Recommended Courses for Research Areas

Below is a list of courses, recommended to students who wish to do research in specific areas. Special topics courses (MATH 597) are also offered in each area and vary from year to year.

• DYNAMICAL SYSTEMS
• Background material:
  
  • Lebesgue measure theory (MATH 501 or equivalent)
  
  • functional analysis (MATH 503 or equivalent)
  
  • linear algebra (MATH 535 or equivalent)
  
  • metric and topological spaces (MATH 527 or equivalent)
  
  Core courses:
  
  MATH 506: Ergodic Theory
  
  MATH 507: Dynamical Systems I
  
  MATH 508: Dynamical Systems II
  
  MATH 515: Classical Mechanics and Variational Methods Additional courses:
  
  MATH 533: Lie Theory I
  
  MATH 534: Lie Theory II

   

• GEOMETRY AND MATHEMATICAL PHYSICS
• Background material:
  
  • linear algebra (MATH 535)
  
  • abstract algebra (rings and modules, homological algebra) (MATH 536 or equivalent)
  
  • classical mechanics (MATH 515 or equivalent)
  
  • quantum mechanics
  
  Core courses:
  
  MATH 527: Metric and Topological Spaces
  
  MATH 528: Differentiable Manifolds
  
  Additional courses:
  
  MATH 529: Algebraic Topology
  
  MATH 533: Lie Theory I
  
  MATH 534: Lie Theory II
  
  MATH 547: Algebraic Geometry I
  
  MATH 548: Algebraic Geometry II

   

• FLUID MECHANICS
• Core courses:
  
  MATH 505: Fluid mechanics

   

• FUNCTIONAL ANALYSIS
• Core courses:
  
  MATH 520: Introduction to Operator Algebras
  
  Additional courses:
  
  MATH 582: C*-algebras
  
  MATH 583: K-thoery
  
  MATH 584: Von-Neumann Algebras

   

• LOGIC AND FOUNDATIONS
• Background material (will be covered in MATH 557 and 558):
  
  • the predicate calculus
  
  • theories and definability
  
  • computability and unsolvability
  
  • elements of axiomatic set theory
  
  Core courses:
  
  MATH 557: Mathematical Logic
  
  MATH 558: Foundations of Mathematics I
  
  MATH 559: Recursion Theory I
  
  MATH 561: Set Theory I
  
  MATH 565: Foundations of Mathematics II
  
  Additional courses:
  
  MATH 574: Topics in Logic

   

• NUMBER THEORY AND COMBINATORICS
• Background material:
  
  • algebra, including groups, rings, modules, fields, Galois theory (MATH 536 or equivalent)
  
  • complex analysis (MATH 502 or equivalent)
  
  • linear algebra (MATH 535 or equivalent)
  
  Core courses:
  
  MATH 567: Number Theory I
  
  MATH 568: Number Theory I
  
  Additional courses:
  
  MATH 569: Algebraic Number Theory I
  
  MATH 570: Algebraic Number Theory I i
  
  MATH 571: Analytic Number Theory I i
  
  MATH 572: Analytic Number Theory IIi
  
  Students interested in algebraic number theory are highly recommend also to take
  
  MATH 547: Algebraic Geometry I
  
  MATH 548: Algebraic Geometry II

   

• NUMERICAL ANALYSIS
• Background material:
  
  • calculus in several space dimensions
  
  • linear algebra and some functional analysis (MATH 535, MATH 524, MATH 503 or equivalent)
  
  • elementary O.D.E. theory, basic computer programming skills
  
  Core courses:
  
  MATH 523: Numerical analysis I
  
  MATH 551 (CSE) Numerical solution of ODEs
  
  MATH 552 (CSE) Numerical solution of PDEs
  
  Additional courses:
  
  MATH 550 (CSE) Numerical Linear Algebra
  
  MATH 553 Introduction to Approximation Theory
  
  MATH 554 Approximation Theory
  
  MATH 555 (CSE) Numerical Optimization
  
  MATH 556 Finite Element Methods

   

• PARTIAL DIFFERENTIAL EQUATIONS
• Background material:
  
  • calculus in several space dimensions
  
  • linear algebra (MATH 535 or equivalent)
  
  • elementary O.D.E. theory
  
  • basics of Lebesgue measure theory and functional analysis (MATH 501, MATH 503 or equivalent)
  
  Core courses:
  
  MATH 513: PDE I
  
  MATH 514: PDE II
  
  Additional courses:
  
  MATH 552 (CSE): Numerical PDE

   

• RIEMANNIAN AND METRIC GEOMETRY
• Background material:
  
  • metric and topological spaces (MATH 527 or equivalent)
  
  • differentiable manifolds (MATH 528 or equivalent)
  
  Core courses:
  
  MATH 530: Differential geometry
  
  MATH 531: Differential topology
  
  Additional courses:
  
  MATH 533: Lie Theory I
  
  MATH 534: Lie Theory II