# Graduate Courses

*Christian Brennecke*

2021 Spring (4 Credits)

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

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Instructor Permissions: **
None

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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

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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*

2021 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 |

*Mihnea Popa*

2020 Fall (4 Credits)

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

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Instructor Permissions: **
None

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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 |

*Curtis McMullen*

2021 Spring (4 Credits)

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Schedule: **
TR 12:00 PM - 1:15 PM

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Instructor Permissions: **
None

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Enrollment Cap: **
n/a

The flexibility of conformal mappings is a cornerstone of geometry and analysis in the complex plane. Conformal invariance also underpins the remarkable properties of scaling limits of discrete random processes in two dimensions. This course will provide an introduction to foundational results and current developments in the field. Topics may include invariants such as the hyperbolic metric, extremal length, harmonic measure and capacity; quasiconformal mappings; and random processes such as Brownian motion, percolation and SLE.

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 Shusterman*

2021 Spring (4 Credits)

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

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Instructor Permissions: **
None

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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 |

*Fabian Gundlach*

2020 Fall (4 Credits)

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Schedule: **
WF 3:00pm-4:15pm

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Instructor Permissions: **
None

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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 |

*Fabian Gundlach*

2021 Spring (4 Credits)

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

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Instructor Permissions: **
None

**
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 |

*Noam D. Elkies*

2021 Spring (4 Credits)

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Schedule: **
TR 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 |

*Cliff Taubes*

2020 Fall (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

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 |

*Joseph D. Harris*

2020 Fall (4 Credits)

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

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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*

2021 Spring (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

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 |

*Elden Elmanto*

2020 Fall (4 Credits)

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

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Instructor Permissions: **
None

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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 |

*Elden Elmanto*

2021 Spring (4 Credits)

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

**
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 |

*Piotr Pstragowski*

2021 Spring (4 Credits)

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

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Instructor Permissions: **
None

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Enrollment Cap: **
n/a

In chromatic homotopy theory, one explores the link between the arithmetic of formal groups and the stable homotopy theory; we will focus on those aspects of this relationship which are visible at finite height. We will begin with the local structure of the moduli of formal groups and discuss how it is reflected and the topological side through the Lubin-Tate spectrum, then describe the current state of knowledge on several open questions on the subject, such as the vanishing and the splitting conjectures. Further topics will depend on the interests of the participants.

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 |

*Shing-Tung Yau*

2020 Fall (4 Credits)

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

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Instructor Permissions: **
None

**
Enrollment Cap: **
n/a

A discussion of ideas of nonlinear partial differential equations and geometric analysis, including their applications to algebraic geometry and general relativity. Construction of metrics and gauge field on complex manifolds will be discussed.

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*

2021 Spring (4 Credits)

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

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Instructor Permissions: **
None

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Enrollment Cap: **
n/a

Free Probability theory was introduced by Voiculescu almost forty years ago in the context of operator algebras. Since then, the theory has found connections to a wide range of areas of mathematics. We will study the general theory and its links to random matrices and representation theory. Specifically, we first discuss general aspects of the probability theory, including combinatorial (free cumulants, non-crossing partitions) and analytical (R-transform, subordination, central limit theorem). Then we apply it to the study of large random matrices. Finally, we discuss application to the representation theory of large groups (symmetric and unitary groups of growing rank).

Additional Course Attributes:

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

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

FAS: Course Level | Primarily for Graduate Studentss |

*Melanie Wood*

2020 Fall (4 Credits)

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

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Instructor Permissions: **
None

**
Enrollment Cap: **
n/a

As we vary over number fields, what distribution of class groups do we see? More generally, as we vary over number fields, how many (and how large) unramified extensions do the number fields have? Starting with the case of quadratic fields, we will learn about what was classically known about class groups of quadratic fields, and the conjectures of Cohen and Lenstra on their distribution, including many different motivations for the conjectures. We will learn about the relationship between class group distributions and counting number fields, and learn about the heuristics on counting number fields. We will learn about probability distributions on abelian groups that arise in these conjectures, and how they are determined by their moments. We will then learn what is known about how this picture generalizes to arbitrary number fields. We will then learn about two major recent research directions in this area: (1) Bounding class groups of number fields, and (2) Applications of topological component counting to class groups of function fields. The lectures will reference many recent papers and also include a significant amount of material that is considered “known by experts” but is not in the literature. There will be weekly problem sets to give students a chance to work with the concepts introduced. Project ideas relating to the course topics will be suggested, and a final project will be required of undergraduates, which can include any blend of exposition, original computations, and computer programming. Some project topics could lead to original research by graduate students beyond the course.

Additional Course Attributes:

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

FAS: Course Level | Primarily for Graduate Studentss |

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

FAS Divisional Distribution | Science & Engineering & Applied Science |

*Curtis McMullen*

2020 Fall (4 Credits)

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

**
Instructor Permissions: **
None

**
Enrollment Cap: **
n/a

A survey of fundamental results and current research. Topics may include: Ergodic theory and unitary operators; hyperbolic manifolds; geodesic and horocycle flows; Mostow rigidity; Kazhdan’s property T; expanding graphs; martingales, random walks and Furstenberg’s theorem; unipotent flows and Ratner rigidity; planes in 3-manifolds and moduli spaces.

Additional Course Attributes:

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

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

FAS: Course Level | Primarily for Graduate Studentss |

*Benjamin Gammage*

2020 Fall (4 Credits)

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

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Instructor Permissions: **
None

**
Enrollment Cap: **
n/a

A survey of some applications of topological quantum field theory to geometry. Possible topics: computations in the Fukaya category with applications to the homological mirror symmetry program, and the geometry of moduli spaces of supersymmetric gauge theories with applications to representation theory.

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 Studentss |

*Horng-Tzer Yau*

2020 Fall (4 Credits)

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

**
Instructor Permissions: **
None

**
Enrollment Cap: **
n/a

We will discuss the following two topics in the class: 1. The classical and current approach to prove some concentration inequalities, including the logarithmic Sobolev inequalities. 2. Elementary properties of SK spin glasses and Parisi’s solution. Depending on the progress of the class, some applications of these topics will also be discussed.

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 Studentss |

*Assaf Shani*

2021 Spring (4 Credits)

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

**
Instructor Permissions: **
None

**
Enrollment Cap: **
n/a

A problem in mathematics is classifying objects up to some notion of isomorphism. Famous examples include: the classification of compact orientable surfaces up to homeomorphism by their genus and classification of Bemoulli shifts up to isomorphism by their entropy. Descriptive set theory allows for a precise study of the complexity of various classification problems and the possible invariants which they admit. Topics: Polish groups and their actions on Polish spaces, definable equivalence relations, classifications problems and invariants, and interactions between these topics and forcing. For example, we will develp Hjorth’s theory of turbulence, which provides a method for showing that certain isomophism problems cannot be classified by any “reasonable invatiants”, and give an equivalent condition in terms of forcing, recently introduced by Larson-Zapletal. The topics will be flexible depending on students’ interests.

Additional Course Attributes:

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

FAS: Course Level | Primarily for Graduate Studentss |

FAS Divisional Distribution | Science & Engineering & Applied Science |

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

*Christopher Gerig*

2021 Spring (4 Credits)

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

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Instructor Permissions: **
None

**
Enrollment Cap: **
n/a

The construction in gauge theory of the Seiberg-Witten invariants, and generalized equations (examples: Vafa-Witten, SL(2,C) connections, multiple spinors, PU(n) monopoles). Along the way some possible applications to the geometry of 3-and 4-manifolds will be mentioned.

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 Studentss |

*Mark Shusterman*

2021 Spring (4 Credits)

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

**
Instructor Permissions: **
None

**
Enrollment Cap: **
n/a

A blend of arithmetic, geometric, and topological techniques will be used to gain insight into positive characteristic versions of classical problems in analytic number such as moments of L-functions, correlations of arithmetic functions, distribution of class groups.

Additional Course Attributes:

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

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

FAS Course Roll | FAS Course Roll |

FAS: Course Level | Primarily for Graduate Students |

*Dori Bejleri*

2020 Fall (4 Credits)

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

**
Instructor Permissions: **
None

**
Enrollment Cap: **
n/a

The classification of algebraic varieties up to birational equivalence is one of the major questions of higher dimensional algebraic geometry. This course will serve as an introduction to the subject, focusing on the minimal model program (MMP). The MMP is the part of the classification program which attempts to describe the “simplist” representatives within a given birational equivalence class. If time permits, we will discuss applications of the MMP to moduli of higher dimensional varieties.

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 Studentss |

*Laura DeMarco*

2021 Spring (4 Credits)

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

**
Instructor Permissions: **
None

**
Enrollment Cap: **
n/a

We will discuss various connections between complex-algebraic dynamical systems and arithmetic geometry.

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 |

*Mihnea Popa*

2021 Spring (4 Credits)

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

**
Instructor Permissions: **
None

**
Enrollment Cap: **
n/a

Topics related to the use of D-modules and Hodge modules in birational and complex geometry: Hodge filtration, V-filtration, Bernstein-Sato polynomial, multiplier ideals, vanishing theorems, Hodge ideals.

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, Brendan Kelly*

2021 Spring (4 Credits)

**
Schedule: **
T 01:30 AM - 02:45 AM

**
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 |