# Courses

- AS.110.595 - Internship
**Credits:**1.00**Instructor:****Term:**Summer 2017**Meetings:****Status:**Open - 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:**Merling, Mona, Mese, Chikako**Term:**Fall 2017**Meetings:**MW 1:30PM - 2:45PM**Status:**Open - 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:**Sire, Yannick**Term:**Fall 2017**Meetings:**MW 12:00PM - 1:15PM**Status:**Open - 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:**Mese, Chikako, Staff**Term:**Fall 2017**Meetings:**TTh 10:30AM - 11:45AM**Status:**Open - AS.110.712 - Topics in Mathematical Physics
**Credits:**0.00**Instructor:**Lindblad, Hans**Term:**Fall 2017**Meetings:**MW 1:30PM - 2:45PM**Status:**Open - AS.110.724 - Topics in Arithmetic Geometry
Topics around the subject of Arithmetic Geometry will be covered in this course.

**Credits:**0.00**Instructor:**Smithling, Brian**Term:**Fall 2017**Meetings:**MW 1:30PM - 2:45PM**Status:**Open - 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:**Brown, Richard, Dodson, Benjamin, Mese, Chikako**Term:**Fall 2017**Meetings:**MW 12:00PM - 1:15PM**Status:**Open - 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:**Riehl, Emily**Term:**Fall 2017**Meetings:**MW 9:00AM - 10:15AM**Status:**Open - 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 2017**Meetings:**MW 1:30PM - 2:45PM**Status:**Approval Required - 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:**Bernstein, Jacob, Mese, Chikako**Term:**Fall 2017**Meetings:**TTh 10:30AM - 11:45AM**Status:**Open - AS.110.727 - Topics/Algebraic Topology
**Credits:**0.00**Instructor:**Merling, Mona**Term:**Fall 2017**Meetings:**TTh 12:00PM - 1:15PM**Status:**Open - 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:**Zheng, Xudong**Term:**Fall 2017**Meetings:**TTh 10:30AM - 11:45AM**Status:**Open - AS.110.742 - Topics In Partial Differential Equations
**Credits:**4.00**Instructor:**Mese, Chikako, Spruck, Joel**Term:**Fall 2017**Meetings:**MW 1:30PM - 2:45PM**Status:**Open - 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:**Mese, Chikako, Wilson, W Stephen**Term:**Fall 2017**Meetings:**M 3:00PM - 5:30PM**Status:**Open - 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:**Shiffman, Bernard**Term:**Fall 2017**Meetings:**TTh 12:00PM - 1:15PM**Status:**Open - 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, Murphy, James**Term:**Fall 2017**Meetings:**W 3:00PM - 4:00PM**Status:**Approval Required - 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:**Consani, Caterina, Mese, Chikako**Term:**Fall 2017**Meetings:**T 4:30PM - 5:30PM**Status:**Open - 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:**Mese, Chikako, Shokurov, Vyacheslav**Term:**Fall 2017**Meetings:**T 4:30PM - 5:30PM**Status:**Open - AS.110.801 - Thesis Research
**Credits:**4.00**Instructor:**Mese, Chikako, Morava, Jack**Term:**Fall 2017**Meetings:****Status:**Open - AS.110.801 - Thesis Research
**Credits:**4.00**Instructor:**Consani, Caterina, Mese, Chikako**Term:**Fall 2017**Meetings:****Status:**Open - AS.110.801 - Thesis Research
**Credits:**4.00**Instructor:**Brown, Richard, Mese, Chikako, Savitt, David Lawrence**Term:**Fall 2017**Meetings:****Status:**Open - 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:**Mese, Chikako, Shiffman, Bernard**Term:**Fall 2017**Meetings:**T 4:30PM - 6:00PM**Status:**Open - AS.110.801 - Thesis Research
**Credits:**4.00**Instructor:**Brown, Richard, Dodson, Benjamin, Mese, Chikako**Term:**Fall 2017**Meetings:****Status:**Open - AS.110.756 - Topics in Algebra
This will be a course in commutative algebra. Topics may include: Noetherian rings and modules, the Nullstellensatz, Hilbert basis theorem, localization, integrality, Noether normalization, primary decomposition, DVRs, Dedekind domains, dimension theory, smoothness and regularity, and homological methods.

**Credits:**0.00**Instructor:**Consani, Caterina**Term:**Fall 2017**Meetings:**TTh 10:30AM - 11:45AM**Status:**Approval Required - AS.110.791 - Seminar in Analysis and Partial Differential Equations
**Credits:**4.00**Instructor:**Sogge, Christopher**Term:**Fall 2017**Meetings:**M 4:00PM - 5:00PM**Status:**Open