Core Areas of Study

Each mathematics major takes courses in a diverse array of different areas of study within the discipline. All students majoring in mathematics take, at minimum, two courses from three of the areas listed below. Students pursuing the intensive major must take courses in the areas of Algebra, Real Analysis, and Complex Analysis.


Analysis

MATH 315b (Intermediate Complex Analysis): This class is a continuation of the introductory course in complex analysis (MATH 310a), studying more advanced properties of holomorphic and meromorphic functions. The course covers the theory of entire functions, including Jensen’s formula and the Hadamard factorization theorem. An in depth study of the Gamma function and the Riemann Zeta function are also among the commonly discussed topics, two functions with important applications in fields across mathematics. Other famous functions, such as Bessel functions and the Airy function, are studied with an eye to their asymptotic behavior, as is the use of complex analysis to study the asymptotic behavior of functions more generally.

In addition to the use of purely analytic techniques, the course highlights the fundamental importance of geometry to complex analysis. Conformal mappings, which preserve certain aspects of the geometry of the complex plane, are of utmost importance and form a central topic of the course. Many famous theorems are studied here, including the Riemann mapping theorem, the Schwarz-Christoffel integral, and Picard’s little theorem. A brief introduction to hyperbolic geometry is also discussed.

Complex analysis is a widely applicable field of mathematics, with applications in many different disciplines of mathematics. Various applications are often discussed in this course, such as elliptic functions and their relation to the theory of elliptic curves.


MATH 325b (Functional Analysis): Functional analysis studies infinite-dimensional vector spaces, like spaces of sequences, continuous functions, and measures, which differ from finite-dimensional vector spaces because topology now becomes crucial to the structure of the space itself.  We need this topology in order to talk about convergence, and in the process we provide a framework that recasts classical results of real analysis, such as Fourier analysis.  There are two important ways such an infinite-dimensional vector space can arise: an inner product leads to the study of Hilbert spaces, while a norm leads to the study of Banach spaces, which are more general.  Our basic examples of these spaces are the $L^p$ spaces, normed spaces of functions defined by the rate at which the constituent functions decay.  The course might also cover the basic structure theory of Banach algebras and $C^*$ algebras, for which there is a celebrated theorem due to Gelfand and Naimark.  It is helpful to have knowledge of Lebesgue integration as well as point set topology; while MATH 320 (Measure Theory) is officially a prerequisite, it is not strictly necessary.  This subject is central to modern real analysis and is important for many subjects such as partial differential equations and quantum mechanics.

You can find the syllabus for the Spring 2018 course here. The course is supplemented by Conway’s A Course in Functional Analysis.
 

Algebra & Number Theory

MATH 354b (Number Theory): This course covers a smorgasbord of introductory number theory topics and can vary depending on who’s teaching it. The class starts with covering fundamental properties of prime numbers and the integers, such as unique prime factorization. After this come various topics related to modular arithmetic and the ring $\mathbb{Z}/n\mathbb{Z}$. These include the Chinese Remainder Theorem, quadratic reciprocity, and the structure of the unit group $(\mathbb{Z}/n\mathbb{Z})^*$. As an application, the basics of the RSA cryptosystem are discussed. The most abstract unit is basic algebraic number theory - the study of fields which extend $\mathbb{Q}$ (i.e. contain $\mathbb{Q}$ as a subfield) and their rings of integers (subrings which are analogous to $\mathbb{Z}$ in many ways). This class only covers quadratic extensions in any depth, and theorems about them are used to characterize solutions to some Diophantine equations. Another topic is the approximation by rational numbers and continued fractions.

This course relies on a fair bit of commutative algebra, but if you’ve taken 350 you should be adequately prepared, and if you’ve taken 370, even more so. Relative to other algebra courses (350, 370, and 380 in particular) this course is on the easy side, and substantially less abstract.
What to take before: 350
What to take after: 373
 

Statistics & Applied Math

Geometry & Topology

MATH 430b (Introduction to Algebraic Topology): This class is a fast paced introduction to the most important techniques of point set topology and one of the most fundamental objects in algebraic topology: the fundamental group. The course follows Munkres’s Topology and spends the first few weeks of the course covering the most essential results of point set topology. Experience with basic point set topology from an analysis course such as MATH 230/231, MATH 300, or  MATH 301 is useful here as the pace is quick and difficult to follow with no knowledge of point set topology. Attention is paid to the definitions of topological spaces, product spaces and quotient spaces, and properties of these spaces such as connectedness and compactness. Helpful examples of unintuitive topological spaces are also provided, leading to a more complete understanding of the material.

With these essential properties established, the course introduces the notions of homotopy and the fundamental group. Examples of explicit computation of the fundamental group, such as that of the circle, are discussed, as are applications of basic homotopy theory such as the Borsuk-Ulam theorem. The Seifert-van Kampen theorem is also introduced, and significant time is devoted to the study of its use in the computation of the fundamental group. A knowledge of elementary group theory, such as that discussed in MATH 350, is useful for this more algebraic part of the course.

The course ends with a discussion of covering spaces, including the construction of the universal cover and the classification of covering spaces. Various applications are also discussed, such as the use of this theory in determining the fundamental group of a graph. A quick proof that subgroups of free groups are free is able to be given with this theory.

Logic & Foundations

Math 244a (Discrete Mathematics): This course is similar to CPSC 202 in that it handles common concepts in discrete mathematics - the study of discrete, usually finite mathematical structures as opposed to continuous ones - except this course does so in a more mathematical way. Still, this course tries to tie in its math concepts with computer science or statistics ones when it can, and often relies on algorithmic kinds of thinking. The course primarily handles three big topics - enumeration, graph theory, and probability. Enumeration and probability rely on how to count things, which is not as straightforward as it might seem. They frequently require some clever thinking and require your logical thought process to be completely sound; otherwise, it is easy to make mistakes. Specific topics include binomial coefficients, principle of inclusion and exclusion, generating functions, probability spaces, expectation, and random variables. Besides these two topics, there is a heavy emphasis on graph theory - the study of sets of vertices and edges, not the usual “graphs” that depict functions. Graphs are abstract in that they represent objects and the relations between them - for example, one can construct a graph where its vertices are “people”, and its edges are “friendships” - and take some time to get used to. The study of graphs is very rich and helps one to understand mathematical structures. Specific topics include connectivity of graphs, trees, coloring, Euler’s formula, algorithms, and big-O notation.

Not much mathematical background is required for this course, except familiarity with sets, functions, induction, proofs, and lots of notation. This course is recommended for all math or computer science majors, as it is very insightful and provides a new kind of thinking that branches out from the typical analysis or algebra ways of thinking. It develops your step-by-step reasoning abilities and encourages you to tackle a problem from multiple angles by considering its structure.

What to take after: 345 (Modern Combinatorics)

Math 345b (Modern Combinatorics): This course covers more modern questions in combinatorics and integrates ideas from linear algebra, analysis, and number theory. Every year, this course changes its primary focus, including areas such as the probabilistic method, random graphs, random matrices, pseudorandomness in graph theory and number theory, Szemeredi’s theorem and lemma, and Green-Tao’s theorem. More specifically, in Spring 2018 it has studied functions from bit strings to a single bit, known as boolean functions. These functions figure prominently in combinatorics, theoretical computer science (notably cryptography), and social choice theory. Regardless of its topic, the class presents a more sophisticated and detailed look into discrete math than Math 244, and it is recommended for math or computer science majors looking to apply their discrete math knowledge to advanced topics. This class assumes familiarity with and makes frequent use of graphs, expectation, probability, and it progresses at a fast and challenging pace.

What to take before: 244 (Discrete Mathematics)