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

### Spring 2020

#### MATH 506

Model Theory I (James Freitag)- PREREQUISITES:
- MATH 502 or PHIL 562.
- DESCRIPTION:
- Elementary embeddings, quantifier elimination, types, saturated and prime models, indiscernibles, Morley's Categoricity Theorem. Same as PHIL 567.

#### MATH 512

Advanced Topics in Logic: TBA (Filippo Calderoni)- PREREQUISITES:
- No prerequisites will be assumed beyond general mathematical maturity.
- DESCRIPTION:
- This course is an introduction to the theory of countable Borel equivalence relation. This subject has a broad impact in several areas of mathematics like ergodic theory, operator algebras, and group theory, and has raised many challenging open problems. We will analyze group actions in the descriptive, topological, and measure theoretic context. We will go through some remarkable results showing that the problem of determining whether two torsion-free abelian groups are isomorphic gets more and more complicated as the rank increases. This demonstrates evidence against satisfactory classification for torsion-free abelian groups of rank bigger than 2.

#### MATH 515

Number Theory II (Ramin Takloo-Bighash)- 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 (Wenliang Zhang)- PREREQUISITES:
- MATH 516.
- DESCRIPTION:
- Rings and algebras, polynomials in several variables, power series rings, tensor products, field extensions, Galois theory, Wedderburn theorems.

#### MATH 535

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

#### MATH 537

Introduction to Harmonic Analysis I (Michael 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 548

Algebraic Topology II (Benjamin 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 549

Differentiable Manifolds I (Daniel 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 553

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

#### MATH 555

Complex Manifolds II (Julius Ross)- PREREQUISITES:
- MATH 517 and MATH 535.
- DESCRIPTION:
- Dolbeault Cohomology, Serre duality, Hodge theory, Kadaira vanishing and embedding theorem, Lefschitz theorem, Complex Tori, Kahler manifolds.

#### MATH 569

Advanced Topics in Geometric and Differential Topology (Alex Furman)- PREREQUISITES:
- TBA
- DESCRIPTION:
- Introduction to Lie Groups and Lie Algebras: Basic examples, correspondence between Lie Groups and Lie algebras, nilpotent solvable, semi-siimple groups/algebras, complex semi-simple Lie Algebras and their root systems.

#### MATH 576

Classical Methods of Partial Differential Equations (Christof Sparber)- 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 581

Special Topics in Fluid Mechanics (Jerry Bona)- PREREQUISITES:
- Background in analysis and functional analysis.
- DESCRIPTION:
- Mathematical and Numerical Analysis of Fluid Mechanics: Derivation of models for water waves, Discontinuous Galerkin Methods for wave equations, Positive Operator Theory, viscous flow, systems of equations in internal wave propagation, Laplace Transform Methods for initial-value problems.

#### MATH 584

Applied Stochastic Models (Gerard 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.

### Fall 2020

Courses for this term have not been posted yet.

### Spring 2021

Courses for this term have not been posted yet.