To help guide you in selecting your math courses, here are some guidelines:

**Remember to consider the professor while choosing courses.**Course material, difficulty, and quality can vary dramatically from year to year depending on who is teaching the course. Thus, we strongly recommend shopping a variety of courses with different professors if possible and considering the professor while selecting courses, in addition to prerequisites and course material. Professors who have consistently received praise from students^{1}include Professors Dennis Gaitsgory, Joe Harris, Peter Kronheimer, Jacob Lurie, Curtis McMullen, Martin Nowak, and Wilfried Schmid. Note that this section only represents general student consensus over the past few years, and your experience with any particular professor may vary depending on your learning style and the course.- Prerequisites listed here are not strictly enforced and may vary from year to year. If you’re not sure whether you’re qualified to take a particular course, talk to the professor and ask what material he/she is assuming you know. It is also perfectly acceptable to learn the material taught in any particular course on your own via reading a textbook or something similar.
- Courses in this section are
**not**ordered by difficulty. They are grouped by subject and numerically ordered. See expected workload and prerequisites for an estimate of difficulty of any particular course.

Courses marked with an asterisk (*) are courses for which we have insufficient information to make a thoroughly accurate recommendation from a student perspective. If you have taken one of these courses, please contact GIIM board and we will incorporate your perspective.

The information here is taken from the course catalog and supplemented with information from students.

### Undergraduate Courses

#### Tutorials and Reading Courses

**Math 60r: Reading Course for Senior Honors Candidates**

*Course Content:* Math 60r is designed for students writing senior theses, in order to provide more time to work on the senior thesis. It is not mandatory to sign up for Math 60r if you are writing a senior thesis; the only real purpose of this course is to free up time to work on the thesis. It can be taken senior fall or spring or both.

**Math 91r: Supervised Reading and Research**

*Course Content:* Math 91r is an opportunity for students who engage in a directed reading course about a topic not typically covered in another math course, under the supervision of a faculty member, Benjamin Peirce fellow, or postdoc. This is a great opportunity to learn about a unique math topic in a different style from the typical problem set/exam cycle of a normal math class. Open to math concentrators in any year. The student needs to take the initiative to ask the faculty member; a group of students can also ask a faculty member to do a reading course together.

**Math 99r: Tutorial**

*Course Content:* Math 99r is a supervised small group (less than 8 students) tutorial that results in a final paper or presentation. Previous topics include Morse Theory, Partitions, Combinatorial Game Theory, Complex Multiplication, and Arithmetic of Quadratic Forms. It may be repeated for course credit with permission from the Director of Undergraduate Studies (Professor Jacob Lurie), but only one tutorial may count for concentration credit. Tutorials usually automatically satisfy the concentration expository requirement.

#### 110s: Analysis Courses

**Math 110: Vector Space Methods for Differential Equations** (spring)

*Course Content:* Math 110 covers the theory of finite and infinite-dimensional inner product spaces and then applies it to solve differential equations, discussing existence and uniqueness theorems, Sturm-Liouville systems, Fourier series and transforms, and eigenvalue problems.

*Prerequisites:* Computational linear algebra and multivariable calculus (Math 19a/b, 21a/b, 23a/b, 25a/b, or 55a/b) and exposure to proofs (Math 23a/b, 25a/b, 55a/b, 101, 112, 121, or 130). Real analysis (Math 23a/b, 25b, 55b, or 112) may be helpful.

*Expected Workload:* 5 – 10 hours/week

**Math 115: Methods of Analysis** (spring)

*Course Content:* Math 115 covers functional and complex analysis with a special emphasis on topics like Fourier analysis, Hilbert spaces, Laplace’s equations, Legendre functions, and other topics of relevance to physics students. This course is a potentially good substitute for Math 113/114 for physics students less interested in pure math.

*Prerequisites:* Real analysis (Math 23a/b, 25b, 55b, or 112) and linear algebra (Math 21b, 23a/b, 25a, or 55a).

*Expected Workload:* 3 – 7 hours/week

**Math 116*: Real Analysis, Convexity, and Optimization** (spring)

*Course Content:* Math 116 discusses the theory of convex sets, normed infinite-dimensional vector spaces, and convex functionals, along with applications to various optimization problems.

*Prerequisites:* Computational linear algebra and multivariable calculus (Math 19a/b, 21a/b, 23a/b, 25a/b, or 55a/b) and exposure to proofs (Math 23a/b, 25a/b, 55a/b, 101, 112, 121, or 130).

*Expected Workload:* 7 – 15 hours/week

**Math 117*: Probability and Random Processes with Economic Applications** (spring)

*Course Content:* Math 117 covers probability theory, including its axioms, Lebesgue integration, conditional probability, and other topics, with special focus on applications towards economics and option pricing.

*Prerequisites:* Single-variable calculus (Math 1 or equivalent), exposure to proofs (Math 23a/b, 25a/b, 55a/b, 101, 112, 121, or 130), and real analysis (Math 23a/b, 25b, 55b, or 112).

*Expected Workload:* 3 – 7 hours/week

**Math 118r*: Dynamical Systems** (spring)

*Course Content:* Math 118r covers an introduction to dynamical systems, including existence and uniqueness of flows, qualitative study of equilibria and attractors, iterated maps, and bifurcation theory.

*Prerequisites:* Computational linear algebra and multivariable calculus (Math 19a/b, 21a/b, 23a/b, or 55a/b).

*Expected Workload:* 5 – 10 hours/week

#### 120s: Algebra Courses

**Math 124: Number Theory** (fall)

*Course Content:* Math 124 covers elementary number theory, including modular arithmetic, factorization and prime numbers, quadratic residues, continued fractions, Pell’s equations and other Diophantine equations, and other topics depending on the year.

*Prerequisites:* Group theory (Math 55a, 101, or 122), which can be taken concurrently.

*Expected Workload:* 5 – 12 hours/week

**Math 129: Number Fields** (spring)

*Course Content:* Math 129 covers introductory algebraic number theory, including the unique factorization in Dedekind domains, splitting and ramification of prime ideals, and the structure of the ideal class group, along with some basic analytic number theory such as Dedekind zeta functions.

*Prerequisites:* Familiarity with Galois theory (Math 123) and elementary number theory (Math 124). Familiarity with complex analysis (Math 113 or 55b) can help but is not essential.

*Expected Workload:* 3 – 7 hours/week

#### 130s: Geometry and Topology Courses

**Math 136*: Differential Geometry** (fall)

*Course Content:* Math 136 covers exterior differential calculus, including applications to curves and shapes in 3-space, various notions of curvature, and Riemannian geometry.

*Prerequisites:* Computational linear algebra (Math 19a/b, 21a/b, 23a, 25a, or 55a).

*Expected Workload:* 5 – 12 hours/week

**Math 137: Algebraic Geometry** (spring)

*Course Content:* Math 137 covers introductory algebraic geometry, including the affine and projective spaces, plane curves, Bezout’s theorem, singularities and genus of a plane curve, and the Riemann-Roch theorem.

*Prerequisites:* Ring theory (Math 123), abstract linear algebra and group theory (Math 55a or 122), and potentially some topology (Math 131 or potentially 55b). Algebraic geometry tends to assume a great deal of background from a variety of fields, so greater mathematical maturity may be helpful.

*Expected Workload:* 7 – 15 hours/week

#### 140s: Logic and Set Theory Courses

Courses in the 140s tend to be particularly professor-dependent and vary dramatically from year to year. Consult with your professor for the most accurate information.

**Math 145a: Set Theory I** (fall)

*Course Content:* Math 145a covers an introduction to set theory including the fundamentals of ZFC, the independence techniques (the constructible universe, forcing, the Solovay model), and the independence of the continuum hypothesis and Suslin’s hypothesis.

*Prerequisites:* No strict mathematical prerequisites, but maturity at the level of Math 19a/21a or higher is recommended. Familiarity with logic may be helpful.

*Expected Workload:* 3 – 8 hours/week

#### 150s: Miscellaneous Courses

**Math 153: Mathematical Biology – Evolutionary Dynamics** (fall)

*Course Content:* Math 153 covers introductory evolutionary dynamics from a mathematical perspective, including game theory, evolutionary games, population dynamics, evolution of languages, and epidemiology.

*Prerequisites:* Familiarity with calculus and differential equations (Math 19a/b, 21a/b, 23a/b, 25a/b, or 55a/b).

*Expected Workload:* 3 – 7 hours/week

**Math 154*: Probability Theory** (spring)

*Course Content:* Math 154 covers probability theory, including discrete and continuous random variables, distribution and density functions, conditional probability, generating functions, weak and strong laws of large numbers, and the central limit theorem. A good alternative to Statistics 110 for mathematically-minded students.

*Prerequisites:* Exposure to proofs (Math 23a, 25a, 55a, 101, 112, 121, or 130).

*Expected Workload:* 5 – 10 hours/week

**Math 155r: Combinatorics: Designs and Groups** (spring)

*Course Content:* Math 155 studies the design, or a collection of subsets of a finite set $S$ whose elements are evenly divided in some sense, such as the collection of edges of a regular graph or the collection of lines in the finite projective plane.

*Prerequisites:* Exposure to proofs (Math 23a/b, 25a/b, 55a/b, 101, 112, 121, or 130) and some computational linear algebra (Math 21b, 23a, 25a, or 55a).

*Expected Workload:* 5 – 10 hours/week

**Math 157*: Mathematics in the World** (spring)

*Course Content:* Math 157 covers the use of math in the real world to solve problems in fields such as logic, information, number theory, probability, and algorithms. This course is recommended for students with interests in more applications of math or in interview-type questions.

*Prerequisites:* Computational linear algebra (Math 19b, 21b, 23a, 25a, or 55a).

*Expected Workload:* 3 – 9 hours/week

### Graduate Courses

Note about graduate courses: These courses are significantly more fast-paced, more rigorous, and assume greater background than the corresponding undergraduate course. Do not take them unless you have a solid background and are willing to spend a significant amount of time on them. It is not necessary to have taken a graduate course to graduate as a math concentrator or go to graduate school.

You can take just one semester of any full-year course, but the fall semester is generally a prerequisite for the spring semester.

#### 210s: Analysis Courses

**Math 212a/br*: Real Analysis** (full year)

*Course Content:* Math 212a (fall) focuses on functional analysis, including Banach and Hilbert spaces, distribution, spectral theory, and the Fourier transform. Math 212br (spring) discusses functional analysis applicable to quantum mechanics, including the Stone-von Neumann theorem, Gruenwald-van Hove theorem, Ruelle’s theorem on the continuous spectrum and scattering states, Agmon’s theorem on the exponential decay of bound states, and scattering theory.

*Prerequisites:* Real and introductory functional analysis (Math 114).

*Expected Workload:* 3 – 7 hours/week

**Math 213a/br: Complex Analysis** (full year)

*Course Content:* Math 213a (fall) discusses advanced complex analysis, including expansions of holomorphic functions, Hadamard’s theorem, the Riemann mapping theorem, elliptic functions, Picard’s theorem, and Nevanlinna theory. Math 213br (spring) discusses the fundamentals of Riemann sheaves including sheaves, cohomology, potential theory, uniformization, and moduli.

*Prerequisites:* Introductory complex analysis (Math 113). Group theory (Math 122), algebraic topology (Math 131), and real analysis (Math 114) are strongly recommended.

*Expected Workload:* 5 – 12 hours/week

#### 220s: Algebra Courses

**Math 221: Algebra** (fall)

*Course Content:* Math 221 is a thorough graduate-level introduction to algebra, including Noetherian rings and modules, the Hilbert basis theorem, principal ideal decomposition, Dedekind domains, the Nullstellensatz, Galois theory, ramification of prime ideals, and representation theory of finite groups.

*Prerequisites:* A strong background in basic abstract algebra, including group theory and abstract linear algebra (Math 123).

*Expected Workload:* 7 – 15 hours/week

**Math 222*: Lie Groups and Lie Algebras** (spring)

*Course Content:* Math 222 is an introduction to Lie theory, including the classification of simple Lie groups and compact Lie groups and their representations.

*Prerequisites:* Functional analysis (Math 114), ring theory (Math 123), and the theory of manifolds (Math 132).

*Expected Workload:* 7 – 15 hours/week

**Math 223a/br*: Algebraic Number Theory** (full year)

*Course Content:* Math 223a (fall) is a graduate-level introduction to algebraic number theory, including the structure of the ideal class group, groups of units, $\zeta$- and L-functions, local fields, Galois cohomology, local class field theory, and local duality. Math 223br (spring) continues with adeles, global class field theory, duality, and cyclotomic systems.

*Prerequisites:* Introductory algebraic number theory (Math 129).

*Expected Workload:* 3 – 7 hours/week

**Math 224*: Representations of Reductive Lie Groups** (spring)

*Course Content:* Math 224 covers the theory of reductive Lie groups, including unitary representations, Harish Chandra modules, characters, the discrete series, and Plancherel’s theorem.

*Prerequisites:* Unknown; please check with your professor.

*Expected Workload:* 5 – 12 hours/week

**Math 229x*: Introduction to Analytic Number Theory** (spring)

*Course Content:* Math 229x covers introductory analytic number theory, including the Riemann zeta function, the Prime Number Theorem, Dirichlet’s theorem, discriminants, sieve methods, and analytic estimates on exponential sums.

*Prerequisites:* Complex analysis (Math 113) and Galois theory (Math 123).

*Expected Workload:* 3 – 7 hours/week

#### 230s: Geometry and Topology Courses

**Math 230a/br*: Differential Geometry** (full year)

*Course Content:* Math 230a (fall) is an introduction to differential geometry, including smooth manifolds, Riemannian geometry, symplectic geometry, Lie groups, and principal bundles. Math 230br (spring) continues with analysis on manifolds, Laplacians, Hodge theory, spin structures, Clifford algebras, Dirac operators, and index theorems.

*Prerequisites:* Smooth manifolds (Math 132) and introductory differential geometry (Math 136).

*Expected Workload:* 7 – 15 hours/week

**Math 231a/br: Algebraic Topology** (full year)

*Course Content:* Math 231a (fall) is an introduction to the key tools of modern algebraic topology, including homology, cohomology, and homotopy. Math 231br (spring) extends these tools to discus more advanced topics, including vector bundles, characteristic classes, and Bott periodicity.

*Prerequisites:* Basic point-set topology and the fundamental group (Math 131). Familiarity with manifolds (Math 132) can help.

*Expected Workload:* 6 – 12 hours/week

**Math 232a/br: Algebraic Geometry** (full year)

*Course Content:* Math 232a (fall) is an introduction to complex algebraic curves, surfaces, and varieties. Math 232br (spring) covers the classification of complex algebraic surfaces.

*Prerequisites:* Ring theory (Math 123). Smooth manifolds (Math 132), introductory algebraic geometry (Math 137), and introductory algebraic topology (Math 131) can also help.

*Expected Workload:* 7 – 15 hours/week

**Math 233a/br*: Scheme Theory** (full year)

*Course Content:* Math 233a (spring) is an introduction to the theory and structure of schemes. Math 233br (uncertain, not offered this year) covers sheaves and sheaf cohomology.

*Prerequisites:* Unknown; please check with your professor.

*Expected Workload:* 10 – 20 hours/week

#### 240s: Miscellaneous Courses

**Math 243*: Evolutionary Dynamics** (spring)

*Course Content:* Math 243 covers advanced topics in evolutionary dynamics, with an emphasis on seminars and research projects.

*Prerequisites:* Mathematical biology (Math 153)

*Expected Workload:* 1 – 5 hours/week

Return to the table of contents.

^{1. This list includes all tenured faculty with Q scores averaging above 4.5 over the past 3 years. ↩}