays a fundamental role in a lot of modern geometry theories. The goal is to understand weak solutions, regularity, and maximum principle etc. of such PDEs.

Suggested Book(s):

1. Partial Differential Equations by L. Evans
2. Elliptic Partial Differential Equations of Second Order by D. Gilbarg and N. Trudinger

One of the central problems in algebraic number theory is to study the extensions of number fields, to which one of the methods is to study the ramification properties of the extensions. In chapter V of Kato’s book, one sees that the desired ramification properties of an abelian extension over the rational number field Q can be achieved via looking into a certain type of known fields, the cyclotomic fields, hence class field theory is involved. Roughly speaking, class field theory is the generalization of the above idea to any abelian extension of number fields, instead of focusing on only abelian extension over Q. It can be stated in different ways while the goal of this project is to understand the one using the Artin map. We will first work with some motivating examples in number theory, such as the two square theorem, then turn to the general theory of number fields. After that, we will explain the significance of the examples with the theory and introduce class field theory.

Prerequisites:

Familiarity with rings, field extensions, and the concept of Galois groups is suggested, but can be regarded as part of the project, hence not required.

Suggested book(s):

1. Number Theory 1: Fermat’s Dream by Kazuya Kato, Nobushige Kurokawa and Takeshi Saito.
2. Number Theory 2: Fermat’s Dream by Kazuya Kato, Nobushige Kurokawa and Takeshi Saito.
3. Algebraic Theory of Numbers by Pierre Samuel.

Riemann surfaces have a wide range of connections with modern subjects in mathematics, and is interesting as a subject itself. During the semester we will start with the basic definition with a mimimal background of complex analysis, then move to applications in differential geometry and topology. The final goal is to introduce the uniformisation theorem, Riemann-Roch thoerem, and moduli space of Riemann surfaces. 

Suggested Book(s):

1. Riemann Surfaces by S. Donaldson

In this DRP, we will read the book Heights of Polynomials and Entropy in Algebraic Dynamics by Thomas Ward and Graham Everest. The height of a univariate polynomial with complex coefficients is a measure (loosely speaking) of the complexity of the polynomial; it is defined to be the maximum of the magnitudes of its coefficients. The entropy of a dynamical system is a measure of the complexity of the orbit structure of the dynamical system, whose definition is a bit lengthy but elementary. This book describes the connections between these two notions of complexity using explicit examples; I hope an undergraduate participant will find it interesting to see how dynamical systems can be used to study number theory.

Suggested Book(s):

1. Heights of Polynomials and Entropy in Algebraic Dynamics by Thomas Ward and Graham Everest