### Current MSCS Graduate Courses ( Fall 2020, Spring 2021 )

### Fall 2020

#### MATH 502

Mathematical Logic (Filippo Calderoni)- PREREQUISITES:
- MATH 430 or consent of the instructor.
- DESCRIPTION:
- First order logic, completeness and incompleteness theorems, introduction to model theory and computability theory. Same as PHIL 562.

#### MATH 514

Number Theory I (Alina Cojocaru)- PREREQUISITES:
- None
- DESCRIPTION:
- Introduction to classical, algebraic, and analytic, number theory. Euclid's algorithm, unique factorization, quadratic reciprocity, and Gauss sums, quadratic forms, real approximations, arithmetic functions, Diophantine equations.

#### MATH 516

Abstract Algebra I (Daniel Groves)- PREREQUISITES:
- MATH 330 and 425.
- DESCRIPTION:
- Structure of groups, Sylow theorems, solvable groups, structure of rings, polynomial rings, projective and injective modules, finitely generated modules over a PID.

#### MATH 520

Commutative and Homological Algebra (Rankeya Datta)- PREREQUISITES:
- MATH 516 and MATH 517; or consent of the instructor.
- DESCRIPTION:
- Commutative rings; primary decomposition; integral closure; valuations; dimension theory; regular sequences; projective and injective dimension; chain complexes and homology; Ext and Tor; Koszul complex; homological study of regular rings.

#### MATH 533

Real Analysis I (Irina Nenciu)- PREREQUISITES:
- MATH 411 or 414 or the equivalent.
- DESCRIPTION:
- Introduction to real analysis. Lebesgue measure and integration, differentiation, L-p classes, abstract integration.

#### MATH 539

Functional Analysis I (Roman Shvydkoy)- PREREQUISITES:
- MATH 533.
- DESCRIPTION:
- Topological vector spaces, Hilbert spaces, Hahn-Banach theorem, open mapping, uniform boundedness principle, linear operators in a Banach space, compact operators.

#### MATH 549

Differentiable Manifolds I (Julius Ross)- PREREQUISITES:
- Math 445; and Math 310 or Math 320 or the equivalent.
- DESCRIPTION:
- Smooth manifolds and maps, tangent and normal bundles, Sard's theorem and transversality, embedding, differential forms, Stokes' theorem, degree theory, vector fields.

#### MATH 552

Algebraic Geometry I (Izzet Coskun)- PREREQUISITES:
- None
- DESCRIPTION:
- Basic commutative algebra, affine and projective varieties, regular and rational maps, function fields, dimension and smoothness, projective curves, schemes, sheaves, and cohomology, positive characteristic.

#### MATH 577

Advanced Partial Differential Equations (Ian Tobasco)- PREREQUISITES:
- MATH 533 and MATH 576 or consent of the instructor.
- DESCRIPTION:
- Linear elliptic theory, maximum principles, fixed point methods, semigroups and nonlinear dynamics, systems of conservation laws, shocks and waves, parabolic equations, bifurcation, nonlinear elliptic theory.

#### MATH 580

Mathematics of Fluid Mechanics (Alexey Cheskidov)- PREREQUISITES:
- Grade of C or better in MATH 410 and grade of C or better in MATH 417 and grade of C or better in MATH 481
- DESCRIPTION:
- Development of concepts and techniques used in mathematical models of fluid motions. Euler and Navier Stokes equations. Vorticity and vortex motion. Waves and instabilities. Viscous fluids and boundary layers. Asymptotic methods

#### MATH 585

Ordinary Differential Equations (Rafail Abramov)- PREREQUISITES:
- C or better on MATH 480
- DESCRIPTION:
- The course focuses on qualitative studies and basic analysis of ordinary differential equations. Topics include classification of ODEs, basic wellposedness of initial value problems, Picar iterations, linear systems with constant and periodic coefficients, Floquet theory, dynamical systems, Poincare-Bendixon theorem, stability, elements of Hamiltonian mechanics, invariant manifolds.

### Spring 2021

Courses for this term have not been posted yet.