### Current MSCS Graduate Courses ( Fall 2017, Spring 2018 )

### Fall 2017

#### MATH 502

Mathematical Logic (Sinapova)- 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 512

Advanced Topics in Logic (Freitag)- PREREQUISITES:
- TBD
- DESCRIPTION:
- TBA

#### MATH 515

Number Theory II (Jones)- PREREQUISITES:
- MATH 514.
- DESCRIPTION:
- Introduction to classical, algebraic, and analytic number theory. Algebraic number fields, units, ideals, and P-adic theory. Riemann Zeta-function, Dirichlet's theorem, prime number theorem.

#### MATH 516

Abstract Algebra I (Ein)- 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 533

Real Analysis I (Cheskidov)- 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 (Bona)- 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 547

Algebraic Topology I (Antieau)- PREREQUISITES:
- MATH 330 and MATH 445.
- DESCRIPTION:
- The fundamental group and its applications, covering spaces, classification of compact surfaces, introduction to homology, development of singular homology theory, applications of homology.

#### MATH 549

Differentiable Manifolds I (Dumas)- 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 551

Riemannian Geometry (Whyte)- PREREQUISITES:
- Math 442 and 549.
- DESCRIPTION:
- Riemannian metrics and Levi-Civita connections, geodesics and completeness, curvature, first and second variation of arc length, comparison theorems.

#### MATH 552

Algebraic Geometry I (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 554

Complex Manifolds I (Paun)- PREREQUISITES:
- MATH 517 and MATH 535.
- DESCRIPTION:
- Holomorphic functions in several variables, Riemann surfaces, Sheaf theory, vector bundles, Stein manifolds, Cartan theorem A and B, Grauert direct image theorem.

#### MATH 569

Advanced Topics in Geometric and Differential Topology: Introduction to geometric group theory (Groves)- PREREQUISITES:
- Math 547 or approval of the department.
- DESCRIPTION:
- I will give an introduction to many of the basic concepts of geometric group theory, such as growth, curvature, isoperimetric functions, quasi-isometries and others. Along the way we will see many of the fundamental examples of the subject and also see some of the topics of current research interest.

#### MATH 582

Special Topics Course (Sparber)- PREREQUISITES:
- MATH 480 and MATH 481; or consent of the instructor.
- DESCRIPTION:
- Study of various linear and nonlinear PDE describing wave phenomena. Topics include: shock formation in hyperbolic equations, weak solutions and entropy conditions, study of the Cauchy problems for linear and nonlinear dispersive equations, solitary waves.

#### MATH 584

Applied Stochastic Models (Awanou)- PREREQUISITES:
- Stat 401 and MATH 417 and 481, or consent of the instructor.
- DESCRIPTION:
- Applications of stochastic models in chemistry, physics, biology, queueing, filtering, and stochastic control, diffusion approximations, Brownian motion, stochastic calculus, stochastically perturbed dynamical systems, first passage times.

#### MATH 585

Ordinary Differential Equations (Shvydkoy)- 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 2018

#### MATH 511

Descriptive Set Theory (Sinapova)- PREREQUISITES:
- Recommended background: MATH 445 or MATH 504 or MATH 533 or MATH 539.
- DESCRIPTION:
- Polish spaces and Baire category; Borel, analytic and coanalytic sets; infinite games and determinacy; coanalytic ranks and scales; dichotomy theorems.

#### MATH 514

Number Theory I (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 517

Second Course in Abstract Algebra II (Tucker)- PREREQUISITES:
- MATH 516.
- DESCRIPTION:
- Rings and algebras, polynomials in several variables, power series rings, tensor products, field extensions, Galois theory, Wedderburn theorems.

#### MATH 525

Advanced Topics in Number Theory (Jones)- PREREQUISITES:
- TBD
- DESCRIPTION:
- TBA

#### MATH 535

Complex Analysis I (Furman)- PREREQUISITES:
- MATH 411.
- DESCRIPTION:
- Analytic functions as mappings. Cauchy theory. Power Series. Partial fractions. Infinite products.

#### MATH 537

Introduction to Harmonic Analysis I (Greenblatt)- PREREQUISITES:
- Math 533, plus Math 417 or Math 535.
- DESCRIPTION:
- Text: J. Duoandikoetxea "Fourier Analysis", AMS, Grad Stud in Math, Vol 29. Excellent concise text on harmonic analysis with a good selection of exercises. Topics will include Fourier Transform, Hardy-Littlewood maximal function, singular integral operators and Hilbert transform, Littlewood-Paley theory, Sobolev and Besov spaces, BMO, and Hardy spaces, Bernstein inequalities, Carleson measures, applications to PDE.

#### MATH 547

Algebraic Topology I (Antieau)- PREREQUISITES:
- MATH 330 and MATH 445.
- DESCRIPTION:
- The fundamental group and its applications, covering spaces, classification of compact surfaces, introduction to homology, development of singular homology theory, applications of homology.

#### MATH 548

Algebraic Topology II (Antieau)- PREREQUISITES:
- MATH 547.
- DESCRIPTION:
- Cohomology theory, universal coefficient theorems, cohomology products and their applications, orientation and duality for manifolds, homotopy groups and fibrations, the Hurewicz theorem, selected topics.

#### MATH 550

Differentiable Manifolds II (TBD)- PREREQUISITES:
- Math 549.
- DESCRIPTION:
- Vector bundles and classifying spaces, Lie groups and Lie algebras, tensors, Hodge theory, Poincare duality. Topics from elliptic operators, Morse theory, cobordism theory, de Rahm theory, characteristic classes.

#### MATH 553

Algebraic Geometry II (Tucker)- PREREQUISITES:
- Math 552.
- DESCRIPTION:
- Divisors and linear systems, differentials, Riemann-Roch theorem for curves, elliptic curves, geometry of curves and surfaces.

#### MATH 571

Advanced Topics in Algebraic Geometry (Zhang)- PREREQUISITES:
- TBD
- DESCRIPTION:
- TBA

#### MATH 576

Classical Methods of Partial Differential Equations (Shvydkoy)- PREREQUISITES:
- MATH 410 and MATH 481 and MATH 533; or consent of instructor
- DESCRIPTION:
- First and second order equations, method of characteristics, weak solutions, distributions, wave, Laplace, Poisson, heat equations, energy methods, regularity problems, Green functions, maximum principles, Sobolev spaces, imbedding theorems

#### MATH 582

Linear and Nonlinear Waves (Sparber)- PREREQUISITES:
- MATH 480 and MATH 481; or consent of the instructor.
- DESCRIPTION:
- Topics to be covered include: Fourier transforms; L^2-based Sobolev spaces and Schwartz space distributions; Well-posedness theory for dispersive equations (mainly Nonlinear Schrödinger and Korteweg de Vries); Energy methods; Existence of Solutions to Semi-linear Wave Equations;

#### MATH 586

Computational Finance (David Nicholls)- PREREQUISITES:
- Grade of C or better in MATH 220 and grade of C or better in STAT 381; or consent of the instructor.
- DESCRIPTION:
- Introduction to the mathematics of financial derivatives; options, asset price random walks, Black-Scholes model; partial differential techniques for option valuation, binomial models, numerical methods; exotic options, interest-rate derivatives.