# Graduate Courses

*Horng-Tzer Yau*

2022 Spring (4 Credits)

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Schedule: **
MW 10:30 AM - 11:45 AM

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Instructor Permissions: **
None

**
Enrollment Cap: **
n/a

Functional analysis and applications. Topics may include the spectral theory of self-adjoint operators, evolution equations and the theorem of Hille-Yosida, distributions, Sobolev spaces and elliptic boundary value problems, calculus of variations with applications to non-linear PDE.

- Recommended Prep:
- Knowledge of the material in Mathematics 114.

- Requirements:
- Prerequisite: Mathematics 114

Additional Course Attributes:

Attribute | Value(s) |
---|---|

All: Cross Reg Availability | Available for Harvard Cross Registration |

FAS: Course Level | Primarily for Graduate Students |

FAS Divisional Distribution | Science & Engineering & Applied Science |

*Yum-Tong Siu*

2020 Fall (4 Credits)

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Schedule: **
TR 03:00 PM - 04:15 PM

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Instructor Permissions: **
None

**
Enrollment Cap: **
n/a

Fundamentals of complex analysis, and further topics such as elliptic functions, canonical products, conformal mappings, the zeta function, and prime number theorem, and Nevanlinna theory. Prerequisites: Basic complex analysis, the topology of covering spaces, differential forms.

- Recommended Prep:
- Basic complex analysis, topology of covering spaces, differential forms.

Additional Course Attributes:

Attribute | Value(s) |
---|---|

FAS: Course Level | Primarily for Graduate Students |

FAS Divisional Distribution | Science & Engineering & Applied Science |

All: Cross Reg Availability | Available for Harvard Cross Registration |

*Yum-Tong Siu*

2022 Spring (4 Credits)

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Schedule: **
TR 03:00 PM - 04:15 PM

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Instructor Permissions: **
None

**
Enrollment Cap: **
n/a

Fundamentals of algebraic curves as complex manifolds of dimension one. Topics may include branched coverings, sheaves and cohomology, potential theory, uniformization and moduli.

- Recommended Prep:
- Knowledge of the material in Mathematics 213a.

Additional Course Attributes:

Attribute | Value(s) |
---|---|

All: Cross Reg Availability | Available for Harvard Cross Registration |

FAS: Course Level | Primarily for Graduate Students |

FAS Divisional Distribution | Science & Engineering & Applied Science |

2021 Fall (4 Credits)

**
Schedule: **
TR 10:30 AM - 11:45 AM

**
Instructor Permissions: **
None

**
Enrollment Cap: **
n/a

A graduate level course in commutative algebra. Topics may include, but are not limited to, Hilbert’s Basis Theorem and Nullstellensatz, ideals, spectra, localization, primary decomposition, Artin-Rees Lemma, flat families and Tor, completions of rings, Noether Normalization, systems of parameters, DVRs, dimension theory, Hilbert-Samuel polynomials, depth, Cohen-Macaulay and regular rings, homological methods.

- Recommended Prep:
- Mathematics 123

Additional Course Attributes:

Attribute | Value(s) |
---|---|

FAS Divisional Distribution | Science & Engineering & Applied Science |

All: Cross Reg Availability | Available for Harvard Cross Registration |

FAS: Course Level | Primarily for Graduate Students |

*Cesar Cuenca*

2021 Fall (4 Credits)

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Schedule: **
MW 03:00 PM - 04:15 PM

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Instructor Permissions: **
None

**
Enrollment Cap: **
n/a

Lie theory, including the classification of semi-simple Lie algebras and/or compact Lie groups and their representations.

- Recommended Prep:
- Knowledge of the material in Mathematics 114, 123 and 132

Additional Course Attributes:

Attribute | Value(s) |
---|---|

All: Cross Reg Availability | Available for Harvard Cross Registration |

FAS Divisional Distribution | Science & Engineering & Applied Science |

FAS: Course Level | Primarily for Graduate Students |

*Mark Shusterman*

2021 Fall (4 Credits)

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Schedule: **
TR 3:00 PM - 4:15 PM

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Instructor Permissions: **
None

**
Enrollment Cap: **
n/a

A graduate introduction to algebraic number theory. Topics: the structure of ideal class groups, groups of units, a study of zeta functions and L-functions, local fields, Galois cohomology, local class field theory, and local duality.

- Recommended Prep:
- Knowledge of the material in Mathematics 129.

Additional Course Attributes:

Attribute | Value(s) |
---|---|

FAS: Course Level | Primarily for Graduate Students |

All: Cross Reg Availability | Available for Harvard Cross Registration |

FAS Divisional Distribution | Science & Engineering & Applied Science |

*Melanie Wood*

2022 Spring (4 Credits)

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Schedule: **
MW 09:00 AM - 10:15 AM

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Instructor Permissions: **
None

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Enrollment Cap: **
n/a

Continuation of Mathematics 223a. Topics: adeles, global class field theory, duality, cyclotomic fields. Other topics may include: Tate’s thesis or Euler systems.

- Recommended Prep:
- Knowledge of the material in Mathematics 223A

Additional Course Attributes:

Attribute | Value(s) |
---|---|

FAS: Course Level | Primarily for Graduate Students |

All: Cross Reg Availability | Available for Harvard Cross Registration |

FAS Divisional Distribution | Science & Engineering & Applied Science |

*Fabian Gundlach*

2022 Spring (4 Credits)

**
Schedule: **
MW 10:30 AM - 11:45 AM

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Instructor Permissions: **
None

**
Enrollment Cap: **
n/a

Fundamental methods, results, and problems of analytic number theory. Riemann zeta function and the Prime Number Theorem; Dirichlet’s theorem on primes in arithmetic progressions; lower bounds on discriminants from functional equations; sieve methods, analytic estimates on exponential sums, and their applications.

- Recommended Prep:
- Knowledge of the material in Mathematics 113 and 123

Additional Course Attributes:

Attribute | Value(s) |
---|---|

FAS Divisional Distribution | Science & Engineering & Applied Science |

All: Cross Reg Availability | Available for Harvard Cross Registration |

FAS: Course Level | Primarily for Graduate Students |

*Martin Lesourd*

2021 Fall (4 Credits)

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Schedule: **
MW 01:30 PM - 02:45 PM

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Instructor Permissions: **
None

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Enrollment Cap: **
n/a

Smooth manifolds (vector fields, differential forms, and their algebraic structures; Frobenius theorem), Riemannian geometry (metrics, connections, curvatures, geodesics), Lie groups, principal bundles and associated vector bundles with their connections, curvature and characteristic classes. Other topics if time permits.

- Recommended Prep:
- Knowledge of the material in Mathematics 132 and 136

Additional Course Attributes:

Attribute | Value(s) |
---|---|

FAS: Course Level | Primarily for Graduate Students |

FAS Divisional Distribution | Science & Engineering & Applied Science |

All: Cross Reg Availability | Available for Harvard Cross Registration |

*Piotr Pstragowski*

2021 Fall (4 Credits)

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Schedule: **
TR 09:00 AM - 10:15 AM

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Instructor Permissions: **
None

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Enrollment Cap: **
n/a

Covering spaces and fibrations. Simplicial and CW complexes, Homology and cohomology, universal coefficients and Künneth formulas. Hurewicz theorem. Manifolds and Poincaré duality.

- Recommended Prep:
- Knowledge of the material in Mathematics 131 and 132

Additional Course Attributes:

Attribute | Value(s) |
---|---|

FAS Divisional Distribution | Science & Engineering & Applied Science |

All: Cross Reg Availability | Available for Harvard Cross Registration |

FAS: Course Level | Primarily for Graduate Students |

*Michael Hopkins*

2022 Spring (4 Credits)

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Schedule: **
MW 01:30 PM - 02:45 PM

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Instructor Permissions: **
None

**
Enrollment Cap: **
n/a

Continuation of Mathematics 231a. Topics will be chosen from: Cohomology products, homotopy theory, bundles, obstruction theory, characteristic classes, spectral sequences, Postnikov towers, and topological applications.

- Recommended Prep:
- Knowledge of the material in Mathematics 231a

Additional Course Attributes:

Attribute | Value(s) |
---|---|

All: Cross Reg Availability | Available for Harvard Cross Registration |

FAS Divisional Distribution | Science & Engineering & Applied Science |

FAS: Course Level | Primarily for Graduate Students |

*Mark Shusterman*

2021 Fall (4 Credits)

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Schedule: **
TR 01:30 PM - 02:45 PM

**
Instructor Permissions: **
None

**
Enrollment Cap: **
n/a

Introduction to complex algebraic curves, surfaces, and varieties.

- Recommended Prep:
- Knowledge of the material in Mathematics 123 and 132 and 137

Additional Course Attributes:

Attribute | Value(s) |
---|---|

All: Cross Reg Availability | Available for Harvard Cross Registration |

FAS Divisional Distribution | Science & Engineering & Applied Science |

FAS: Course Level | Primarily for Graduate Students |

*Mark Shusterman*

2022 Spring (4 Credits)

**
Schedule: **
TR 03:00 PM - 04:15 PM

**
Instructor Permissions: **
None

**
Enrollment Cap: **
n/a

Introduction to the theory and language of schemes. Textbooks: Algebraic Geometry by Robin Hartshorne and Geometry of Schemes by David Eisenbud and Joe Harris. Weekly homework will constitute an important part of the course.

- Recommended Prep:
- Knowledge of the material in Mathematics 232a

Additional Course Attributes:

Attribute | Value(s) |
---|---|

All: Cross Reg Availability | Available for Harvard Cross Registration |

FAS Divisional Distribution | Science & Engineering & Applied Science |

FAS: Course Level | Primarily for Graduate Students |

*Martin Nowak*

2021 Fall (4 Credits)

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Schedule: **
TR 03:00 PM - 04:15 PM

**
Instructor Permissions: **
None

**
Enrollment Cap: **
n/a

This course introduces basic concepts of mathematical biology and evolutionary dynamics: reproduction, selection, mutation, genetic drift, quasi-species, finite and infinite population dynamics, game dynamics, evolution of cooperation, language, spatial models, evolutionary graph theory, infection dynamics, virus dynamics, somatic evolution of cancer.

- Recommended Prep:
- Mathematics 19a,b or 21a,b or 22a,b or 23a,b or 25a,b or 55a,b; or an equivalent background in mathematics.

:

Attribute | Value(s) |
---|---|

All: Cross Reg Availability | Available for Harvard Cross Registration |

FAS Divisional Distribution | Science & Engineering & Applied Science |

FAS: Course Level | For Undergraduate and Graduate Students |

*Martin Nowak*

2022 Spring (4 Credits)

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Schedule: **
TR 03:00 PM - 04:15 PM

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Instructor Permissions: **
None

**
Enrollment Cap: **
n/a

Research seminar on evolutionary dynamics, spanning mathematical and computational models of evolution in biological and social systems. Students attend a weekly lecture and conduct an original research project.

- Recommended Prep:
- Experience with mathematical biology at the level of Mathematics 153

:

Attribute | Value(s) |
---|---|

All: Cross Reg Availability | Available for Harvard Cross Registration |

FAS Divisional Distribution | Science & Engineering & Applied Science |

FAS: Course Level | Primarily for Graduate Students |

*Lauren Williams*

2021 Fall (4 Credits)

**
Schedule: **
MW 09:00 AM - 10:15 AM

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Instructor Permissions: **
None

**
Enrollment Cap: **
n/a

This course will survey one of the most exciting recent developments in algebraic combinatorics, namely, Fomin and Zelevinsky’s theory of cluster algebras. Cluster algebras are a class of combinatorially defined commutative rings that provide a unifying structure for phenomena in a variety of algebraic and geometric contexts. Introduced in 2001, cluster algebras have already been shown to be related to a host of other fields of math, such as quiver representations, Teichmuller theory, Poisson geometry, and total positivity. Cluster structures in Grassmannians have in particular been linked to integrable systems and physics. In the first part of the course I will cover the basics of cluster algebras and total positivity. In the second part of the class I will discuss recent developments and applications of the theory, including positroids, amplituhedra, and KP solitons.

Additional Course Attributes:

Attribute | Value(s) |
---|---|

FAS Divisional Distribution | Science & Engineering & Applied Science |

All: Cross Reg Availability | Available for Harvard Cross Registration |

FAS: Course Level | Primarily for Graduate Students |

*Horng-Tzer Yau*

2021 Fall (4 Credits)

**
Schedule: **
TR 10:30 AM - 11:45 AM

**
Instructor Permissions: **
None

**
Enrollment Cap: **
n/a

The first half of this class consists of reviews and proofs of several commonly used concentration inequalities in high dimensional probability theory. These include Bernstein’s and Chernoff’s inequality and the logarithmic Sobolev inequality. The second half of the class will address applications of these inequalities in graph theory, spin glasses and random matrices. References for concentration inequalities include Vershynin’s note on “High-Dimensional Probability: an introduction with applications in data science” and the chapter on the logarithmic Sobolev inequality in my book “Dynamical approach to random matrix theory”.

Additional Course Attributes:

Attribute | Value(s) |
---|---|

All: Cross Reg Availability | Available for Harvard Cross Registration |

FAS: Course Level | Primarily for Graduate Students |

FAS Divisional Distribution | Science & Engineering & Applied Science |

*Dori Bejleri*

2022 Spring (4 Credits)

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Schedule: **
MW 12:00 PM - 01:15 PM

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Instructor Permissions: **
None

**
Enrollment Cap: **
n/a

Given a collection of algebraic equations, can we find a parametrization of the set of solutions by rational functions? This is one of the most fundamental questions in algebraic geometry and leads to the notion of rationality and its various generalizations (stable rationality, unirationality, rational connectedness). In this class we will survey some classical results, recent breakthroughs and open problems in the study of rationality. Possible topics include the Lüroth problem, decomposition of the diagonal, the degeneration method, the behavior of rationality in families, (stable) rationality of hypersurfaces, etc.

Additional Course Attributes:

Attribute | Value(s) |
---|---|

All: Cross Reg Availability | Available for Harvard Cross Registration |

FAS: Course Level | Primarily for Graduate Students |

FAS Divisional Distribution | Science & Engineering & Applied Science |

*Noam D. Elkies*

2021 Fall (4 Credits)

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Schedule: **
MW 12:00 PM - 01:15 PM

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Instructor Permissions: **
None

**
Enrollment Cap: **
n/a

In number theory and algebraic geometry one often constructs a remarkable object without giving a good way to compute it explicitly, even when the object is as down-to-earth as an integer or a polynomial. We develop some of the techniques, tools, tactics, and tricks that often let us exhibit and study such objects. Most of our motivating examples are low-dimensional moduli spaces of various kinds and the structures they parametrize.

Additional Course Attributes:

Attribute | Value(s) |
---|---|

All: Cross Reg Availability | Available for Harvard Cross Registration |

FAS: Course Level | Primarily for Graduate Students |

FAS Divisional Distribution | Science & Engineering & Applied Science |

2022 Spring (4 Credits)

**
Schedule: **
TR 09:00 AM - 10:15 AM

**
Instructor Permissions: **
None

**
Enrollment Cap: **
n/a

We will start with recalling basic results on Hodge structures (pure, mixed), variations of Hodge structures and their period domains, nilpotent and sl2 orbit thorem. We will then study Hodge loci and their algebraicity through o-minimality techniques . Then we will discuss the important case of Shimura varieties and their special subvarieties. If time permits, we might also study typical and atypical intersections questions with a view towards Zilber-Pink conjectures.

Additional Course Attributes:

Attribute | Value(s) |
---|---|

All: Cross Reg Availability | Available for Harvard Cross Registration |

FAS: Course Level | Primarily for Graduate Students |

FAS Divisional Distribution | Science & Engineering & Applied Science |

*Christopher Eur*

2022 Spring (4 Credits)

**
Schedule: **
MW 03:00 PM - 04:15 PM

**
Instructor Permissions: **
None

**
Enrollment Cap: **
n/a

Tropical geometry sits at a junction between combinatorics and algebraic geometry. We survey some fundamental results and current research. Topics may include: tropicalizations of embedded varieties, tropicalizations of curves, matroids (a.k.a. tropical linear spaces), tropical compactifications and intersection theory, and tropical Hodge theory.

Additional Course Attributes:

Attribute | Value(s) |
---|---|

All: Cross Reg Availability | Available for Harvard Cross Registration |

FAS: Course Level | Primarily for Graduate Students |

FAS Divisional Distribution | Science & Engineering & Applied Science |

*Austin Conner*

2022 Spring (4 Credits)

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Schedule: **
TR 01:30 PM - 02:45 PM

**
Instructor Permissions: **
None

**
Enrollment Cap: **
n/a

A discussion of topics in complexity theory using the tools of algebraic geometry and representation theory. Possible topics: rank and border rank of tensors and the complexity of matrix multiplication; circuit complexity of linear maps and matrix rigidity; circuit complexity of polynomials and the permanent vs determinant problem. A strong background in linear algebra is required. Some experience in algebraic geometry and/or representation theory would be helpful but is not required.

Additional Course Attributes:

Attribute | Value(s) |
---|---|

All: Cross Reg Availability | Available for Harvard Cross Registration |

FAS: Course Level | Primarily for Graduate Students |

FAS Divisional Distribution | Science & Engineering & Applied Science |

*Cesar Cuenca*

2022 Spring (4 Credits)

**
Schedule: **
TR 03:00 PM - 04:15 PM

**
Instructor Permission: **
None

**
Enrollment Cap: **
n/a

The class will cover the theory of the Riemann-Hilbert problem that is sufficient for the study of the asymptotic expansion of orthogonal polynomials. This will lead to the universal local behavior of certain log-gas systems with a general class of potentials. If time permits, we will also discuss the discrete version of the Riemann-Hilbert problem and its applications to asymptotic representation theory.

Additional Course Attributes:

Attribute | Value(s) |
---|---|

FAS Divisional Distribution | Science & Engineering & Applied Science |

All: Cross Reg Availability | Available for Harvard Cross Registration |

FAS: Course Level | Primarily for Graduate Students |

*Joseph D. Harris*

2021 Fall (4 Credits)

**
Schedule: **
MW 10:30 AM - 11:45 AM

**
Instructor Permissions: **
None

**
Enrollment Cap: **
n/a

The theory of algebraic curves is one of the oldest and best-understood in algebraic geometry. At the same time, it’s also an area of tremendous recent activity and attractive open problems. In this course, I hope to describe what’s known about algebraic curves, and also to talk about areas of current research. The course should be accessible to anyone with a year of algebraic geometry (e.g., Chapter 4 of Hartshorne is more than enough), and lots of thesis problems will be available for any student who wants one.

Additional Course Attributes:

Attribute | Value(s) |
---|---|

FAS Divisional Distribution | Science & Engineering & Applied Science |

All: Cross Reg Availability | Available for Harvard Cross Registration |

FAS: Course Level | Primarily for Graduate Students |

*Joseph D. Harris*

2022 Spring (4 Credits)

**
Schedule: **
TR 12:00 PM - 01:15 PM

**
Instructor Permissions: **
None

**
Enrollment Cap: **
n/a

The theory of algebraic curves is one of the oldest and best-understood in algebraic geometry. At the same time, it’s also an area of tremendous recent activity and attractive open problems. In this course, I hope to describe what’s known about algebraic curves, and also to talk about areas of current research. The course should be accessible to anyone with a year of algebraic geometry (e.g., Chapter 4 of Hartshorne is more than enough), and lots of thesis problems will be available for any student who wants one.

Additional Course Attributes:

Attribute | Value(s) |
---|---|

FAS Divisional Distribution | Science & Engineering & Applied Science |

All: Cross Reg Availability | Available for Harvard Cross Registration |

FAS: Course Level | Primarily for Graduate Students |

*Fabian Gundlach*

2021 Fall (4 Credits)

**
Schedule: **
TR 12:00 PM - 01:15 PM

**
Instructor Permissions: **
None

**
Enrollment Cap: **
n/a

We explore some algorithms in algebra and number theory. The focus of the course will be theory, not implementation. Possible topics include polynomial factorization, algorithms on number fields, elliptic curves, computational algebraic geometry. We might also discuss some computational group theory or representation theory.

Additional Course Attributes:

Attribute | Value(s) |
---|---|

FAS Divisional Distribution | Science & Engineering & Applied Science |

All: Cross Reg Availability | Available for Harvard Cross Registration |

FAS: Course Level | Primarily for Graduate Students |

*Mark Kisin*

2021 Fall (4 Credits)

**
Schedule: **
TBD

**
Instructor Permissions: **
Instructor

**
Enrollment Cap: **
n/a

We will study recent advances in number theory. One of the aims of the class is to help participants improve their expository skills by giving talks on recent number theory papers, or on their own work in this subject.

Additional Course Attributes:

Attribute | Value(s) |
---|---|

FAS Divisional Distribution | Science & Engineering & Applied Science |

All: Cross Reg Availability | Available for Harvard Cross Registration |

FAS: Course Level | Primarily for Graduate Students |

*Robin Gottlieb*

2021 Fall (4 Credits)

**
Schedule: **
TBD

**
Instructor Permissions: **
Instructor

**
Enrollment Cap: **
n/a

Become an effective instructor. This course focuses on observation, practice, feedback, and reflection providing insight into teaching and learning. Involves iterated videotaped micro-teaching sessions, accompanied by individual consultations. Required of all mathematics graduate students.

Additional Course Attributes:

Attribute | Value(s) |
---|---|

All: Cross Reg Availability | Not Available for Harvard Cross Registration |

FAS: Course Level | Graduate Course |

FAS Divisional Distribution | None |