# 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****GEOMETRY AND MATHEMATICAL PHYSICS****FLUID MECHANICS****FUNCTIONAL ANALYSIS****LOGIC AND FOUNDATIONS****NUMBER THEORY AND COMBINATORICS****NUMERICAL ANALYSIS****PARTIAL DIFFERENTIAL EQUATIONS****RIEMANNIAN AND METRIC GEOMETRY**

**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

**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

**Core courses:**

MATH 505: Fluid mechanics

**Core courses:**

MATH 520: Introduction to Operator Algebras

**Additional courses:**

MATH 582: C*-algebras

MATH 583: K-thoery

MATH 584: Von-Neumann Algebras

**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

**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

**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

**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

**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