# Graduate Courses

*Curtis McMullen*

2019 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

Functional analysis and applications. Topics may include distributions, elliptic regularity, spectral theory, operator algebras, unitary representations, and ergodic theory.

- 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 |

*Peter Kronheimer*

2019 Fall (4 Credits)

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Schedule: **
WF 12:00 PM - 01:15 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 |

*Shing-Tung Yau*

2020 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

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) |
---|---|

FAS: Course Level | Primarily for Graduate Students |

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

*Brooke Ullery*

2019 Fall (4 Credits)

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

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Instructor Permissions: **
None

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Enrollment Cap: **
n/a

The first course in Algebra: Noetherian rings and modules, Hilbert basis theorem, Cayley-Hamilton theorem, integral dependence, Galois theory, Noether normalization, the Nullstellensatz, localization, primary decomposition. Representation theory of finite groups. Introduction to Lie groups and Lie algebras: definitions, the exponential maps, semi-simple Lie algebras, examples.

- 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 |

*Dennis Gaitsgory*

2020 Spring (4 Credits)

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Schedule: **
TR 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*

2019 Fall (4 Credits)

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

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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 |

*Noam D. Elkies*

2020 Spring (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

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 |

*Sebastien Picard*

2019 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

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 |

*Valentino Tosatti*

2020 Spring (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

A continuation of Mathematics 230a. Topics in complex differential geometry: Complex Manifolds. Kahler metrics. Ricci curvature. Calabi Conjecture and it’s proof. Miyaoka-Yau Chern number inequalities and uniformization. The uniqueness of the Kahler structure of projective spaces. Calabi-Yau manifolds and their moduli.

- Recommended Prep:
- Knowledge of the material in Mathematics 230a

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 |

*Alexander Kupers*

2019 Fall (4 Credits)

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

*Eylem Yildiz*

2020 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

Continuation of Mathematics 231a. Topics may include stable homotopy theory, topological or algebraic K-theory, characteristic classes and vector bundles, cobordism, and categorical homotopy theory.

- 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 |

*Man-Wai Cheung*

2019 Fall (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

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 |

*Man-Wai Cheung*

2020 Spring (4 Credits)

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

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Instructor Permissions: **
None

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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*

2020 Spring (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

Advanced topics of evolutionary dynamics. Seminars and research projects.

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

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 |

*Yum-Tong Siu*

2019 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

A discussion of recent techniques and results and open problems involving holomorphic pluricanonical sections and jet differentials of complex manifolds. Topics: The techniques introduced for the deformational invariance of plurigenera and the solution of a number of conjectures on optimal constants in analysis arising from such techniques. The analytic approach to the finite generation of the canonical ring and the abundance conjecture. The hyperbolicty of a generic high-degree complex hypersurface in a complex projective space and more generally the second main theorem in Nevanlinna theory for an entire holomorphic curve in a complex projective space and its counting function for a smooth complex hypersurface.

- Recommended Prep:
- Prerequisite for the course is basic knowledge in complex analysis, for example, at the level of Mathematics 213a.

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 |

*Dori Bejleri*

2019 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

The existence and contruction of moduli spaces parametrizing geometric objects is a central problem in algebraic geometry. We will study the various tools and techniques used to address this problem, as well as applications. Possible topics include Hilbert schemes, geometric invariant theory, algebraic stacks, and compactifications of moduli spaces of varieties.

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 Gerig*

2019 Fall (4 Credits)

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

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Instructor Permissions: **
None

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Enrollment Cap: **
n/a

ECH is one of the 3 isomorphic Floer homologies which apply to 3-manifolds, (symplectic) 4-manifolds, and relations between them. It is defined by “counting” holomorphic curves and periodic orbits, and a big part of this course will be to study its foundations and difficulties (which plague other contact homologies. Some applications of ECH to be described include 1) distinguishing contact 3-manifolds and symplectic 4manifolds, 2) the Weinstein conjecture on the existence of periodic orbits, and 3) the relations to SeibergWitten 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 Students |

*Shing-Tung Yau*

2019 Fall (4 Credits)

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

**
Instructor Permissions: **
None

**
Enrollment Cap: **
n/a

We will discuss methods from nonlinear analytics to construct metrics and connections on bundles in Kahler and nonlinear geometry.

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 |

*Elana Kalashnikov*

2020 Spring (4 Credits)

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

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Instructor Permissions: **
None

**
Enrollment Cap: **
n/a

GIT quotient construction of toric varieties, relation to fan and polytope constructions, divisors, smoothness. Quantum cohomology and the moduli space of stable maps. J functions and Givental’s Lagrangian cone. Quantum Lefschetz. The I function and the Hori-Vafa mirror for toric complete intersections. Non toric GIT quotients, especially quiver flag varieties. The Abelian/non-Abelian correspondence. Laurent polynomial mirrors for toric complete intersections and their properties. Mirrors of non-toric Fano varieties and the Fano classification programme.

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 |

*Elden Elmanto*

2020 Spring (4 Credits)

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

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Instructor Permissions: **
None

**
Enrollment Cap: **
n/a

Algebraic cobordism is a cohomology theory for schemes constructed by Vladimir Voevodsky in the context of stable motivic homotopy theory, and greatly expanded by Marc Levine and Fabien Morel. It is, in some sense, simultaneously a generalization of “Fulton-style” intersection theory for Chow groups and a simplification of the latter theory. The course will revolbe around a theorem of Levine and Morel which recovers the Chow groups from algebraic cobordism. To do this, we will explain the notion of an oriented theory in algebraic geometry and prove that algebraic cobordism is the “universal” oriented theory. From there, we will prove the basic theorems (such as localization, homotopy invariance, computation of the coefficient ring and the degree formula) and give a sampler of computations. Further topics will depend on the interests of the participants.

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 |

*Lauren Williams*

2020 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

This course will provide an elementary introduction to the combinatorial aspects of Schubert calculus, the part of enumerative geometry dealing with classical varieties such as Grassmanians, flag varieties, and their Schubert varieties. A classical example of a Schubert calculus question is the following: given a generic configuration of four 2-dimensional subspaces in a complex 4-dimensional space, how many 2dimensional subspaces intersect each of these four in a line? To be able to answer this and related questions, one needs to concretely understand the structure of the cohomology ring of the Grassmanian. In this course we will develop the necessary combinatorial machinery to answer such enumerative questions, including Young tableaux, the Bruhat order, symmetric functions, and Schubert polynomials. More advancd topics may include (time-permitting): quantum cohomology rings, toric Schur functions, real Schubert calculus.

Additional Course Attributes:

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

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

FAS: Course Level | Primarily for Graduate Studentss |

FAS Divisional Distribution | Science & Engineering & Applied Science |

*Michael Hopkins*

2020 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

This course will explain the role of algebraic topology in the classification of topological phases of matter. We will cover basic solid-state physics, lattice models, topological quantum field theories, reflection positivity, and the classification of invertible topological field theories using homotopy 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 |

*Noam D. Elkies*

2019 Fall (4 Credits)

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

**
Instructor Permissions: **
None

**
Enrollment Cap: **
n/a

Construction and uses of modular forms from theta functions (with harmonic and periodic weights) attached to rational quadratic forms. Applications include radial and angular distribution of lattice vectors, classification of integral quadratic forms of low rank and discriminant, and connections with error correcting codes.

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 |

*Marius Lemm*

2020 Spring (4 Credits)

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

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Instructor Permissions: **
None

**
Enrollment Cap: **
n/a

Quantum spin systems lie at the interface of theoretical physics and mathematical spectral theory. They allow us to study true many-body phenomena, like phase transitions, within a solid mathematical framework. In this topics class, we will study the basic notions, main results, and open problems, of quantum spin systems. Two highlights will be the derivation of a spectral gap in the AKLT chain and the proof of spontaneous breaking of SU(2) symmetry in the Heisenberg antiferromagnet by Dyson, Lieb, and Simon. We will mainly use methods from spectral theory and matrix analysis, as well as some representation theory of SU(2).

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 |

*Sebastien Picard*

2020 Spring (4 Credits)

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

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Instructor Permissions: **
None

**
Enrollment Cap: **
n/a

This is a topics course on non-Kahler complex geometry, with a focus on non-Kahler Calabi-Yau threefolds. These objects were proposed as heterotic string compactifications by C.Hull and A. Strominger. Some of the mathematical structures which emerge include Hermitian-Yang-Mills connections, special holonomy constraints, Michelsohn’s notion of balanced metrics, and Hitchin’s generalized complex geometry. We will also discuss analytic methods in this field, as developed by J. Li, J.-X. Fu, and S.T. Yau.

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 |

*Fabian Gundlach*

2020 Spring (4 Credits)

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

**
Instructor Permissions: **
None

**
Enrollment Cap: **
n/a

We will investigate different counting problems in algebraic number theory. For example, how frequent are number fields of a given degree or given Galois group? We will give an overview of different approaches, including techniques from the geometry of numbers and class field theory. Time permitting, related algorithmic problems will be discussed.

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 |

2019 Fall (4 Credits)

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

**
Instructor Permissions: **
None

**
Enrollment Cap: **
n/a

The main goal will be to study smooth 4-manifolds. The plan is to start with knots in 3-manifolds and their interactions with smooth 4-manifold theory. Topics will be chosen from the topoloogy of 4-manifolds via their handlebodies, such as various constructions of 3-and 4-manifolds, by using techniques from gluing, carving, roping, corks, plugs, Gluck construction, and applications of some 4-manifold invariants to these constructions, and state some open problems.

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 |

*Jameel Al-Aidroos and Robin Gottlieb*

2019 Fall (4 Credits)

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

**
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.

- Class Notes:
- Robin Gottlieb and Brendan Kelly

Additional Course Attributes:

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

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

FAS Divisional Distribution | None |

FAS: Course Level | Graduate Course |