Column one has the course number and section. Other columns show the course title, days offered, instructor's name, room number, if the course is cross-referenced with another program, and a option to view additional course information in a pop-up window.

# Riemannian Geometry

AS.110.645 (01)

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**Level:**Graduate**Status:**Open**Seats Available:**10/20**PosTag(s):**n/a

# Real Variables

AS.110.605 (01)

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**Level:**Graduate**Status:**Open**Seats Available:**8/20**PosTag(s):**n/a

# Partial Differential Equations II

AS.110.632 (01)

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:**0.00**Level:**Graduate**Status:**Open**Seats Available:**19/20**PosTag(s):**n/a

# Riemann Surfaces

AS.110.608 (01)

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**Level:**Graduate**Status:**Canceled**Seats Available:**30/30**PosTag(s):**n/a

# Algebraic Topology

AS.110.615 (01)

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**Level:**Graduate**Status:**Open**Seats Available:**8/20**PosTag(s):**n/a

# Number Theory

AS.110.617 (01)

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**Level:**Graduate**Status:**Open**Seats Available:**18/20**PosTag(s):**n/a

# Functional Analysis

AS.110.637 (01)

**Credits:**0.00**Level:**Graduate**Status:**Open**Seats Available:**5/19**PosTag(s):**n/a

# Algebraic Geometry

AS.110.643 (01)

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**Level:**Graduate**Status:**Open**Seats Available:**15/19**PosTag(s):**n/a

# High-Dimensional Approximation, Probability, and Statistical Learning

AS.110.675 (01)

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:**0.00**Level:**Graduate**Status:**Canceled**Seats Available:**40/40**PosTag(s):**n/a

# Algebra

AS.110.601 (01)

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**Level:**Graduate**Status:**Open**Seats Available:**9/20**PosTag(s):**n/a

# Topics in Algebraic Topology

AS.110.727 (01)

**Credits:**0.00**Level:**Graduate**Status:**Open**Seats Available:**11/15**PosTag(s):**n/a

# Topics in Mathematical Physics

AS.110.712 (01)

**Credits:**0.00**Level:**Graduate**Status:**Canceled**Seats Available:**10/10**PosTag(s):**n/a

# Topics Algebraic Geometry

AS.110.737 (01)

**Credits:**0.00**Level:**Graduate**Status:**Open**Seats Available:**27/30**PosTag(s):**n/a

# Thesis Research

AS.110.801 (02)

**Credits:**4.00**Level:**Graduate**Status:**Open**Seats Available:**25/25**PosTag(s):**n/a

# Thesis Research

AS.110.801 (03)

**Credits:**4.00**Level:**Graduate**Status:**Open**Seats Available:**23/25**PosTag(s):**n/a

# Thesis Research

AS.110.801 (04)

**Credits:**4.00**Level:**Graduate**Status:**Open**Seats Available:**24/25**PosTag(s):**n/a

# Topics in Differential Geometry

AS.110.749 (01)

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**Level:**Graduate**Status:**Open**Seats Available:**11/15**PosTag(s):**n/a

# Topics in Representation Theory

AS.110.750 (01)

**Credits:**0.00**Level:**Graduate**Status:**Approval Required**Seats Available:**7/10**PosTag(s):**n/a

# Thesis Research

AS.110.801 (01)

**Credits:**4.00**Level:**Graduate**Status:**Open**Seats Available:**24/25**PosTag(s):**n/a

# Topics In Alg Num Theory

AS.110.733 (01)

**Credits:**0.00**Level:**Graduate**Status:**Open**Seats Available:**3/8**PosTag(s):**n/a

# Thesis Research

AS.110.801 (11)

**Credits:**4.00**Level:**Graduate**Status:**Open**Seats Available:**1/5**PosTag(s):**n/a

# Thesis Research

AS.110.801 (05)

**Credits:**4.00**Level:**Graduate**Status:**Open**Seats Available:**25/25**PosTag(s):**n/a

# Thesis Research

AS.110.801 (13)

**Credits:**4.00**Level:**Graduate**Status:**Open**Seats Available:**4/5**PosTag(s):**n/a

# Thesis Research

AS.110.801 (06)

**Credits:**4.00**Level:**Graduate**Status:**Open**Seats Available:**23/25**PosTag(s):**n/a

# Thesis Research

AS.110.801 (07)

**Credits:**4.00**Level:**Graduate**Status:**Open**Seats Available:**25/25**PosTag(s):**n/a

# Thesis Research

AS.110.801 (09)

**Credits:**4.00**Level:**Graduate**Status:**Open**Seats Available:**10/10**PosTag(s):**n/a

# Thesis Research

AS.110.801 (14)

**Credits:**4.00**Level:**Graduate**Status:**Canceled**Seats Available:**5/5**PosTag(s):**n/a

# Thesis Research

AS.110.801 (08)

**Credits:**4.00**Level:**Graduate**Status:**Open**Seats Available:**24/25**PosTag(s):**n/a

# Thesis Research

AS.110.801 (12)

**Credits:**4.00**Level:**Graduate**Status:**Open**Seats Available:**5/5**PosTag(s):**n/a

# Thesis Research

AS.110.801 (10)

**Credits:**4.00**Level:**Graduate**Status:**Open**Seats Available:**5/5**PosTag(s):**n/a

Course # (Section) | Title | Day/Times | Instructor | Room | PosTag(s) | Info |
---|---|---|---|---|---|---|

AS.110.645 (01) | Riemannian Geometry | TTh 10:30AM - 11:45AM | Mese, Chikako | Krieger 180 | ||

AS.110.605 (01) | Real Variables | MW 12:00PM - 1:15PM | Bernstein, Jacob | Hodson 315 | ||

AS.110.632 (01) | Partial Differential Equations II | MW 12:00PM - 1:15PM | Sire, Yannick | Krieger 204 | ||

AS.110.608 (01) | Riemann Surfaces | MW 12:00PM - 1:15PM | Staff | |||

AS.110.615 (01) | Algebraic Topology | MW 10:30AM - 11:45AM | Kitchloo, Nitya | Krieger 204 | ||

AS.110.617 (01) | Number Theory | TTh 10:30AM - 11:45AM | Consani, Caterina | |||

AS.110.637 (01) | Functional Analysis | MW 9:00AM - 10:15AM | Dodson, Benjamin | Gilman 75 | ||

AS.110.643 (01) | Algebraic Geometry | TTh 9:00AM - 10:15AM | Han, Jingjun | Krieger Laverty | ||

AS.110.675 (01) | High-Dimensional Approximation, Probability, and Statistical Learning | MW 1:30PM - 2:45PM | Maggioni, Mauro | |||

AS.110.601 (01) | Algebra | TTh 12:00PM - 1:15PM | Shokurov, Vyacheslav | Krieger 180 | ||

AS.110.727 (01) | Topics in Algebraic Topology | MW 9:00AM - 10:15AM | Kitchloo, Nitya | Krieger 204 | ||

AS.110.712 (01) | Topics in Mathematical Physics | MW 1:30PM - 2:45PM | Staff | |||

AS.110.737 (01) | Topics Algebraic Geometry | TTh 10:30AM - 11:45AM | Shokurov, Vyacheslav | Krieger 204 | ||

AS.110.801 (02) | Thesis Research | Consani, Caterina | ||||

AS.110.801 (03) | Thesis Research | Lindblad, Hans | ||||

AS.110.801 (04) | Thesis Research | Sogge, Christopher | ||||

AS.110.749 (01) | Topics in Differential Geometry | MW 12:00PM - 1:15PM | Wang, Yi | Gilman 217 | ||

AS.110.750 (01) | Topics in Representation Theory | TTh 12:00PM - 1:15PM | Staff | Krieger Laverty | ||

AS.110.801 (01) | Thesis Research | Riehl, Emily | ||||

AS.110.733 (01) | Topics In Alg Num Theory | MW 10:30AM - 11:45AM | Savitt, David Lawrence | Gilman 77 | ||

AS.110.801 (11) | Thesis Research | Savitt, David Lawrence | ||||

AS.110.801 (05) | Thesis Research | Sakellaridis, Ioannis | ||||

AS.110.801 (13) | Thesis Research | Sire, Yannick | ||||

AS.110.801 (06) | Thesis Research | Maggioni, Mauro | ||||

AS.110.801 (07) | Thesis Research | Kitchloo, Nitya | ||||

AS.110.801 (09) | Thesis Research | Wang, Yi | ||||

AS.110.801 (14) | Thesis Research | Staff | ||||

AS.110.801 (08) | Thesis Research | Shokurov, Vyacheslav | ||||

AS.110.801 (12) | Thesis Research | Dodson, Benjamin | ||||

AS.110.801 (10) | Thesis Research | Bernstein, Jacob |