# 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
**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
**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