Graduate Courses

To see a complete list of courses offered and their descriptions, visit the academic catalog.

For current course schedule information and registration visit SIS or consult the table below.

Fall 2022

Algebra I
AS.110.601 (01)

The first of a two semester algebra sequence to provide the student with the foundations for Number Theory, Algebraic Geometry, Representation Theory, and other areas. Topics include refined elements of group theory, commutative algebra, Noetherian rings, local rings, modules, and rudiments of category theory, homological algebra, field theory, Galois theory, and non-commutative algebras.

Credits: 0.00
Level: Graduate
Days/Times: TTh 12:00PM - 1:15PM
Instructor: Staff
Room: Croft Hall G02
Status: Open
Seats Available: 8/15
PosTag(s): n/a

Real Variables
AS.110.605 (01)

This course covers the theory of the Lebesgue theory of integration in d-dimensional Euclidean space, and offers a brief introduction to the theory of Hilbert spaces. Topics include the Lebesgue measure on Euclidean space, the Lebesgue integral, classical convergence results for the Lebesgue integral, Fubini's theorem, the spaces of L^1 and L^2 functions.

Credits: 0.00
Level: Graduate
Days/Times: MW 12:00PM - 1:15PM
Instructor: Sogge, Chris
Room: Gilman 313
Status: Open
Seats Available: 11/15
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
Days/Times: TTh 1:30PM - 2:45PM
Instructor: Mese, CHIKAKO
Room: Maryland 104
Status: Open
Seats Available: 14/20
PosTag(s): n/a

Algebraic Topology I
AS.110.615 (01)

Singular homology theory, cohomology and products, category theory and homological algebra, Künneth and universal coefficient theorems, Poincaré and Alexander duality theorems, Lefschetz fixed-point theorem, covering spaces and fundamental groups. Prerequsites: the equivalent of one semester in both Abstract Algebra and Real Analysis (specifically, point set topology).

Credits: 0.00
Level: Graduate
Days/Times: MW 1:30PM - 2:45PM
Instructor: Kitchloo, Nitya
Room: Gilman 217
Status: Open
Seats Available: 10/15
PosTag(s): n/a

Number Theory I
AS.110.617 (01)

Elements of advanced algebra and number theory. Possible topics for the year-long sequence include local and global fields, Galois cohomology, semisimple algebras, class field theory, elliptic curves, modular and automorphic forms, integral representations of L-functions, adelic geometry and function fields, fundamental notions in arithmetic geometry (including Arakelov and diophantine geometry).

Credits: 0.00
Level: Graduate
Days/Times: TTh 9:00AM - 10:15AM
Instructor: Iyengar, Ashwin
Room: Krieger 204
Status: Open
Seats Available: 12/15
PosTag(s): n/a

Partial Differential Equations I
AS.110.631 (01)

This course is the first in the sequence about the general theory of PDEs. The beginning of the course will describe several important results of functional analysis which are instrumental for the study of PDEs: Hahn-Banach theorem, Uniform boundedness and closed graph theorems, reflexive spaces and weak topologies, elements of semi-group theory. Then we will describe the basic theory of Sobolev spaces and the standard existence theory for (initial) boundary value problems of elliptic/parabolic type. Finally, the rest of the course will be devoted to finer properties of solutions of elliptic equations such as maximum principles, Harnack principles and regularity.

Credits: 0.00
Level: Graduate
Days/Times: MW 10:30AM - 11:45AM
Instructor: Dodson, Benjamin
Room: Krieger 411
Status: Open
Seats Available: 9/15
PosTag(s): n/a

Functional Analysis
AS.110.637 (01)

This class will explore basic aspects of functional analysis, focusing mostly on normed vector spaces. This will include the Hahn-Banach and open mapping theorems, a discussion of strong and weak topologies, the theory of compact operators, and spaces of integrable functions and Sobolev spaces, with applications to the study of some partial differential equations. Prerequisite: Real Analysis

Credits: 0.00
Level: Graduate
Days/Times: MW 12:00PM - 1:15PM
Instructor: Lindblad, Hans
Room: Gilman 381
Status: Open
Seats Available: 8/15
PosTag(s): n/a

Algebraic Geometry I
AS.110.643 (01)

Introduction to affine varieties and projective varieties. Hilbert's theorems about polynomials in several variables with their connections to geometry. Abstract algebraic varieties and projective geometry. Dimension of varieties and smooth varieties. Sheaf theory and some notions of cohomology. Applications of sheaves to geometry; e.g., theory of divisors, rudiments of scheme theory for the understanding of the Riemann-Roch theorem for curves and surfaces. Other topics may include Jacobian varieties, resolution of singularities, birational geometry on surfaces, schemes, connections with complex analytic geometry and topology. 

Credits: 0.00
Level: Graduate
Days/Times: TTh 12:00PM - 1:15PM
Instructor: Shokurov, Vyacheslav
Room: Krieger 411
Status: Open
Seats Available: 4/10
PosTag(s): n/a

Riemannian Geometry I
AS.110.645 (01)

This course is a graduate-level introduction to foundational material in Riemannian Geometry. Riemannian manifolds, a smooth manifold equipped with a Riemannian metric. Topics include connections, geodesics, Jacobi fields, submanifold theory including the second fundamental form and Gauss equations, manifolds of constant curvature, comparison theorems, Morse index theorem, Hadamard theorem and Bonnet-Myers theorem.

Credits: 0.00
Level: Graduate
Days/Times: TTh 10:30AM - 11:45AM
Instructor: Duncan, Jonah Alexander Jacob
Room: Maryland 110
Status: Open
Seats Available: 7/15
PosTag(s): n/a

Stochastic Differential Equations: An Introduction With Applications
AS.110.653 (01)

This course is an introduction to stochastic differential equations and applications. Basic topics to be reviewed include Ito and Stratonovich integrals, Ito formula, SDEs and their integration. The course will focus on diffusion processes and diffusion theory, with topics include Markov properties, generator, Kolmogrov’s equations (Fokker-Planck equation), Feynman-Kac formula, the martingale problem, Girsanov theorem, stability and ergodicity. The course will briefly introduce applications, with topics include statistical inference of SDEs, filtering and control.

Credits: 0.00
Level: Graduate
Days/Times: TTh 10:30AM - 11:45AM
Instructor: Lu, Fei
Room: Hodson 301
Status: Open
Seats Available: 21/30
PosTag(s): n/a

What is... Seminar
AS.110.710 (01)

This is a professional development course for graduate students, where they will learn, practice, or enhance their skills at giving math talks. The course will run in the format of a "What is... Seminar", where each week one of the participants will present a 1 hour talk on an accessible and relatively self-contained topic, titled What is (insert your math notion of choice). In preparation for their talk, students will meet with the instructor at least once, where they will receive guidance and detailed advice to help them give a great talk. Although the definition of a "great talk" is subjective, participants should be willing to follow the instructors' advice. Graduate students at any stage of their PhD are encouraged to attend, regardless of their experience giving talks.

Credits: 0.00
Level: Graduate
Days/Times: T 4:30PM - 5:45PM
Instructor: Sarazola Duarte, Maru Eugenia
Room: Latrobe 107
Status: Open
Seats Available: 2/8
PosTag(s): n/a

Topics in Mathematical Physics
AS.110.712 (01)

The course will be about scattering for Nonlinear Klein Gordon, Nonlinear Schroedinger and Nonlinear wave equations with variable coefficients in one space dimension. The ultimate goal is to show stability of stationary solutions, so called kinks, as well as moving soliton solutions for certain nonlinear equations of mathematical physics. Recently there has been a lot of activity in this area which uses some interesting math like spectral theory, estimates of Fourier Multipliers, twisted Fourier transform,... in order to study operators with variable coefficients.

Credits: 0.00
Level: Graduate
Days/Times: TTh 1:30PM - 2:45PM
Instructor: Lindblad, Hans
Room: Krieger 411
Status: Open
Seats Available: 6/10
PosTag(s): n/a

Topics In Alg Num Theory
AS.110.733 (01)

Credits: 0.00
Level: Graduate
Days/Times: TTh 12:00PM - 1:15PM
Instructor: Savitt, David Lawrence
Room: Krieger 413
Status: Open
Seats Available: 7/10
PosTag(s): n/a

Independent Study-Graduates
AS.110.800 (01)

This is an independent study course for students interested in a working with a professor on a specific topic.

Credits: 0.00
Level: Graduate Independent Academic Work
Days/Times:
Instructor: Riehl, EMILY
Room:
Status: Open
Seats Available: 4/5
PosTag(s): n/a

Independent Study-Graduates
AS.110.800 (02)

This is an independent study course for students interested in a working with a professor on a specific topic.

Credits: 0.00
Level: Graduate Independent Academic Work
Days/Times:
Instructor: Mramor, Alex Everest
Room:
Status: Open
Seats Available: 5/5
PosTag(s): n/a

Independent Study-Graduates
AS.110.800 (03)

This is an independent study course for students interested in a working with a professor on a specific topic.

Credits: 0.00
Level: Graduate Independent Academic Work
Days/Times:
Instructor: Kitchloo, Nitya
Room:
Status: Open
Seats Available: 5/5
PosTag(s): n/a

Independent Study-Graduates
AS.110.800 (04)

This is an independent study course for students interested in a working with a professor on a specific topic.

Credits: 0.00
Level: Graduate Independent Academic Work
Days/Times:
Instructor: Sogge, Chris
Room:
Status: Open
Seats Available: 4/5
PosTag(s): n/a

Independent Study-Graduates
AS.110.800 (14)

This is an independent study course for students interested in a working with a professor on a specific topic.

Credits: 0.00
Level: Graduate Independent Academic Work
Days/Times:
Instructor: Gepner, David James
Room:
Status: Open
Seats Available: 3/5
PosTag(s): n/a

Thesis Research
AS.110.801 (01)