### Current MSCS Graduate Courses ( Spring 2018, Fall 2018, Spring 2019 )

### 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 (Ross)- 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 (Furman)- 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 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.

### Fall 2018

#### MATH 504

Set Theory I (Sinapova)- PREREQUISITES:
- Math 430 or 502 or Phil 562.
- DESCRIPTION:
- Same as Phil 565. Naïve and axiomatic set theory. Independence of the continuum hypothesis and the axiom of choice.

#### MATH 512

Advanced Topics in Logic: Model Theory of Valued Fields (Marker)- PREREQUISITES:
- None.
- DESCRIPTION:
- Introduction to the model theory of valued fields including the work of Ax-Kochen and Ershov on the p-adics and Artin’s Conjecture and more recent work on algebrarically closed valued fields and their applications. This course will be relatively self-contained and both logical and algebraic prerequisites will be kept to a minimum.

#### 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 516

Abstract Algebra I (Zhang)- 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 (Switala)- 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 (Furman)- 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 (Sparber)- 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 546

Advanced Topics in Analysis: TBD (Furman)- PREREQUISITES:
- TBD
- DESCRIPTION:
- TBD

#### MATH 549

Differentiable Manifolds I (Groves)- 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 (Tucker)- 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 568

Topics in Algebraic Topology: Noncommutative Motives and Algebraic K-Theory (Antieau)- PREREQUISITES:
- None.
- DESCRIPTION:
- The topics will include algebraic K-theory, dg categories, Hochschild homology, noncommutative motives, ad t-structures.

#### MATH 571

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

#### MATH 577

Advanced Partial Differential Equations (Bona)- 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 (Cheskiodv)- 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 (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.

#### MATH 589

Teaching and Presentation of Mathematics (Bode)- PREREQUISITES:
- Approval of the Department
- DESCRIPTION:
- Strategies and techniques for effective teaching in college and for mathematical consulting. Observation and evaluation, classroom management, presenting mathematics in multidisciplinary research teams. Required for teaching assistants in MSCS. No graduation credit awarded for students enrolled in the Master of Science in the Teaching of Mathematics degree program.

### Spring 2019

#### MATH 512

Advanced Topics in Logic: Sinapova (Sinapova)- PREREQUISITES:
- TBD
- DESCRIPTION:
- TBD

#### 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 517

Second Course in Abstract Algebra II (Cojocaru)- 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: TBD (Takloo-Bighash)- PREREQUISITES:
- TBD
- DESCRIPTION:
- TBD

#### MATH 535

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

#### MATH 546

Advanced Topics in Analysis: Advanced Functional Analysis (Sparber)- PREREQUISITES:
- Math 533 and Math 539.
- DESCRIPTION:
- This course will focus on more advanced topics in functional analysis, usually not covered in the MATH 539. Among them are: The spectral theorem for bounded and unbounded self-adjoint operators; semi-groups of operators and applications to PDE; basic properties of locally convex spaces; perturbation theory for Schrödinger operators and application in quantum mechanics.

#### MATH 547

Algebraic Topology I (Whyte)- 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 550

Differentiable Manifolds II (Dumas)- 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 (Coskun)- PREREQUISITES:
- Math 552.
- DESCRIPTION:
- Divisors and linear systems, differentials, Riemann-Roch theorem for curves, elliptic curves, geometry of curves and surfaces.

#### MATH 576

Classical Methods of Partial Differential Equations (Dai)- 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