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
• Days/Times: TTh 10:30AM - 11:45AM
• Instructor: Consani, Caterina
• Room:  
• Status: Open
• Seats Available: 18/20

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
• Days/Times: MW 12:00PM - 1:15PM
• Instructor: Staff
• Room:  
• Status: Canceled
• Seats Available: 30/30

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
• Days/Times: TTh 12:00PM - 1:15PM
• Instructor: Shokurov, Vyacheslav
• Room: Krieger 180
• Status: Open
• Seats Available: 7/20

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
• Days/Times: MW 10:30AM - 11:45AM
• Instructor: Kitchloo, Nitya
• Room: Latrobe 120
• Status: Open
• Seats Available: 4/20

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
• Days/Times: TTh 10:30AM - 11:45AM
• Instructor: Mese, Chikako
• Room: Krieger 180
• Status: Open
• Seats Available: 5/20

Functional Analysis
AS.110.637 (01)

• Credits: 0.00
• Level: Graduate
• Days/Times: TTh 9:00AM - 10:15AM
• Instructor: Dodson, Benjamin
• Room: Krieger 180
• Status: Open
• Seats Available: 8/19

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
• Days/Times: TTh 9:00AM - 10:15AM
• Instructor: Han, Jingjun
• Room: Krieger Laverty
• Status: Open
• Seats Available: 16/19

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
• Days/Times: MW 1:30PM - 2:45PM
• Instructor: Maggioni, Mauro
• Room:  
• Status: Canceled
• Seats Available: 40/40

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
• Days/Times: MW 2:00PM - 3:15PM
• Instructor: Sire, Yannick
• Room: Krieger 204
• Status: Open
• Seats Available: 17/20

Topics in Algebraic Topology
AS.110.727 (01)

• Credits: 0.00
• Level: Graduate
• Days/Times: MW 9:00AM - 10:15AM
• Instructor: Kitchloo, Nitya
• Room:  
• Status: Open
• Seats Available: 14/15

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
• Days/Times: MW 12:00PM - 1:15PM
• Instructor: Bernstein, Jacob
• Room: Hodson 315
• Status: Open
• Seats Available: 12/20

Topics in Mathematical Physics
AS.110.712 (01)

• Credits: 0.00
• Level: Graduate
• Days/Times: MW 1:30PM - 2:45PM
• Instructor: Staff
• Room:  
• Status: Canceled
• Seats Available: 10/10

Thesis Research
AS.110.801 (14)

• Credits: 4.00
• Level: Graduate
• Days/Times:
• Instructor: Smithling, Brian
• Room:  
• Status: Open
• Seats Available: 5/5

Topics Algebraic Geometry
AS.110.737 (01)

• Credits: 0.00
• Level: Graduate
• Days/Times: TTh 10:30AM - 11:45AM
• Instructor: Shokurov, Vyacheslav
• Room: Krieger 204
• Status: Open
• Seats Available: 27/30

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
• Days/Times: MW 12:00PM - 1:15PM
• Instructor: Wang, Yi
• Room: Gilman 217
• Status: Open
• Seats Available: 9/15

Thesis Research
AS.110.801 (01)

• Credits: 4.00
• Level: Graduate
• Days/Times:
• Instructor: Riehl, Emily
• Room:  
• Status: Open
• Seats Available: 22/25

Independent Study-Graduates
AS.110.800 (01)

• Credits: 0.00
• Level: Graduate
• Days/Times:
• Instructor: Riehl, Emily
• Room:  
• Status: Open
• Seats Available: 4/5

Topics in Representation Theory
AS.110.750 (01)

• Credits: 0.00
• Level: Graduate
• Days/Times: TTh 12:00PM - 1:15PM
• Instructor: Sakellaridis, Ioannis
• Room: Krieger Laverty
• Status: Closed
• Seats Available: 3/10

Thesis Research
AS.110.801 (10)

• Credits: 4.00
• Level: Graduate
• Days/Times:
• Instructor: Bernstein, Jacob
• Room:  
• Status: Open
• Seats Available: 5/5

Topics In Alg Num Theory
AS.110.733 (01)

• Credits: 0.00
• Level: Graduate
• Days/Times: MW 10:30AM - 11:45AM
• Instructor: Savitt, David Lawrence
• Room: Gilman 77
• Status: Open
• Seats Available: 1/8

Thesis Research
AS.110.801 (11)

• Credits: 4.00
• Level: Graduate
• Days/Times:
• Instructor: Savitt, David Lawrence
• Room:  
• Status: Open
• Seats Available: 1/5

Thesis Research
AS.110.801 (03)

• Credits: 4.00
• Level: Graduate
• Days/Times:
• Instructor: Lindblad, Hans
• Room:  
• Status: Open
• Seats Available: 23/25

Thesis Research
AS.110.801 (12)

• Credits: 4.00
• Level: Graduate
• Days/Times:
• Instructor: Dodson, Benjamin
• Room:  
• Status: Open
• Seats Available: 5/5

Thesis Research
AS.110.801 (02)

• Credits: 4.00
• Level: Graduate
• Days/Times:
• Instructor: Consani, Caterina
• Room:  
• Status: Open
• Seats Available: 24/25

Thesis Research
AS.110.801 (04)

• Credits: 4.00
• Level: Graduate
• Days/Times:
• Instructor: Sogge, Christopher
• Room:  
• Status: Open
• Seats Available: 24/25

Thesis Research
AS.110.801 (08)

• Credits: 4.00
• Level: Graduate
• Days/Times:
• Instructor: Shokurov, Vyacheslav
• Room:  
• Status: Open
• Seats Available: 24/25

Thesis Research
AS.110.801 (05)

• Credits: 4.00
• Level: Graduate
• Days/Times:
• Instructor: Sakellaridis, Ioannis
• Room:  
• Status: Open
• Seats Available: 25/25

Thesis Research
AS.110.801 (13)

• Credits: 4.00
• Level: Graduate
• Days/Times:
• Instructor: Sire, Yannick
• Room:  
• Status: Open
• Seats Available: 4/5

Thesis Research
AS.110.801 (07)

• Credits: 4.00
• Level: Graduate
• Days/Times:
• Instructor: Kitchloo, Nitya
• Room:  
• Status: Open
• Seats Available: 25/25

Thesis Research
AS.110.801 (06)

• Credits: 4.00
• Level: Graduate
• Days/Times:
• Instructor: Maggioni, Mauro
• Room:  
• Status: Open
• Seats Available: 22/25

Thesis Research
AS.110.801 (09)

• Credits: 4.00
• Level: Graduate
• Days/Times:
• Instructor: Wang, Yi
• Room:  
• Status: Open
• Seats Available: 9/10

Number Theory
AS.110.618 (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, semi-simple algebras over local and number fields, adele geometry.

• Credits: 0.00
• Level: Graduate
• Days/Times: TTh 12:00PM - 1:15PM
• Instructor: Sakellaridis, Ioannis
• Room: Gilman 400
• Status: Open
• Seats Available: 13/15

Lie Groups and Lie Algebras
AS.110.619 (01)

Lie groups and Lie algebras, classification of complex semi-simple Lie algebras, compact forms, representations and Weyl formulas, symmetric Riemannian spaces.

• Credits: 0.00
• Level: Graduate
• Days/Times: TTh 9:00AM - 10:15AM
• Instructor: Staff
• Room: Krieger Laverty
• Status: Open
• Seats Available: 16/20

Algebra
AS.110.602 (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, and rudiments of category theory, homological algebra, field theory, Galois theory, and non-commutative algebras. Recommended Course Background: AS.110.401-AS.110.402

• Credits: 0.00
• Level: Graduate
• Days/Times: MW 12:00PM - 1:15PM
• Instructor: Shokurov, Vyacheslav
• Room: Greenhouse 113
• Status: Open
• Seats Available: 13/15

Partial Differential Equations I
AS.110.631 (01)

An introductory graduate course in partial differential equations. Classical topics include first order equations and characteristics, the Cauchy-Kowalewski theorem, Laplace’s equations, heat equation, wave equation, fundamental solutions, weak solutions, Sobolev spaces, maximum principles.

• Credits: 0.00
• Level: Graduate
• Days/Times: MW 1:30PM - 2:45PM
• Instructor: Sire, Yannick
• Room: Mattin Center 161
• Status: Open
• Seats Available: 13/15

Algebraic Geometry
AS.110.644 (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, schemes, connections with complex analytic geometry and topology.

• Credits: 0.00
• Level: Graduate
• Days/Times: TTh 10:30AM - 11:45AM
• Instructor: Han, Jingjun
• Room: Gilman 77
• Status: Open
• Seats Available: 9/10

Riemannian Geometry
AS.110.646 (01)

The goal is to give a self-contained course on mean curvature flow, starting with the basic linear heat equation in Euclidean space and – hopefully – getting to topics of current research. Mean curvature flow is a geometric heat equation that shares many properties with Ricci flow, harmonic map heat flow, Yang-Mills flow and the Navier-Stokes equations. Recommended Course Background: AS.110.605 and an undergraduate course in differential geometry; AS.110.645 and AS.110.631

• Credits: 0.00
• Level: Graduate
• Days/Times: MW 1:30PM - 2:45PM
• Instructor: Bernstein, Jacob
• Room: Gilman 381
• Status: Open
• Seats Available: 12/15

Topics in Analysis
AS.110.726 (01)

The topics covered will involve the theory of calculus of Functors applied to Geometric problems like Embedding theory. Other related areas will be covered depending on the interest of the audience.

• Credits: 0.00
• Level: Graduate
• Days/Times: MW 3:00PM - 4:15PM
• Instructor: Lindblad, Hans
• Room: Bloomberg 172
• Status: Open
• Seats Available: 9/12

Harmonic Analysis
AS.110.633 (01)

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

• Credits: 0.00
• Level: Graduate
• Days/Times: MW 12:00PM - 1:15PM
• Instructor: Dodson, Benjamin
• Room: Maryland 202
• Status: Open
• Seats Available: 9/15

Complex Variables
AS.110.607 (01)

Analytic functions of one complex variable. Topics include Mittag-Leffler Theorem, Weierstrass factorization theorem, elliptic functions, Riemann-Roch theorem, Picard theorem, and Nevanlinna theory. Recommended Course Background: AS.110.311, AS.110.405

• Credits: 0.00
• Level: Graduate
• Days/Times: TTh 10:30AM - 11:45AM
• Instructor: Wang, Yi
• Room: Shaffer 301
• Status: Open
• Seats Available: 13/15

Algebraic Topology
AS.110.616 (01)

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

• Credits: 0.00
• Level: Graduate
• Days/Times: MW 10:30AM - 11:45AM
• Instructor: Kitchloo, Nitya
• Room: Krieger 204
• Status: Open
• Seats Available: 12/15

Topics In Alg Num Theory
AS.110.733 (01)

• Credits: 0.00
• Level: Graduate
• Days/Times: MW 3:00PM - 4:15PM
• Instructor: Savitt, David Lawrence
• Room: Smokler Center Library
• Status: Open
• Seats Available: 11/12

Nash-Moser iteration and Kolmogorov-Arnold-Moser (KAM) theory
AS.110.741 (01)

KAM theory is aiming at constructing invariant (under the flow of a dynamical system in finite or infinite dimensions) tori (as in invariant manifold) for Hamiltonian systems. I will start by reminders about symplectic geometry and topology. Then I will present an easy proof of the Siegel theorem in complex analysis using an easy version of the KAM scheme. Several lectures will be dedicated to “hard implicit function theorems” via Nash-Moser iterations in Frechet spaces. I will then state the KAM theorem in its original form and its modern variants and I will provide the three different proofs of Kolmogorov, Arnold and Moser in finite dimensions. Finally, I will explain the major difficulties in the case of Hamiltonian PDEs to adapt the previous proofs and will move on to some known results such as the ones of Kuksin, Poeschel, Bourgain and De La Llave and myself. I will probably explain in detail the results of Bourgain for nonlinear Schrodinger.

• Credits: 0.00
• Level: Graduate
• Days/Times: MW 12:00PM - 1:15PM
• Instructor: Sire, Yannick
• Room: Mattin Center 161
• Status: Open
• Seats Available: 10/12

Independent Study-Graduates
AS.110.800 (01)

• Credits: 0.00
• Level: Graduate
• Days/Times:
• Instructor: Maggioni, Mauro
• Room:  
• Status: Approval Required
• Seats Available: 3/3

Thesis Research
AS.110.801 (04)

• Credits: 0.00
• Level: Graduate
• Days/Times:
• Instructor: Sogge, Christopher
• Room:  
• Status: Open
• Seats Available: 5/5

Thesis Research
AS.110.801 (01)

• Credits: 0.00
• Level: Graduate
• Days/Times:
• Instructor: Riehl, Emily
• Room:  
• Status: Open
• Seats Available: 5/5

Thesis Research
AS.110.801 (02)

• Credits: 0.00
• Level: Graduate
• Days/Times:
• Instructor: Consani, Caterina
• Room:  
• Status: Open
• Seats Available: 5/5

Thesis Research
AS.110.801 (03)

• Credits: 0.00
• Level: Graduate
• Days/Times:
• Instructor: Lindblad, Hans
• Room:  
• Status: Open
• Seats Available: 3/5

Thesis Research
AS.110.801 (05)

• Credits: 0.00
• Level: Graduate
• Days/Times:
• Instructor: Sakellaridis, Ioannis
• Room:  
• Status: Open
• Seats Available: 5/5

Topics in Stochastic Dynamical Systems
AS.110.757 (01)

The course will present an introduction to stochastic dynamical systems and some applications in model reduction and data assimilation. The main focus will be on stability and ergodicity of stochastic dynamical systems, including stochastic differential equations driven by white and fractional noise, and their numerical approximations. We will then discuss model reduction, focusing on Mori-Zwanzig formalism and approximation of the generalized Langevin equation, and methods on the parametric inference of related stochastic systems. Data assimilation and stochastic control will also be briefly introduced.

• Credits: 0.00
• Level: Graduate
• Days/Times: TTh 10:30AM - 11:45AM
• Instructor: Lu, Fei
• Room: Shriver Hall 104
• Status: Open
• Seats Available: 2/12

Thesis Research
AS.110.801 (06)

• Credits: 0.00
• Level: Graduate
• Days/Times:
• Instructor: Maggioni, Mauro
• Room:  
• Status: Open
• Seats Available: 5/5

Thesis Research
AS.110.801 (09)

• Credits: 0.00
• Level: Graduate
• Days/Times:
• Instructor: Wang, Yi
• Room:  
• Status: Open
• Seats Available: 4/5

Thesis Research
AS.110.801 (08)

• Credits: 0.00
• Level: Graduate
• Days/Times:
• Instructor: Shokurov, Vyacheslav
• Room:  
• Status: Open
• Seats Available: 5/5

Thesis Research
AS.110.801 (07)

• Credits: 0.00
• Level: Graduate
• Days/Times:
• Instructor: Kitchloo, Nitya
• Room:  
• Status: Open
• Seats Available: 5/5

Thesis Research
AS.110.801 (11)

• Credits: 0.00
• Level: Graduate
• Days/Times:
• Instructor: Savitt, David Lawrence
• Room:  
• Status: Open
• Seats Available: 4/5

Thesis Research
AS.110.801 (12)

• Credits: 0.00
• Level: Graduate
• Days/Times:
• Instructor: Dodson, Benjamin
• Room:  
• Status: Open
• Seats Available: 5/5

Thesis Research
AS.110.801 (10)

• Credits: 0.00
• Level: Graduate
• Days/Times:
• Instructor: Bernstein, Jacob
• Room:  
• Status: Open
• Seats Available: 5/5

Thesis Research
AS.110.801 (13)

• Credits: 0.00
• Level: Graduate
• Days/Times:
• Instructor: Sire, Yannick
• Room:  
• Status: Open
• Seats Available: 4/5