### Current MSCS Graduate Courses ( Spring 2019, Fall 2019, Spring 2020 )

### Spring 2019

#### MATH 512

Advanced Topics in Logic: Combinatorial Set Theory (Sinapova)- PREREQUISITES:
- DESCRIPTION:
- This course is on applications of forcing and large cardinals to infinitary combinatorics. We will start with an introduction to cardinal arithmetic, especially at singular cardinals, large cardinals, and forcing techniques. Then we will analyze their interactions with combinatorial principles like square, the tree property, and strengthenings of the tree property such as ITP.

#### 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

### Fall 2019

#### MATH 502

Mathematical Logic (Dima 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 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 525

Advanced Topics in Number Theory (Nathan Jones)- PREREQUISITES:
- MATH 515; or consent of the instructor
- DESCRIPTION:
- Introduction to topics at the forefront of research in number theory. Topics will vary and may include elliptic curves, automorphic forms, diophantine geometry or sieve methods. ATH

#### MATH 533

Real Analysis I (Alex 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 (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 547

Algebraic Topology I (Ben 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 551

Riemannian Geometry (Wouter Van Limbeek)- 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 (Howard Nuer)- 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 (Izzet Coskun)- 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 571

Advanced Topics in Algebraic Geometry (Wenliang Zhang)- PREREQUISITES:
- Approval of the department
- DESCRIPTION:
- Various topics such as algebraic curves, surfaces, higher dimensional geometry, singularities theory, moduli problems, vector bundles, intersection theory, arithematical algebraic geometry, and topologies of algebraic varieties.

#### MATH 580

Mathematics of Fluid Mechanics (Mimi Dai)- 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 (Alexey Cheskidov)- 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 (Brooke Shipley)- 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 2020

Courses for this term have not been posted yet.