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.

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.

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: Shokurov, Vyacheslav
  • Room:    
  • Status: Open
  • Seats Available: 16/20
  • 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, Christopher
  • Room: Krieger 302  
  • Status: Open
  • Seats Available: 14/20
  • 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
  • Days/Times: MW 12:00PM - 1:15PM
  • Instructor: Bernstein, Jacob
  • Room:    
  • Status: Open
  • Seats Available: 17/20
  • 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: Gepner, David James
  • Room:    
  • Status: Open
  • Seats Available: 16/20
  • 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 10:30AM - 11:45AM
  • Instructor: Staff
  • Room:    
  • Status: Open
  • Seats Available: 16/20
  • 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: TTh 1:30PM - 2:45PM
  • Instructor: Sire, Yannick
  • Room:    
  • Status: Open
  • Seats Available: 19/20
  • 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 1:30PM - 2:45PM
  • Instructor: Lindblad, Hans
  • Room: Hackerman 320  
  • Status: Open
  • Seats Available: 11/20
  • 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 10:30AM - 11:45AM
  • Instructor: Shokurov, Vyacheslav
  • Room:    
  • Status: Open
  • Seats Available: 17/20
  • PosTag(s): n/a

Riemannian Geometry II
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: Wang, Yi
  • Room: Gilman 186  
  • Status: Open
  • Seats Available: 12/20
  • 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 216  
  • Status: Open
  • Seats Available: 5/25
  • PosTag(s): n/a

Topics In Homotopy Type Theory
AS.110.721 (01)

Homotopy type theory (HoTT) is a new proposed foundation system for mathematics that extends Martin-Löf's dependent type theory with Voevodsky's univalence axiom. Dependent type theory is a formal system for constructive mathematics, in which a theorem is proven by constructing a term in the type that encodes its statement. In Homotopy type theory, types are thought of as spaces and terms as points in those spaces. A proof that two terms in a common type are equal is now interpreted as a path between two points in a space. In particular, types might have interesting higher homotopical structure, which can be thought of as revealing fundamental differences between two proofs of a common proposition. One advantage of this foundation system is its amenability to computer formalization, which this course will illustrate by introducing the computer proof assistant Agda.

  • Credits: 0.00
  • Level: Graduate
  • Days/Times: MW 1:30PM - 2:45PM
  • Instructor: Riehl, Emily
  • Room: Maryland 109  
  • Status: Open
  • Seats Available: 3/15
  • PosTag(s): n/a

Topics in Partial Differential Equations
AS.110.741 (01)

  • Credits: 0.00
  • Level: Graduate
  • Days/Times: MW 1:30PM - 2:45PM
  • Instructor: Dodson, Benjamin
  • Room:    
  • Status: Open
  • Seats Available: 6/10
  • PosTag(s): n/a

Independent Study-Graduates
AS.110.800 (01)

  • Credits: 0.00
  • Level: Graduate
  • Days/Times:
  • Instructor: Riehl, Emily
  • Room:    
  • Status: Open
  • Seats Available: 5/5
  • PosTag(s): n/a

Independent Study-Graduates
AS.110.800 (02)

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

Independent Study-Graduates
AS.110.800 (03)

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

Independent Study-Graduates
AS.110.800 (14)

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

Thesis Research
AS.110.801 (01)

  • Credits: 0.00
  • Level: Graduate
  • Days/Times:
  • Instructor: Riehl, Emily
  • Room:    
  • Status: Open
  • Seats Available: 23/25
  • PosTag(s): n/a

Thesis Research
AS.110.801 (02)

  • Credits: 0.00
  • Level: Graduate
  • Days/Times:
  • Instructor: Consani, Caterina
  • Room:    
  • Status: Open
  • Seats Available: 24/25
  • PosTag(s): n/a

Thesis Research
AS.110.801 (03)

  • Credits: 0.00
  • Level: Graduate
  • Days/Times:
  • Instructor: Lindblad, Hans
  • Room:    
  • Status: Open
  • Seats Available: 22/25
  • PosTag(s): n/a

Thesis Research
AS.110.801 (04)

  • Credits: 0.00
  • Level: Graduate
  • Days/Times:
  • Instructor: Sogge, Christopher
  • Room:    
  • Status: Open
  • Seats Available: 25/25
  • PosTag(s): n/a

Thesis Research
AS.110.801 (05)

  • Credits: 0.00
  • Level: Graduate
  • Days/Times:
  • Instructor: Sakellaridis, Ioannis
  • Room:    
  • Status: Open
  • Seats Available: 25/25
  • PosTag(s): n/a

Thesis Research
AS.110.801 (06)

  • Credits: 0.00
  • Level: Graduate
  • Days/Times:
  • Instructor: Maggioni, Mauro
  • Room:    
  • Status: Open
  • Seats Available: 25/25
  • PosTag(s): n/a

Thesis Research
AS.110.801 (07)

  • Credits: 0.00
  • Level: Graduate
  • Days/Times:
  • Instructor: Kitchloo, Nitya
  • Room:    
  • Status: Open
  • Seats Available: 25/25
  • PosTag(s): n/a

Thesis Research
AS.110.801 (08)

  • Credits: 0.00
  • Level: Graduate
  • Days/Times:
  • Instructor: Shokurov, Vyacheslav
  • Room:    
  • Status: Open
  • Seats Available: 24/25
  • PosTag(s): n/a

Thesis Research
AS.110.801 (09)

  • Credits: 0.00
  • Level: Graduate
  • Days/Times:
  • Instructor: Wang, Yi
  • Room:    
  • Status: Open
  • Seats Available: 9/10
  • PosTag(s): n/a

Thesis Research
AS.110.801 (10)

  • Credits: 0.00
  • Level: Graduate
  • Days/Times:
  • Instructor: Bernstein, Jacob
  • Room:    
  • Status: Open
  • Seats Available: 4/5
  • PosTag(s): n/a

Thesis Research
AS.110.801 (11)

  • Credits: 0.00
  • Level: Graduate
  • Days/Times:
  • Instructor: Savitt, David Lawrence
  • Room:    
  • Status: Open
  • Seats Available: 2/5
  • PosTag(s): n/a

Thesis Research
AS.110.801 (12)

  • Credits: 0.00
  • Level: Graduate
  • Days/Times:
  • Instructor: Dodson, Benjamin
  • Room:    
  • Status: Open
  • Seats Available: 5/5
  • PosTag(s): n/a

Thesis Research
AS.110.801 (13)

  • Credits: 4.00
  • Level: Graduate
  • Days/Times:
  • Instructor: Sire, Yannick
  • Room:    
  • Status: Open
  • Seats Available: 4/5
  • PosTag(s): n/a

Thesis Research
AS.110.801 (15)

  • Credits: 0.00
  • Level: Graduate
  • Days/Times:
  • Instructor: Lu, Fei
  • Room:    
  • Status: Open
  • Seats Available: 2/5
  • PosTag(s): n/a

Course # (Section) Title Day/Times Instructor Room PosTag(s) Info
AS.110.601 (01)Algebra ITTh 12:00PM - 1:15PMShokurov, Vyacheslav 
AS.110.605 (01)Real VariablesMW 12:00PM - 1:15PMSogge, ChristopherKrieger 302
AS.110.608 (01)Riemann SurfacesMW 12:00PM - 1:15PMBernstein, Jacob 
AS.110.615 (01)Algebraic Topology IMW 1:30PM - 2:45PMGepner, David James 
AS.110.617 (01)Number Theory ITTh 10:30AM - 11:45AMStaff 
AS.110.631 (01)Partial Differential Equations ITTh 1:30PM - 2:45PMSire, Yannick 
AS.110.637 (01)Functional AnalysisMW 1:30PM - 2:45PMLindblad, HansHackerman 320
AS.110.643 (01)Algebraic Geometry ITTh 10:30AM - 11:45AMShokurov, Vyacheslav 
AS.110.645 (01)Riemannian Geometry IITTh 10:30AM - 11:45AMWang, YiGilman 186
AS.110.653 (01)Stochastic Differential Equations: An Introduction With ApplicationsTTh 10:30AM - 11:45AMLu, FeiHodson 216
AS.110.721 (01)Topics In Homotopy Type TheoryMW 1:30PM - 2:45PMRiehl, EmilyMaryland 109
AS.110.741 (01)Topics in Partial Differential EquationsMW 1:30PM - 2:45PMDodson, Benjamin 
AS.110.800 (01)Independent Study-GraduatesRiehl, Emily 
AS.110.800 (02)Independent Study-GraduatesMramor, Alexander Everest 
AS.110.800 (03)Independent Study-GraduatesKitchloo, Nitya 
AS.110.800 (14)Independent Study-GraduatesGepner, David James 
AS.110.801 (01)Thesis ResearchRiehl, Emily 
AS.110.801 (02)Thesis ResearchConsani, Caterina 
AS.110.801 (03)Thesis ResearchLindblad, Hans 
AS.110.801 (04)Thesis ResearchSogge, Christopher 
AS.110.801 (05)Thesis ResearchSakellaridis, Ioannis 
AS.110.801 (06)Thesis ResearchMaggioni, Mauro 
AS.110.801 (07)Thesis ResearchKitchloo, Nitya 
AS.110.801 (08)Thesis ResearchShokurov, Vyacheslav 
AS.110.801 (09)Thesis ResearchWang, Yi 
AS.110.801 (10)Thesis ResearchBernstein, Jacob 
AS.110.801 (11)Thesis ResearchSavitt, David Lawrence 
AS.110.801 (12)Thesis ResearchDodson, Benjamin 
AS.110.801 (13)Thesis ResearchSire, Yannick 
AS.110.801 (15)Thesis ResearchLu, Fei