Course Descriptions

Elements of topology on the real line. Rigorous treatment of limits, continuity, differentiation, and the Riemann integral. Taylor series. Introduction to metric spaces. Pointwise and uniform convergence for sequences and series of functions. Applications.

Recitation/ Problem Session: Monday, 7:10-8:30 (following the course)

Prerequisites:

A good background in linear algebra, and some experience with writing computer programs (in MATLAB, Python or another language). MATLAB will be used as the main language for the course. Alternatively, you can also use Python for the homework assignments. You are encouraged but not required to learn and use a compiled language.

Description:

This course is part of a two-course series meant to introduce graduate students in mathematics to the fundamentals of numerical mathematics (but any Ph.D. student seriously interested in applied mathematics should take it). It will be a demanding course covering a broad range of topics. There will be extensive homework assignments involving a mix of theory and computational experiments, and an in-class final. Topics covered in the class include floating-point arithmetic, solving large linear systems, eigenvalue problems, interpolation and quadrature (approximation theory), nonlinear systems of equations, linear and nonlinear least squares, nonlinear optimization, and Fourier transforms. This course will not cover differential equations, which form the core of the second part of this series, Numerical Methods II.

Recommended Text (Springer books are available online from the NYU network):

• Deuflhard, P. & Hohmann, A. (2003). Numerical Analysis in Modern Scientific Computing. Texts in Applied Mathematiks [Series, Bk. 43]. New York, NY: Springer-Verlag.

Further Reading (available on reserve at the Courant Library):

• Bau III, D., & Trefethen, L.N. (1997). Numerical Linear Algebra. Philadelphia, PA: Society for Industrial & Applied Mathematics.
• Quarteroni, A., Sacco, R., & Saleri, F. (2006). Numerical Mathematics (2^nd ed.). Texts in Applied Mathematics [Series, Bk. 37]. New York, NY: Springer-Verlag.

If you want to brush up your MATLAB:

• Gander, W., Gander, M.J., & Kwok, F. (2014). Scientific Computing – An Introduction Using Maple and MATLAB. Texts in Computation Science and Engineering [Series, Vol. 11]. New York, NY: Springer-Verlag.
• Moler, C. (2004). Numerical Computing with Matlab. SIAM. Available online.

Cross-listing: CSCI-GA 2420.001

Prerequisites:

Graduate Linear Algebra, Numerical Methods, PDEs. In case of doubt, please contact instructor.

Description:

This course provides an introduction to inverse problems that are governed by systems of partial differential equations (PDEs), and to their numerical solution. The focus of the course is on variational formulations, ill-posedness, regularization, variational discretization, large-scale solution algorithms for inverse problems and the computation of derivatives using adjoint methods. Depending on the interest of the participants, the course will provide an introduction to the Bayesian framework for inverse problems additionally to the deterministic approach, and will draw connections between the two. Examples will be drawn from different areas of science and engineering, including image processing, continuum mechanics, and geophysics.

Recommended texts:

• C. Vogel: Computational Methods for Inverse Problems, SIAM 2002.
• H. Engl, M. Hanke, A. Neubauer: Regularization of Inverse Problems, Dordrecht, 2nd edition, 1996.
• F. Troeltzsch: Optimal Control of Partial Differential Equations: Theory, Methods and Applications, Graduate Studies in Mathematics, Vol. 112, AMS, 2010.
• A. Tarantola: Inverse Problem Theory and Methods for Model Parameter Estimation, SIAM 2005.

Description:

This six-week course will be structured in an unusual way, following the pattern of last year's course but with new topics.  Each of our six meetings will be independent.  At each meeting, the first hour will be a lecture aimed at anyone interested in numerical analysis at a high level, organized around a well-known topic and mixing historical perspectives, recent developments, and always some new mathematics. The second hour will be for enrolled students only, a hands-on work session making use of Matlab and/or Chebfun.  In the final two weeks each student will complete a small project.

Tentative schedule of topics:

•    Lecture 1, 04 Sep: Series solutions of Laplace problems
•    Lecture 2, 11 Sep: Pseudospectra
•    Lecture 3, 18 Sep: Block operators and spectral discretizations
•    Lecture 4, 25 Sep: Exponential integrators for stiff PDEs
•    Lecture 5, 02 Oct: Minimax, CF, and Hankel norm approximation
•    Lecture 6, 09 Oct: Polynomials and multivariate polynomials

[Possible alternative topic: Contour integrals and hyperfunctions]

Description:

This course will cover advanced numerical techniques for solving PDEs, with a particular focus on fluid dynamics. This includes advection-diffusion-reaction equations, compressible and incompressible Navier-Stokes equations, and fluid-structure coupling. Basic familiarity with temporal integrators for ODEs (multistep, Runge-Kutta), methods for solving PDEs (finite difference, finite volume, finite elements for parabolic and elliptic problems), iterative solvers for linear systems, and the Navier-Stokes equations will be assumed. Topics covered will include:

•    higher-order spatio-temporal discretizations for advection-diffusion equations
•    artificial dissipation and dispersion
•    conservation laws (limiters, high-resolution advection)
•    incompressible flow (projection methods, Stokes solvers, spectral methods)
•    fluid-structure coupling (boundary-integral formulations, immersed boundary methods)
•    geo-physical dynamics (shallow water, wave equations, turbulent flows)

This course will introduce students to the software development process, including applications in financial asset trading, research, hedging, portfolio management, and risk management. Students will use the Java programming language to develop object-oriented software, and will focus on the most broadly important elements of programming - superior design, effective problem solving, and the proper use of data structures and algorithms. Students will work with market and historical data to run simulations and test strategies. The course is designed to give students a feel for the practical considerations of software development and deployment. Several key technologies and recent innovations in financial computing will be presented and discussed.

Prerequisites: Undergraduate multivariate calculus and linear algebra. Programming experience strongly recommended but not required.

Overview: This course is intended to provide a practical introduction to computational problem solving. Topics covered include: the notion of well-conditioned and poorly conditioned problems, with examples drawn from linear algebra; the concepts of forward and backward stability of an algorithm, with examples drawn from floating point arithmetic and linear-algebra; basic techniques for the numerical solution of linear and nonlinear equations, and for numerical optimization, with examples taken from linear algebra and linear programming; principles of numerical interpolation, differentiation and integration, with examples such as splines and quadrature schemes; an introduction to numerical methods for solving ordinary differential equations, with examples such as multistep, Runge Kutta and collocation methods, along with a basic introduction of concepts such as convergence and linear stability; An introduction to basic matrix factorizations, such as the SVD; techniques for computing matrix factorizations, with examples such as the QR method for finding eigenvectors; Basic principles of the discrete/fast Fourier transform, with applications to signal processing, data compression and the solution of differential equations.

This is not a programming course but programming in homework projects with MATLAB/Octave and/or C is an important part of the course work. As many of the class handouts are in the form of MATLAB/Octave scripts, students are strongly encouraged to obtain access to and familiarize themselves with these programming environments.

Recommended Texts:

• Bau III, D., & Trefethen, L.N. (1997). Numerical Linear Algebra. Philadelphia, PA: Society for Industrial & Applied Mathematics
• Quarteroni, A.M., & Saleri, F. (2006). Texts in Computational Science & Engineering [Series, Bk. 2]. Scientific Computing with MATLAB and Octave (2^nd ed.). New York, NY: Springer-Verlag
• Otto, S.R., & Denier, J.P. (2005). An Introduction to Programming and Numerical Methods in MATLAB. London: Springer-Verlag London

Cross-listing: CSCI-GA 2112.001

Prerequisites: Continuous Time Finance or permission of instructor.

The classical curriculum of mathematical finance programs generally covers the link between linear parabolic partial differential equations (PDEs) and stochastic differential equations (SDEs), resulting from Feynmam-Kac's formula. However, the challenges faced by today's practitioners mostly involve nonlinear PDEs. The aim of this course is to provide the students with the mathematical tools and computational methods required to tackle these issues, and illustrate the methods with practical case studies such as American option pricing, uncertain volatility, uncertain mortality, different rates for borrowing and lending, calibration of models to market smiles, credit valuation adjustment (CVA), portfolio optimization, transaction costs, illiquid markets, super-replication under delta and gamma constraints, etc. 

We will strive to make this course reasonably comprehensive, and to find the right balance between ideas, mathematical theory, and numerical implementations. We will spend some time on the theory: optimal stopping, stochastic control, backward stochastic differential equations (BSDEs), McKean SDEs, branching diffusions. But the main focus will deliberately be on ideas and numerical examples, which we believe help a lot in understanding the tools and building intuition.

Recommended text: Guyon, J. and Henry-Labordère, P.: Nonlinear Option Pricing, Chapman & Hall/CRC Financial Mathematics Series, 2014.

Prerequisites: Derivative Securities, Risk & Portfolio Management with Econometrics, and Computing in Finance (or equivalent programming experience).

A rigorous background in Bayesian statistics geared towards applications in finance, including decision theory and the Bayesian approach to modeling, inference, point estimation, and forecasting, sufficient statistics, exponential families and conjugate priors, and the posterior predictive density. A detailed treatment of multivariate regression including Bayesian regression, variable selection techniques, multilevel/hierarchical regression models, and generalized linear models (GLMs). Inference for classical time-series models, state estimation and parameter learning in Hidden Markov Models (HMMs) including the Kalman filter, the Baum-Welch algorithm and more generally, Bayesian networks and belief propagation. Solution techniques including Markov Chain Monte Carlo methods, Gibbs Sampling, the EM algorithm, and variational mean field. Real world examples drawn from finance to include stochastic volatility models, portfolio optimization with transaction costs, risk models, and multivariate forecasting.

Prerequisites:  Risk & Portfolio Management with Econometrics, Scientific Computing in Finance (or Scientific Computing) and Computing in Finance (or equivalent programming experience.

This is a full semester course focusing on practical aspects of alternative data, machine learning and data science in quantitative finance. Homework and hands-on projects form an integral part of the course, where students get to explore real-world datasets and software.

The course begins with an overview of the field, its technological and mathematical foundations, paying special attention to differences between data science in finance and other industries. We review the software that will be used throughout the course.

We examine the basic problems of supervised and unsupervised machine learning, and learn the link between regression and conditioning. Then we deepen our understanding of the main challenge in data science – the curse of dimensionality – as well as the basic trade-off of variance (model parsimony) vs. bias (model flexibility).

Demonstrations are given for real world data sets and basic data acquisition techniques such as web scraping and the merging of data sets. As homework each student is assigned to take part in downloading, cleaning, and testing data in a common repository, to be used at later stages in the class.

We examine linear and quadratic methods in regression, classification and unsupervised learning. We build a BARRA-style implicit risk-factor model and examine predictive models for county-level real estate, economic and demographic data, and macro economic data. We then take a dive into PCA, ICA and clustering methods to develop global macro indicators and estimate stable correlation matrices for equities. 

In many real-life problems, one needs to do SVD on a matrix with missing values. Common applications include noisy image-recognition and recommendation systems. We discuss the Expectation Maximization algorithm, the L1-regularized Compressed Sensing algorithm, and a naïve gradient search algorithm. 

The rest of the course focuses on non-linear or high-dimensional supervised learning problems. First, kernel smoothing and kernel regression methods are introduced as a way to tackle non-linear problems in low dimensions in a nearly model-free way. Then we proceed to generalize the kernel regression method in the Bayesian Regression framework of Gaussian Fields, and for classification as we introduce Support Vector Machines, Random Forest regression, Neural Nets and Universal Function Approximators.

Prerequisites: Undergraduate linear algebra or permission of the instructor.

Linear spaces, subspaces. Linear dependence, linear independence; span, basis, dimension, isomorphism. Quotient spaces. Linear functionals, Dual spaces. Linear mappings, null space, range, fundamental theorem of linear algebra. Underdetermined systems of linear equations. Composition, inverse, transpose of linear maps, algebra of linear maps. Similarity transformations. Matrices, matrix Multiplication, Matrix Inverse, Matrix Representation of Linear Maps, Determinant, Laplace expansion, Cramer's rule. Eigenvalue problem, eigenvalues and eigenvectors, characteristic polynomial, Cayley-Hamilton theorem. Diagonalization.

Text: Friedberg, S.H., Insel, A.J., & Spence, L.E. (2003). Linear Algebra (4^th ed.). Upper Saddle River, NJ: Prentice-Hall/ Pearson Education. 

Recommended Text: Lipschutz, S., & Lipson, M. (2012). Schaum’s Outlines [Series]. Schaum’s Outline of Linear Algebra (5^th ed.). New York, NY: McGraw-Hill.

Note: Extensive lecture notes keyed to these texts will be issued by the instructor.

Prerequisites: Undergraduate linear algebra.

Linear algebra is two things in one: a general methodology for solving linear systems, and a beautiful abstract structure underlying much of mathematics and the sciences. This course will try to strike a balance between both. We will follow the book of our own Peter Lax, which does a superb job in describing the mathematical structure of linear algebra, and complement it with applications and computing. The most advanced topics include spectral theory, convexity, duality, and various matrix decompositions.

Text: Lax, P.D. (2007). Pure and Applied Mathematics: A Wiley Series of Texts, Monographs and Tracts [Series, Bk. 78]. Linear Algebra and Its Applications (2^nd ed.). Hoboken, NJ: John Wiley & Sons/ Wiley-Interscience.

Recommended Text: Strang, G. (2005). Linear Algebra and Its Applications (4^th ed.). Stamford, CT: Cengage Learning.

Prerequisites: Elements of linear algebra and the theory of rings and fields.

Basic concepts of groups, rings and fields. Symmetry groups, linear groups, Sylow theorems; quotient rings, polynomial rings, ideals, unique factorization, Nullstellensatz; field extensions, finite fields.

Recommended Texts:

• Artin, M. (2010). Featured Titles for Abstract Algebra [Series]. Algebra (2^nd ed.). Upper Saddle River, NJ: Pearson
• Chambert-Loir, A. (2004). Undergraduate Texts in Mathematics [Series]. A Field Guide to Algebra (2005 ed.). New York, NY: Springer-Verlag
• Serre, J-P. (1996). Graduate Texts in Mathematics [Series, Vol. 7]. A Course in Arithmetic (Corr. 3^rd printing 1996 ed.). New York, NY: Springer-Verlag

Prerequisites: Any knowledge of groups, rings, vector spaces and multivariable calculus is helpful. Undergraduate students planning to take this course must have V63.0343 Algebra I or permission of the Department.

After introducing metric and general topological spaces, the emphasis will be on the algebraic topology of manifolds and cell complexes. Elements of algebraic topology to be covered include fundamental groups and covering spaces, homotopy and the degree of maps and its applications. Some differential topology will be introduced including transversality and intersection theory. Some examples will be taken from knot theory.

Recommended Texts:

• Hatcher, A. (2002). Algebraic Topology. New York, NY: Cambridge University Press
• Munkres, J. (2000). Topology (2^nd ed.). Upper Saddle River, NJ: Prentice-Hall/ Pearson Education
• Guillemin, V., Pollack, A. (1974). Differential Topology. Englewood Cliffs, NJ: Prentice-Hall
• Milnor, J.W. (1997). Princeton Landmarks in Mathematics [Series]. Topology from a Differentiable Viewpoint (Rev. ed.). Princeton, NJ: Princeton University Press

Prerequisites: Multivariable calculus and linear algebra.

Differentiable manifolds, tangent bundle, embedding theorems, vector fields and differential forms. Introduction to Riemannian metrics, connections and geodesics.

Text: Lee, J.M. (2009). Graduate Studies in Mathematics [Series, Vol. 107]. Manifolds and Differential Geometry. Providence, RI: American Mathematical Society.

•    Lecture 1 Functional fields, Rationality and  Stable rationality
•    Lecture 2 Algebraic groups. Basic results of Invariant theory for algebaric groups
•    Lecture 3 Obstructions to rationality.
•    Lecture 4 Basic reaults on Group cohomology.
•    Lecture 5 Nonramified group cohomology. and obstructions to rationality.
•    Lecture 6 Nonrationality results based on the cohomology obstructions
•    Lecture 7 Recent approaches to rationality problems using degnerations.

Description:

This course will cover topics in the quantitative geometry of manifolds, groups, and spaces.  Quantitative geometry uses tools from geometry and analysis to study the asymptotics of a space:  how curves and surfaces in the space behave at different scales, for instance, or how geometric invariants such as systoles affect the shape of a space.  In this course, we will develop tools to measure how the geometry of a space changes as its scale or complexity increases and use these tools to describe spaces arising from topology and geometric group theory.  

Topics to be covered may include: filling inequalities; systolic geometry; asymptotic cones; embedding problems; and uniform rectifiability and its applications to geometry.

Prerequisites:

Riemannian geometry, algebraic topology

Description:

The course starts with basic geometry of n-dimensional spaces on the emphasis on probability features of these spaces, with basic examples coming from convexity, Banach spaces and Riemannian geometry. Then we shall turn to examples of spaces motivated by statistical mechanics, molecular biology and optimization problem in the machine learning theory.

Note: Master's students need permission of course instructor before registering for this course.

Prerequisites: A familiarity with rigorous mathematics, proof writing, and the epsilon-delta approach to analysis, preferably at the level of MATH-GA 1410, 1420 Introduction to Mathematical Analysis I, II.

Measure theory and integration. Lebesgue measure on the line and abstract measure spaces. Absolute continuity, Lebesgue differentiation, and the Radon-Nikodym theorem. Product measures, the Fubini theorem, etc. L^p spaces, Hilbert spaces, and the Riesz representation theorem. Fourier series.

Main Text: Folland's Real Analysis: Modern Techniques and Their Applications

Secondary Text: Bass' Real Analysis for Graduate Students

Prerequisites: Advanced calculus (or equivalent).

Complex numbers; analytic functions, Cauchy-Riemann equations; linear fractional transformations; construction and geometry of the elementary functions; Green's theorem, Cauchy's theorem; Jordan curve theorem, Cauchy's formula; Taylor's theorem, Laurent expansion; analytic continuation; isolated singularities, Liouville's theorem; Abel's convergence theorem and the Poisson integral formula.

Text: Brown, J., & Churchill, R. (2008). Complex Variables and Applications (8^th ed.). New York, NY: McGraw-Hill.

Note: Master's students need permission of course instructor before registering for this course.

Prerequisites: Complex Variables I (or equivalent) and MATH-GA 1410 Introduction to Math Analysis I.

Complex numbers, the complex plane. Power series, differentiability of convergent power series. Cauchy-Riemann equations, harmonic functions. conformal mapping, linear fractional transformation. Integration, Cauchy integral theorem, Cauchy integral formula. Morera's theorem. Taylor series, residue calculus. Maximum modulus theorem. Poisson formula. Liouville theorem. Rouche's theorem. Weierstrass and Mittag-Leffler representation theorems. Singularities of analytic functions, poles, branch points, essential singularities, branch points. Analytic continuation, monodromy theorem, Schwarz reflection principle. Compactness of families of uniformly bounded analytic functions. Integral representations of special functions. Distribution of function values of entire functions.

Text: Ahlfors, L. (1979). International Series in Pure and Applied Mathematics [Series, Bk. 7]. Complex Analysis (4thin ed.). New York, NY: McGraw-Hill.

Note: Master's students should consult course instructor before registering for PDE II in the spring.

Prerequisites: Knowledge of undergraduate level linear algebra and ODE; also some exposure to complex variables (can be taken concurrently).

A basic introduction to PDEs, designed for a broad range of students whose goals may range from theory to applications. This course emphasizes examples, representation formulas, and properties that can be understood using relatively elementary tools. We will take a broad viewpoint, including how the equations we consider emerge from applications, and how they can be solved numerically. Topics will include: the heat equation; the wave equation; Laplace's equation; conservation laws; and Hamilton-Jacobi equations. Methods introduced through these topics will include: fundamental solutions and Green's functions; energy principles; maximum principles; separation of variables; Duhamel's principle; the method of characteristics; numerical schemes involving finite differences or Galerkin approximation; and many more.

See the syllabus for more information (including a tentative semester plan).

Recommended Texts:

• Guenther, R.B., & Lee, J.W. (1996). Partial Differential Equations of Mathematical Physics and Integral Equations. Mineola, NY: Dover Publications.
• Evans, L.C. (2010). Graduate Studies in Mathematics [Series, Bk. 19]. Partial Differential Equations (2^nd ed.). Providence, RI: American Mathematical Society.

Description:

Elliptic regularity theory: Cacciopoli inequality, Schauder estimates, De Giorgi-Nash Theory. Variational methods. Homogenization. Basics on hyperbolic equations, dispersive equations and viscosity solutions.

Prerequisites: Linear algebra, real variables (including measure theory), and basic complex analysis.

Topics: Banach spaces. Functionals and operators. Principle of uniform boundedness, open mapping and closed graph theorems. Duality and weak topologies. Alaoglu's theorem. Invariant subspaces. Spectral theorem for compact operators on Banach spaces and self-adjoint operators on Hilbert spaces. Hilbert-Schmidt operators. Semigroups. Fixed-point theorem. Applications to differential equations and harmonic analysis.

The course will concentrate on both general theory and concrete examples. Working knowledge of measure and integral is expected.

Recommended (optional) texts:

• Lax, P.D. (2002). Functional Analysis, Wiley.
• Reed, M., & Simon, B. (1972). Functional Analysis, Academic Press.
• Conway, J.B. (1985). A Course in Functional Analysis, Springer.

Description:

The course will present various ideas and methods in the theory of elliptic equations with particular emphasizes on a Priori and quantitative estimates. Topics to be discussed including: Mean-Value property and consequences, Hope and Alexandroff maximum principles, Harnack type inequalities, De Giorgi and Krylov estimates, viscosity solutions, boundary value problems and eigenvalue problems.

References:

1. Q.Han and F.H.Lin, Lectures on Elliptic PDEs, Courant Lecture Notes Vol. 1, Second Ed. AMS 2011.
2. D.Gilbarg and N.Trudinger: Elliptic Partial Differential Equations of Second Order, Second Ed. Springer, Berlin-New York, 1983.

Description:

This course is an introduction to the differentiable theory of dynamical systems. No prior knowledge of the subject is assumed. One way to "classify" dynamical systems is to view them on an order-disorder spectrum. While most dynamical systems lie somewhere in the middle, it is, not surprisingly, the kinds at the two ends of the spectrum that are best understood. This course is mostly about dynamical systems at the chaotic end. I will discuss stable and unstable manifolds, horseshoes, homoclinic orbits and attractors, the existence and persistence of chaotic behavior, transient vs observable chaos. To give a more balanced perspective, I will also spend a few weeks on (nonchaotic) quasi-periodic dynamics.

Prerequisite:

linear algebra and analysis on R^n; knowledge of manifolds helpful but not required.

Recommended texts:

• Brin and Stuck, Introduction to Dynamical Systems
• Katok and Hasselblatt, Introduction to the Modern Theory of Dynamical Systems

Prerequisites: Elementary linear algebra and differential equations.

This is a first-year course for all incoming PhD and Masters students interested in pursuing research in applied mathematics. It provides a concise and self-contained introduction to advanced mathematical methods, especially in the asymptotic analysis of differential equations. Topics include scaling, perturbation methods, multi-scale asymptotics, transform methods, geometric wave theory, and calculus of variations

Recommended Texts:

• Barenblatt, G.I. (1996). Cambridge Texts in Applied Mathematics [Series, Bk. 14]. Scaling, Self-similarity, and Intermediate Asymptotics: Dimensional Analysis and Intermediate Asymptotics. New York, NY: Cambridge University Press
• Hinch, E.J. (1991). Cambridge Texts in Applied Mathematics [Series, Bk. 6]. Perturbation Methods. New York, NY: Cambridge University Press
• Bender, C.M., & Orszag, S.A. (1999). Advanced Mathematical Methods for Scientists and Engineers [Series, Vol. 1]. Asymptotic Methods and Perturbation Theory. New York, NY: Springer-Verlag
• Whitham, G.B. (1999). Pure and Applied Mathematics: A Wiley Series of Texts, Monographs and Tracts [Series Bk. 42]. Linear and Nonlinear Waves (Reprint ed.). New York, NY: John Wiley & Sons/ Wiley-Interscience
• Gelfand, I.M., & Fomin, S.V. (2000). Calculus of Variations. Mineola, NY: Dover Publications

Prerequisites: Introductory complex variable and partial differential equations.

The course will expose students to basic fluid dynamics from a mathematical and physical perspectives, covering both compressible and incompressible flows. Topics: conservation of mass, momentum, and Energy. Eulerian and Lagrangian formulations. Basic theory of inviscid incompressible and compressible fluids, including the formation of shock waves. Kinematics and dynamics of vorticity and circulation. Special solutions to the Euler equations: potential flows, rotational flows, irrotational flows and conformal mapping methods. The Navier-Stokes equations, boundary conditions, boundary layer theory. The Stokes Equations.

Text: Childress, S. Courant Lecture Notes in Mathematics [Series, Bk. 19]. An Introduction to Theoretical Fluid Mechanics. Providence, RI: American Mathematical Society/ Courant Institute of Mathematical Sciences.

Recommended Text: Acheson, D.J. (1990). Oxford Applied Mathematics & Computing Science Series [Series]. Elementary Fluid Dynamics. New York, NY: Oxford University Press.

Prerequisites: Derivative Securities, Scientific Computing, Computing for Finance, and Stochastic Calculus.

The term "statistical arbitrage" covers any trading strategy that uses statistical tools and time series analysis to identify approximate arbitrage opportunities while evaluating the risks inherent in the trades (considering the transaction costs and other practical aspects). This course starts with a review of Time Series models and addresses econometric aspects of financial markets such as volatility and correlation models. We will review several stochastic volatility models and their estimation and calibration techniques as well as their applications in volatility based trading strategies. We will then focus on statistical arbitrage trading strategies based on co-integration, and review pairs trading strategies. We will present several key concepts of market microstructure, including models of market impact, which will be discussed in the context of developing strategies for optimal execution. We will also present practical constraints in trading strategies and further practical issues in simulation techniques. Finally, we will review several algorithmic trading strategies frequently used by practitioners.

Prerequisites: Univariate statistics, multivariate calculus, linear algebra, and basic computing (e.g. familiarity with MATLAB or co-registration in Computing in Finance).

A comprehensive introduction to the theory and practice of portfolio management, the central component of which is risk management. Econometric techniques are surveyed and applied to these disciplines. Topics covered include: factor and principal-component models, CAPM, dynamic asset pricing models, Black-Litterman, forecasting techniques and pitfalls, volatility modeling, regime-switching models, and many facets of risk management, both theory and practice.

Students in the Mathematics in Finance program conduct research projects individually or in small groups under the supervision of finance professionals. The course culminates in oral and written presentations of the research results.

A first course in derivatives. History and market infrastructure, Forwards and Futures, Term structures of futures and forward rates, ETFs based on rolling futures, Options, Put-Call Parity, synthetic forwards, Black-Scholes Formula, Numerical Pricing of American-style options, The Greeks, Interest-rate options, Swaptions, Caps & Floors, Basic option strategies, Risk-management of derivatives.

Prerequisites:

Derivative Securities and Stochastic Calculus, or equivalent.

Description:

This is a second course in arbitrage-based pricing of derivative securities. Concerning equity and FX models: We discuss numerous approaches that are used in practice in these markets, such as the local volatility model, Heston, SABR, and stochastic local volatility. The discussion will include calibration and hedging issues and the pricing of the most common structured products.

Concerning interest rate models: We start with a thorough discussion of one-factor short-rate models (Vasicek, CIR, Hull-White) then proceed to more advanced topics such as two-factor Hull-White, forward rate models (HJM) and the LIBOR market model. Throughout, the pricing of specific payoffs will be considered and practical examples and insights will be provided. We give an introduction to inflation models.

We cover a few special topics: We provide an introduction to stochastic optimal control with applications, as well as optimal stopping time theory and its application to American options pricing. We introduce Cox default processes and discuss their applications to unilateral and bilateral CVA/DVA.

Prerequisites: Computing in Finance (or equivalent programming skills) and Derivative Securities (familiarity with Black-Scholes interest rate models)

This half-semester class focuses on the practical workings of the fixed-income and rates-derivatives markets. The course content is motivated by a representative set of real-world trading, investment, and hedging objectives. Each situation will be examined from the ground level and its risk and reward attributes will be identified. This will enable the students to understand the link from the underlying market views to the applicable product set and the tools for managing the position once it is implemented. Common threads among products – structural or model-based – will be emphasized. We plan on covering bonds, swaps, flow options, semi-exotics, and some structured products.

A problem-oriented holistic view of the rate-derivatives market is a natural way to understand the line from product creation to modeling, marketing, trading, and hedging. The instructors hope to convey their intuition about both the power and limitations of models and show how sell-side practitioners manage these constraints in the context of changes in market backdrop, customer demands, and trading parameters.

Prerequisites: Derivate Securities and Computing in Finance (or equivalent familiarity with financial models and computing skills)

This half-semester course introduces the institutional market for bonds and loans subject to default risk and develops concepts and quantitative frameworks useful for modeling the valuation and risk management of such fixed income instruments and their associated derivatives. Emphasis will be put on theoretical arbitrage restrictions on the relative value between related instruments and practical applications in hedging, especially with credit derivatives. Some attention will be paid to market convention and related terminology, both to ensure proper interpretation of market data and to prepare students for careers in the field.

We will draw on the fundamental theory of derivatives valuation in complete markets and the probabilistic representation of the associated valuation operator. As required, this will be extended to incomplete markets in the context of doubly stochastic jump-diffusion processes. Specific models will be introduced, both as examples of the underlying theory and as tools that can be (and are) used to make trading and portfolio management decisions in real world markets.

Prerequisites: Advanced Risk Management, Derivative Securities (or equivalent familiarity with market and credit risk models), and Computing in Finance (or equivalent programming experience)
 
Course description: This class explores technical and regulatory aspects of counterparty credit risk, with an emphasis on model building and computational methods. The first part of the class will provide technical foundation, including the mathematical tools needed to define and compute valuation adjustments such as CVA and DVA. The second part of the class will move from pricing to regulation, with an emphasis on the computational aspects of regulatory credit risk capital under Basel 3. A variety of highly topical subjects will be discussed during the course, including: funding costs, XVA metrics, initial margin, credit risk mitigation, central clearing, and balance sheet management. Students will get to build a realistic computer system for counterparty risk management of collateralized fixed income portfolios, and will be exposed to modern frameworks for interest rate simulation and capital management.

Description:

This class will explore how two mathematical frameworks: optimal transport and large deviation theory, can be combined to provide tools and methods of analysis for some central tasks in data science: classification, clustering and regression, dimensionality reduction, conditional probability estimation, prediction of rare events. Recurring themes will include the power of the composition of simple functions to capture complex behavior - as demonstrated by neural networks - and of the characterization of both the available observations of the system under study and the parameters tuned to model it in terms of underlying probability distributions.

Description:

An important emerging scientific issue in many practical problems ranging from climate and weather prediction to neural and material sciences involves the real time filtering through observations of noisy turbulent signals for complex dynamical systems with many degrees of freedom, as well as the statistical accuracy of various strategies in this context. This is a research expository course on modern applied mathematics involving the blending of rigorous mathematical theories, qualitative and quantitative modeling, and novel numerical procedures driven by the goal of understanding physical phenomena which are of central importance. These ideas include classical stability analysis for PDE’s and their finite difference approximations, suitable versions of Kalman filtering, ensemble Kalman filters, and instructive stochastic qualitative models from turbulence theory and concrete models from climate atmosphere ocean science. The course will begin with an elementary introduction to these topics filling in the necessary background with elementary scalar and low dimensional models with eventual applications to fully turbulent and chaotic, linear and nonlinear, large dimensional systems. New development in mathematical theories and algorithms currently developed at CIMS will also be discussed at the end of this course.

Audience:

The course should be interesting for graduate students, and postdocs in pure and applied mathematics, physics, engineering, and climate, atmosphere, ocean science interested in filtering and data assimilation for complex turbulent dynamical systems.

Course Organization:

• Main reference:
  • Majda, A.J., & Harlim, J. (2012). Filtering Complex Turbulent Systems. New York, NY: Cambridge University Press. (Online version available through the NYU Library.)
• Course style: A lecture reading course with active participation of registered students and active discussion led by Prof. Andrew Majda and his postdocs.
• Prerequisite: Elementary background in ODEs and PDEs.
• Grading: All active registered participants will pass the course, no exams or problem set homework.
• (see also http://cims.nyu.edu/~qidi/courseinfo18.html  for supplementary course materials)

As part of our new NSF research training group (RTG) in Modeling & Simulation, we will be organizing a lunchtime group meeting for students, postdocs, and faculty working in applied mathematics who do modeling & simulation. The aim is to create a space to discuss applied mathematics research in an informal setting: to (a) give students and postdocs a chance to present their research (or a topic of common interest) and get feedback from the group, (b) learn about other ongoing and future research activities in applied math at the Institute, and (c) discuss important open problems and research challenges.

The meetings will be Thursdays from 12:30-2:00, in room 1314, starting Sept. 6th (here is the weekly schedule). Graduate students that wish to get credit for this will be required to attend the working group in both Fall of 2018 and Spring of 2019, and can register for MATH-GA-2830.004 in the fall but will get their grades in spring. Students enrolled in the course will be required to attend regularly and participate in discussions, as well as present at least once during the academic year.

Prerequisites: Calculus through partial derivatives and multiple integrals; no previous knowledge of probability is required.

The course introduces the basic concepts and methods of probability.

Topics include: probability spaces, random variables, distributions, law of large numbers, central limit theorem, random walk, Markov chains and martingales in discrete time, and if time allows diffusion processes including Brownian motion.

Recommeded text: Probability Essentials, by J.Jacod and P.Protter. Springer, 2004.

Prerequisites: MATH-GA 2901 Basic Probability or equivalent.

Review of basic probability and useful tools. Bernoulli trials and random walk. Law of large numbers and central limit theorem. Conditional expectation and martingales. Brownian motion and its simplest properties. Diffusion in general: forward and backward Kolmogorov equations, stochastic differential equations and the Ito calculus. Feynman-Kac and Cameron-Martin Formulas. Applications as time permits.

Optional Problem Session: Thursday, 5:30-7:00

Text: Durrett, R. (1996). Probability and Stochastics Series [Series, Bk. 6]. Stochastic Calculus: A Practical Introduction. New York, NY: CRC Press.

Prerequisites: A first course in probability, familiarity with Lebesgue integral, or MATH-GA 2430 Real Variables as mandatory co-requisite.

First semester in an annual sequence of Probability Theory, aimed primarily for Ph.D. students. Topics include laws of large numbers, weak convergence, central limit theorems, conditional expectation, martingales and Markov chains.

Recommended Text: S.R.S. Varadhan, Probability Theory (2001)

Description:

The course will center around two very recent breakthroughs on sparse random regular graphs. The first part would aim to go through the recent paper of Ding-Sly-Sun (Acta Math 2017) on maximal independent sets in random regular graphs, via belief propagation and tools from statistical physics. The second part would be devoted to the spectrum of such graphs, beginning with delocalization in arbitrary regular graphs with large girth due to Brooks-Lindenstrauss (Israel J. Math. 2013) and Anantharaman-Le Masson (Duke Math J. 2015), followed by the recent delocalization result by Bauerschmidt, Huang and Yau (arXiv:1609.09052).

Description:

I will cover elements of the theory of random matrices, with particular emphasis on global behavior of eigenvalues. I will review classical tools (moments and their combinatorics, Stieltjes transform, concentration inequalities) and then will focus on the following topics.

1. Free probability and its use in random matrices. The linearization trick and control of operator norms.
2. Non-normal matrices and the hermitization trick. The single ring theorem, outliers.
3. Stability and instability of spectra.

Description: This will be a seminar style course covering two related topics.

Topic 1: Polya's Approach to the Riemann Hypothesis (RH): We will review the history of results about whether the zeros in the complex plane of the Laplace transform H_\lambda of \exp(\lambda u^2) \Phi (u) are all pure imaginary. Here, \lambda and u are real and \Phi is a specific function such that the Riemann Hypothesis (RH) is valid if and only if the pure imaginary zeros property holds for H_0. Results of Polya (1920's), de Bruijn (1950) and Newman (1976) proved the existence of \Lambda \in (-\infty, 1/2] such that the pure imaginary zeros property is valid if and only if \lambda \geq \Lambda and thus RH is valid if and only if \Lambda \leq 0. Rodgers and Tao recently (Jan., 2018) proved the 1976 conjecture that \Lambda \geq 0 (which was formulated as a version of the statment that "the RH, if true, is only barely so").

Topic 2: Stochastic Processes and RH: We will survey a number of results from the statistical mechanics literature (beginning with a 1952 theorem of Lee and Yang) that can be used to generate stochastic processes X(t) such that for h(t) \geq 0, the random variable X_h defined as \int X(t) h(t) dt has moment generating function, E(\exp(z X_h)), whose zeros in the complex plane are all pure imaginary. As per Topic 1 above, if such an X_h has a very specific distribution, i.e., with probability density proportional to the above \Phi, the Riemann Hypothesis would follow.

This course serves as an introduction to the fundamentals of geophysical fluid dynamics. No prior knowledge of fluid dynamics will be assumed, but the course will move quickly into the subtopic of rapidly rotating, stratified flows. Topics to be covered include (but are not limited to): the advective derivative, momentum conservation and continuity, the rotating Navier-Stokes equations and non-dimensional parameters, equations of state and thermodynamics of Newtonian fluids, atmospheric and oceanic basic states, the fundamental balances (thermal wind, geostrophic and hydrostatic), the rotating shallow water model, vorticity and potential vorticity, inertia-gravity waves, geostrophic adjustment, the quasi-geostrophic approximation and other small-Rossby number limits, Rossby waves, baroclinic and barotropic instabilities, Rayleigh and Charney-Stern theorems, geostrophic turbulence. Students will be assigned bi-weekly homework assignments and some computer exercises, and will be expected to complete a final project. This course will be supplemented with out-of-class instruction.

Recommended Texts:

• Vallis, G.K. (2006). Atmospheric and Oceanic Fluid Dynamics: Fundamentals and Large-scale Circulation. New York, NY: Cambridge University Press.
• Salmon, R. (1998). Lectures on Geophysical Fluid Dynamics. New York, NY: Oxford University Press.
• Pedlosky, J. (1992). Geophysical Fluid Dynamics (2^nd ed.). New York, NY: Springer-Verlag.