Current Courses

AS.110.608 - Riemann Surfaces

Abstract Riemann surfaces. Examples: algebraic curves, elliptic curves and functions on them. Holomorphic and meromorphic functions and differential forms, divisors and the Mittag-Leffler problem. The analytic genus. Bezout's theorem and applications. Introduction to sheaf theory, with applications to constructing linear series of meromorphic functions. Serre duality, the existence of meromorphic functions on Riemann surfaces, the equality of the topological and analytic genera, the equivalence of algebraic curves and compact Riemann surfaces, the Riemann-Roch theorem. Period matrices and the Abel-Jacobi mapping, Jacobi inversion, the Torelli theorem. Uniformization (time permitting).

Credits: 0.00
Instructor: Xu, Hang
Term: Fall 2018
Meetings: MW 12:00PM - 1:15PM
Status: Open
Level: Graduate
Departments: AS Mathematics

AS.110.637 - Functional Analysis

Credits: 0.00
Instructor: Lu, Fei
Term: Fall 2018
Meetings: MW 9:00AM - 10:15AM
Status: Open
Level: Graduate
Departments: AS Mathematics

AS.110.605 - Real Variables

Measure and integration on abstract and locally compact spaces (extension of measures, decompositions of measures, product measures, the Lebesgue integral, differentiation, Lp-spaces); introduction to functional analysis; integration on groups; Fourier transforms.

Credits: 4.00
Instructor: Bernstein, Jacob
Term: Fall 2018
Meetings: MW 12:00PM - 1:15PM
Status: Open
Level: Graduate
Departments: AS Mathematics

AS.110.643 - Algebraic Geometry

Affine varieties and commutative algebra. Hilbert’s theorems about polynomials in several variables with their connections to geometry. General varieties and projective geometry. Dimension theory and smooth varieties. Sheaf theory and cohomology. Applications of sheaves to geometry; e.g., the Riemann-Roch theorem. Other topics may include Jacobian varieties, resolution of singularities, geometry on surfaces, connections with complex analytic geometry and topology, schemes.

Credits: 4.00
Instructor: Shokurov, Vyacheslav
Term: Fall 2018
Meetings: TTh 10:30AM - 11:45AM
Status: Open
Level: Graduate
Departments: AS Mathematics

AS.110.615 - Algebraic Topology

Polyhedra, simplicial and singular homology theory, Lefschetz fixed-point theorem, cohomology and products, homological algebra, Künneth and universal coefficient theorems, Poincaré and Alexander duality theorems.

Credits: 4.00
Instructor: Kitchloo, Nitya
Term: Fall 2018
Meetings: MW 1:30PM - 2:45PM
Status: Open
Level: Graduate
Departments: AS Mathematics

AS.110.617 - Number Theory

Topics in advanced algebra and number theory, including local fields and adeles, Iwasawa-Tate theory of zeta functions and connections with Hecke’s treatment, semisimple algebras over local and number fields, adeles geometry.

Credits: 4.00
Instructor: Staff
Term: Fall 2018
Meetings: TTh 10:30AM - 11:45AM
Status: Open
Level: Graduate
Departments: AS Mathematics

AS.110.632 - Partial Differential Equations II

An introductory graduate course in partial differential equations. Classical topics include first order equations and characteristics, the Cauchy-Kowalevski theorem, Laplace's equation, heat equation, wave equation, fundamental solutions, weak solutions, Sobolev spaces, maximum principles. The second term focuses on special topics such as second order elliptic theory.

Credits: 4.00
Instructor: Luehrmann, Jonas
Term: Fall 2018
Meetings: MW 12:00PM - 1:15PM
Status: Open
Level: Graduate
Departments: AS Mathematics

AS.110.633 - Harmonic Analysis

Fourier multipliers, oscillatory integrals, restriction theorems, Fourier integral operators, pseudodifferential operators, eigenfunctions. Undergrads need instructor's permission.

Credits: 0.00
Instructor: Staff
Term: Fall 2018
Meetings: MW 10:15AM - 11:45AM
Status: Canceled
Level: Graduate
Departments: AS Mathematics

AS.110.675 - High-Dimensional Approximation, Probability, and Statistical Learning

The course covers fundamental mathematical ideas for certain approximation and statistical learning problems in high dimensions. We start with basic approximation theory in low-dimensions, in particular linear and nonlinear approximation by Fourier and wavelets in classical smoothness spaces, and discuss applications in imaging, inverse problems and PDE’s. We then introduce notions of complexity of function spaces, which will be important in statistical learning. We then move to basic problems in statistical learning, such as regression and density estimation. The interplay between randomness and approximation theory is introduced, as well as fundamental tools such as concentration inequalities, basic random matrix theory, and various estimators are constructed in detail, in particular multi scale estimators. At all times we consider the geometric aspects and interpretations, and will discuss concentration of measure phenomena, embedding of metric spaces, optimal transportation distances, and their applications to problems in machine learning such as manifold learning and dictionary learning for signal processing.

Credits: 4.00
Instructor: Maggioni, Mauro
Term: Fall 2018
Meetings: MW 1:30PM - 2:45PM
Status: Open
Level: Graduate
Departments: AS Mathematics

AS.110.645 - Riemannian Geometry

Differential manifolds, vector fields, flows, Frobenius’ theorem. Differential forms, deRham’s theorem, vector bundles, connections, curvature, Chern classes, Cartan structure equations. Riemannian manifolds, Bianchi identities, geodesics, exponential maps. Geometry of submanifolds, hypersurfaces in Euclidean space. Other topics as time permits, e.g., harmonic forms and Hodge theorem, Jacobi equation, variation of arc length and area, Chern-Gauss-Bonnet theorems.

Credits: 4.00
Instructor: Mese, Chikako
Term: Fall 2018
Meetings: TTh 10:30AM - 11:45AM
Status: Open
Level: Graduate
Departments: AS Mathematics

AS.110.601 - Algebra

An introductory graduate course on fundamental topics in algebra to provide the student with the foundations for number theory, algebraic geometry, and other advanced courses. Topics include group theory, commutative algebra, Noetherian rings, local rings, modules, rudiments of category theory, homological algebra, field theory, Galois theory, and non-commutative algebras.

Credits: 4.00
Instructor: Smithling, Brian
Term: Fall 2018
Meetings: TTh 12:00PM - 1:15PM
Status: Open
Level: Graduate
Departments: AS Mathematics

AS.110.712 - Topics in Mathematical Physics

Credits: 0.00
Instructor: Lindblad, Hans
Term: Fall 2018
Meetings: MW 1:30PM - 2:45PM
Status: Open
Level: Graduate
Departments: AS Mathematics

AS.110.724 - Topics in Arithmetic Geometry

Topics around the subject of Arithmetic Geometry will be covered in this course.

Credits: 0.00
Instructor: Kitchloo, Nitya
Term: Fall 2018
Meetings: MW 1:30PM - 2:45PM
Status: Canceled
Level: Graduate
Departments: AS Mathematics

AS.110.799 - Seminar in Algebraic Geometry

For graduate students only. Presentations of current research papers by faculty, graduate students and invited guest speakers.

Credits: 4.00
Instructor: Zheng, Xudong
Term: Fall 2018
Meetings: T 4:30PM - 5:30PM
Status: Open
Level: Graduate
Departments: AS Mathematics

AS.110.793 - Seminar in Topology

For graduate students only. Presentations of current research papers by faculty, graduate students and invited guest speakers.

Credits: 4.00
Instructor: Kitchloo, Nitya
Term: Fall 2018
Meetings: M 3:00PM - 5:30PM
Status: Open
Level: Graduate
Departments: AS Mathematics

AS.110.801 - Thesis Research

Credits: 4.00
Instructor: Brown, Richard, Mese, Chikako, Savitt, David Lawrence
Term: Fall 2018
Meetings:
Status: Open
Level: Graduate
Departments: AS Mathematics

AS.110.733 - Topics In Alg Num Theory

Credits: 0.00
Instructor: Savitt, David Lawrence
Term: Fall 2018
Meetings: MW 10:30AM - 11:45AM
Status: Open
Level: Graduate
Departments: AS Mathematics

AS.110.798 - Seminar in Number Theory

Presentations of current research papers by faculty, graduate students and invited guest speakers. For graduate students only.

Credits: 0.00
Instructor: Smithling, Brian
Term: Fall 2018
Meetings: T 4:30PM - 5:30PM
Status: Open
Level: Graduate
Departments: AS Mathematics

AS.110.791 - Seminar in Analysis and Partial Differential Equations

Presentations of current research papers by faculty, graduate students and invited guest speakers. For graduate students only.

Credits: 4.00
Instructor: Bernstein, Jacob
Term: Fall 2018
Meetings: M 4:00PM - 5:00PM
Status: Open
Level: Graduate
Departments: AS Mathematics

AS.110.801 - Thesis Research

Credits: 4.00
Instructor: Brown, Richard, Dodson, Benjamin, Mese, Chikako
Term: Fall 2018
Meetings:
Status: Open
Level: Graduate
Departments: AS Mathematics

AS.110.794 - Seminar in Category Theory

Presentations of current research papers by faculty, graduate students and invited guest speakers. For graduate students only.

Credits: 0.00
Instructor: Riehl, Emily
Term: Fall 2018
Meetings: Th 4:00PM - 6:30PM
Status: Approval Required
Level: Graduate
Departments: AS Mathematics

AS.110.727 - Topics in Algebraic Topology

Credits: 0.00
Instructor: Merling, Mona
Term: Fall 2018
Meetings: TTh 12:00PM - 1:15PM
Status: Open
Level: Graduate
Departments: AS Mathematics

AS.110.790 - Seminar in Complex Geometry

Presentations of current research papers by faculty, graduate students and invited guest speakers. For graduate students only.

Credits: 4.00
Instructor: Shiffman, Bernard, Xu, Hang
Term: Fall 2018
Meetings: T 4:30PM - 6:00PM
Status: Open
Level: Graduate
Departments: AS Mathematics

AS.110.795 - Seminar in Data Analysis

Presentations of current research papers by faculty, graduate students and invited guest speakers. For graduate students only.

Credits: 0.00
Instructor: Maggioni, Mauro
Term: Fall 2018
Meetings: W 3:00PM - 4:00PM
Status: Approval Required
Level: Graduate
Departments: AS MathematicsEN Applied Mathematics & Statistics

AS.110.737 - Topics Algebraic Geometry

Credits: 0.00
Instructor: Shokurov, Vyacheslav
Term: Fall 2018
Meetings: MW 9:00AM - 10:15AM
Status: Open
Level: Graduate
Departments: AS Mathematics

AS.110.801 - Thesis Research

Credits: 4.00
Instructor: Brown, Richard, Dodson, Benjamin, Mese, Chikako
Term: Fall 2018
Meetings:
Status: Open
Level: Graduate
Departments: AS Mathematics

AS.110.749 - Topics in Differential Geometry

In this class, we will study Aaron Naber and Jeff Cheeger's recent result on proving codimension four conjecture. We plan to talk about some early results of the structure on manifolds with lower Ricci bound by Cheeger and Colding. We will prove quantitative splitting theorem, volume convergence theorem, and the result that almost volume cone implies almost metric cone. Then we will discuss regularity of Einstein manifolds and the codimension four conjecture.

Credits: 0.00
Instructor: Bernstein, Jacob
Term: Fall 2018
Meetings: TTh 12:00PM - 1:15PM
Status: Open
Level: Graduate
Departments: AS Mathematics

AS.110.801 - Thesis Research

Credits: 4.00
Instructor: Mese, Chikako, Morava, Jack
Term: Fall 2018
Meetings:
Status: Open
Level: Graduate
Departments: AS Mathematics

AS.110.801 - Thesis Research

Credits: 4.00
Instructor: Consani, Caterina, Mese, Chikako
Term: Fall 2018
Meetings:
Status: Open
Level: Graduate
Departments: AS Mathematics