Course Descriptions

Description:

Differentiation and integration for vector-valued functions of one and several variables: curves, surfaces, manifolds, inverse and implicit function theorems, integration on manifolds, Stokes' theorem, applications.

Description:

Measure theory and Lebesgue integration on the Euclidean space. Convergence theorems. L^p spaces and Hilbert spaces. Fourier series. Introduction to abstract measure theory and integration.

Recitation/ Problem Session:

7:10-9:00 (following the course).

Prerequisites:

Numerical linear algebra, elements of ODE and PDE.

Description:

This course will cover fundamental methods that are essential for the numerical solution of differential equations. It is intended for students familiar with ODE and PDE and interested in numerical computing; computer programming assignments form an essential part of the course. The course will introduce students to numerical methods for (1) nonlinear equations, Newton's method; (2) ordinary differential equations, Runge-Kutta and multistep methods, convergence and stability; (3) finite difference, finite element and integral equation methods for elliptic partial differential equations; (4) fast solvers, multigrid methods; and (5) parabolic and hyperbolic partial differential equations.

Text:

LeVeque, R. (2007). Classics in Applied Mathematics [Series]. Finite Difference Methods for Ordinary and Partial Differential Equations. Philadelphia, PA: Society for Industrial and Applied Mathematics.

Cross-listing:

CSCI-GA 2421.001.

Prerequisites:

Risk and Portfolio Management with Econometrics, Derivative Securities, and Computing in Finance

Description:

This is a version of the course Scientific Computing (MATH-GA 2043.001) designed for applications in quantitative finance.  It covers software and algorithmic tools necessary to practical numerical calculation for modern quantitative finance.  Specific material includes IEEE arithmetic, sources of error in scientific computing, numerical linear algebra (emphasizing PCA/SVD and conditioning), interpolation and curve building with application to bootstrapping, optimization methods, Monte Carlo methods, and the solution of differential equations.

Please Note: Students may not receive credit for both MATH-GA 2043.001 and MATH-GA 2048.001

Prerequisites:

Undergraduate linear algebra or permission of the instructor.

Description:

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:

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.

Prerequisites:

Linear Algebra I or permission of the instructor.

Description:

Review of: Trace, determinant, characteristic and minimal polynomial. Eigenvalues, diagonalization, spectral theorem and spectral mapping theorem. When diagonalization fails: nilpotent operators and their structure, generalized eigenspace decomposition, Jordan canonical form. Polar and singular value decomposition. Complexification and smooth surfaces in R^n and their tangent spaces. Intro to matrix Lie algebras and Lie groups.

Text:

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

Plus: Extensive instructor’s class notes.

Description:

Representations of finite groups. Characters, orthogonality of characters of irreducible representations, a ring of representations. Induced representations, Artin’s theorem, Brauer’s theorem. Representations of compact groups and the Peter-Weyl theorem. Lie groups, examples of Lie groups, representations and characters of Lie group. Lie algebras associated with Lie groups. Applications of the group representations in algebra and physics. Elements of algebraic geometry.

Text:

Artin, M. (2010). Algebra (2nd ed.). Upper Saddle River, NJ: Prentice-Hall/ Pearson Education.

Recommended Texts:

• Lang, S. (2005). Graduate Texts in Mathematics [Series, Bk. 211]. Algebra (3^rd ed.). New York, NY: Springer-Verlag.
• Serre, J.P. (1977). Graduate Texts in Mathematics [Series, Bk. 42]. Linear Representations of Finite Groups. New York, NY: Springer-Verlag.
• Reid, M. (1989). London Mathematical Society Student Texts [Series]. Undergraduate Algebraic Geometry. New York, NY: Cambridge University Press.
• James, G., & Lieback, M. (1993). Cambridge Mathematical Textbooks [Series]. Representations and Characters of Groups. New York, NY: Cambridge University Press.
• Fulton, W., Harris, J. (2008). Graduate Texts in Mathematics/Readings in Mathematics [Series, Bk. 129]. Representation Theory: A First Course (Corrected ed.). New York, NY: Springer-Verlag.
• Sagan, B.E. (1991). Wadsworth & Brooks/Cole Mathematics Series [Series]. The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions. Pacific Grove, CA: Wadsworth & Brooks/Cole.
• Brocker, T., & Dieck, T. (2003). Graduate Texts in Mathematics [Series, Bk. 98]. Representations of Compact Lie Groups. New York, NY: Springer-Verlag.

Prerequisites:

A background in commutative algebra is required.

Description:

This graduate course aims at covering several fundamental constructions in Algebraic geometry, from the point of view of a first course in scheme theory. Topics include: algebraic varieties, schemes, local and global properties of schemes, coherent sheaves, divisors, cohomology, Riemann-Roch theorem. 

Prerequisites:

Undergraduate elementary number theory, abstract algebra, including groups, rings and ideals, fields, and Galois theory (e.g. undergraduate Algebra I and II). A background in complex analysis, as well as in algebra, is required.

Description:

This graduate course will cover several analytic techniques in number theory,  as well as properties of number fields and their rings of integers. Topics include: primes in arithmetic progressions, zeta-function, prime number theorem, number fields, rings of integers, Dedekind zeta-function, introduction to analytic techniques: circle method, sieves.

Description:

Homology and cohomology from simplicial, singular, cellular, axiomatic and differential form viewpoints. Axiomatic characterizations and applications to geometrical problems of embedding and fixed points. Manifolds and Poincaré duality. Products and ring structures. Vector bundles, tangent bundles, DeRham cohomology and differential forms.

Description:

Differential Geometry II will focus on Riemannian geometry. Topics to be covered may include: second variation of arc length, Rauch comparison theorem and applications, Toponogov's theorem, invariant metrics on Lie groups, Morse theory, cut locus, the sphere theorem, complete manifolds of nonnegative curvature.

Recommended Texts:

• John Milnor, Morse Theory (Princeton University Press, 1963).
• John M. Lee, Riemannian Manifolds: An Introduction to Curvature (Springer, 1997).

The course will be concerned with Ricci curvature including almost rigidity and structure of Gromov-Hausdorff limit spaces.

Prerequisites:

Some Riemannian geometry, algebraic topology, and group theory. Some familiarity with Lie groups may be helpful.

Description:

Geometric group theory is based on studying discrete groups by looking at geometric objects on which they act. A single group can act on many different spaces, but all of these spaces share certain large-scale geometric properties and asymptotic behavior. In this course, we will introduce some of the techniques of geometric group theory and study their applications. This is a broad field, so we will study an assortment of topics, possibly including: Bass-Serre theory and groups acting on trees; hyperbolic surfaces and the mapping class group; the geometry and topology of 3-manifolds; filling invariants and the Dehn function; asymptotic cones; Gromov's polynomial growth theorem; symmetric spaces and buildings; and/or other topics determined by interest.

References:

• B. Bowditch, A course on geometric group theory
• S. Levy, W. Thurston, Three-Dimensional Geometry and Topology
• C. Druţu, M. Kapovich, Geometric group theory

Prerequisites:

Complex Variables I (or equivalent).

Description:

The fundamental theorem of algebra, the argument principle; calculus of residues, Fourier transform; the Gamma and Zeta functions, product expansions; Schwarz principle of reflection and Schwarz-Christoffel transformation; elliptic functions, Riemann surfaces; conformal mapping and univalent functions; maximum principle and Schwarz's lemma; the Riemann mapping theorem.

Text:

Ahlfors, L. (1979). International Series in Pure and Applied Mathematics [Series, Bk. 7]. Complex Analysis (3^rd ed.). New York, NY: McGraw-Hill.

Prerequisites:

Undergraduate background in analysis, linear algebra and complex variables.

Description:

Existence and uniqueness of initial value problems. Linear equations. Complex analytic equations. Boundary value problems and Sturm-Liouville theory. Comparison theorem for second order equations. Asymptotic behavior of nonlinear systems. Perturbation theory and Poincaré-Bendixson theorems.

Recommended Text:

Teschl, G. (2012). Graduate Studies in Mathematics [Series, Vol. 140]. Ordinary Differential Equations and Dynamical Systems. Providence, RI: AMS Chelsea Publishing/ American Mathematical Society.

Prerequisites:

MATH-GA 2490.001 PDE I and MATH-GA 2430.001 Real Variables, or the equivalent.

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

Description:

This course is a continuation of MATH-GA 2490 and is designed for students who are interested in analysis and PDEs. The course gives an introduction to Sobolev spaces, Holder spaces, and the theory of distributions; elliptic equations, harmonic functions, boundary value problems, regularity of solutions, Hilbert space methods, eigenvalue problems; general hyperbolic systems, initial value problems, energy methods; linear evaluation equations, Fourier methods for solving initial value problems, parabolic equations, dispersive equations; nonlinear elliptic problems, variational methods; Navier-Stokes and Euler equations.

Recommended Texts:

• Garabedian, P.R. (1998). Partial Differential Equations (2^nd Rev. ed.). Providence, RI: AMS Chelsea Publishing/ American Mathematical Society.
• Evans, L.C. (2010). Graduate Studies in Mathematics [Series, Bk. 19]. Partial Differential Equations (2^nd ed.). Providence, RI: American Mathematical Society.
• John, F. (1995). Applied Mathematical Sciences [Series, Vol. 1]. Partial Differential Equations (4^th ed.). New York, NY: Springer-Verlag.

Prerequisites:

Real analysis; basic knowledge of complex variables and functional analysis.

Description:

Classical Fourier Analysis: Fourier series on the circle, Fourier transform on the Euclidean space, Introduction to Fourier transform on LCA groups. Stationary phase. Topics in real variable methods: maximal functions, Hilbert and Riesz transforms and singular integral operators. Time permitting: Introduction to Littlewood-Paley theory, time-frequency analysis, and wavelet theory.

Recommended Text:

Muscalu, C. and Schlag, W. (2013). Cambridge Studies in Advanced Mathematics [Series, Bk. 137]. Classical and Multilinear Harmonic Analysis (Vol.1). New York, NY: Cambridge University Press. (Online version available to NYU users through Cambridge University Press.)

Prerequisite: some knowledge of RV II, Functional Analysis and PDE. 

Variational problems, which consist in minimizing a certain ”energy" functional among a suitable class of functions, arise in the study of PDEs as well as in optimization problems related to many branches of science (e.g. physics, materials, chemistry, economics, operation research). This course will review the basics of solving variational problems and their associated PDEs, and present the notion of Gamma-convergence used for analyzing asymptotics of variational problems. It will emphasize general methods as well as a diversity of examples. 

A rough outline includes: (1) from energy to PDEs: computing variations; (2) the direct method of the calculus of variations; (3) quasiconvexity and applications; (4) constrained problems, the example of the obstacle problem; (5) relaxation, the example of optimal transportation; Gamma-convergence: definition and properties; (6) application to homogenization; (7) application to interface problems: the examples of Allen-Cahn and Aviles-Giga; (8) nonminimizing critical points: minmax principles, Palais-Smale condition, Noether's theorem; (9) harmonic maps and monotonicity formula. 

Suggested texts: Variational Methods, M. Struwe; Direct methods in the calculus of variations, B. DaCorogna, Modern Methods in the calculus of variations, I. Fonseca, G. Leoni, Gamma-convergence for Beginners, A. Braides, Conservation laws and moving frames, F. Helein, Harmonic maps
 

In this course we will show how to analyze invariant random matrix ensembles using Riemann-Hilbert methods. The main goal will be to prove universality for such ensembles.

Prerequisites:

Real analysis at the graduate level.

Description:

This course is an introduction to ergodic theory, a probabilistic approach to dynamical systems. No prior knowledge of the subject is assumed. Topics include ergodicity, the Ergodic Theorems, mixing properties, entropy; ergodic theory of continuous and differentiable maps including Lyapunov exponents.

Recommended Texts:

Walters, P. (2000). Graduate Texts in Mathematics [Series, Bk. 79]. An Introduction to Ergodic Theory. New York, NY: Springer-Verlag.

Strichartz estimates are L^p estimates which quantify dispersion for, say, a solution of the linear Schrodinger equation. I will present different contexts where such estimates can be proved: Euclidean space, Euclidean space with a potential, flat tori, compact manifolds. The theory of these estimates is intimately tied to a number of mathematical fields: Harmonic Analysis (restriction conjecture and Bochner-Riesz conjecture), Number Theory (l² decoupling conjecture recently proved by Bourgain and Demeter), Geometric Measure Theory (Kakeya conjecture). I will also sketch applications to nonlinear PDE.

Prerequisites:

Basic Probability (or equivalent masters-level probability course), Linear Algebra (graduate course), and (beginning graduate-level) knowledge of ODEs, PDEs, and analysis.

Description:

This is a graduate class that will introduce the major topics in stochastic analysis from an applied mathematics perspective.  Topics to be covered include Markov chains, stochastic processes, stochastic differential equations, numerical algorithms, and asymptotics. It will pay particular attention to the connection between stochastic processes and PDEs, as well as to physical principles and applications. The class will attempt to strike a balance between rigour and heuristic arguments: it will assume that students have some familiarity with measure theory and analysis and will make occasional reference to these, but many results will be derived through other arguments. The target audience is PhD students in applied mathematics, who need to become familiar with the tools or use them in their research.

Text:

• Stochastic Processes and Applications, by G. A. Pavliotis.
• C. Gardiner, Stochastic Methods: A Handbook for the Natural and Social Sciences

Prerequisites:

Computing in Finance, and Risk Portfolio Management with Econometrics, or equivalent.

Description:

In this course we develop a quantitative investment and trading framework. In the first part of the course, we study the mechanics of trading in the financial markets, some typical trading strategies, and how to work with and model high frequency data. Then we turn to transaction costs and market impact models, portfolio construction and robust optimization, and optimal betting and execution strategies. In the last part of the course, we focus on simulation techniques, back-testing strategies, and performance measurement. We use advanced econometric tools and model risk mitigation techniques throughout the course. Handouts and/or references will be provided on each topic.

This course provides brief mathematical introductions to elasticity, classical mechanics, and statistical mechanics -- topics at the interface where differential equations and probability meet physics and materials science. For students preparing to do research on physical applications, the class provides an introduction to crucial concepts and tools; for students of analysis the class provides valuable context by exploring some central applications. No prior exposure to mechanics or physics is assumed.

The segment on elasticity (about 6 weeks) will include: one-dimensional models (strings and rods); buckling as a bifurcation; nonlinear elasticity for 3D solids; and linear elasticity. The segment on classical mechanics (about 5 weeks) will include: basic examples; alternative formulations including action minimization and Hamilton's equations; relations to the Calculus of Variations including Hamilton-Jacobi equations, optimal control, and geodesics; stability and parametric resonance. The segment on statistical mechanics (about 3 weeks) will include basic concepts such as the microcanonical and canonical ensembles, entropy, and the equilibrium distribution; some simple examples; and the numerical method known as Metropolis sampling.

Description:

Risk Management is arguably one of the most important tools for managing a trading book and quantifying the effects of leverage and diversification (or lack thereof).

This course is an introduction to risk-management techniques for portfolios of (i) equities and delta-1 securities and futures (ii) equity derivatives (iii) fixed income securities and derivatives, including credit derivatives, and  (iv) mortgage-backed securities.

A systematic approach to the subject is adopted, based on selection of risk factors, econometric analysis, extreme-value theory for tail estimation, correlation analysis, and copulas to estimate joint factor distributions. We will cover the construction of risk-measures (e,g. VaR and Expected Shortfall) and historical back-testing of portfolios. We also review current risk-models and practices used by large financial institutions and clearinghouses.

If time permits, the course will also cover models for managing the liquidity risk of portfolios of financial instruments.

Prerequisites:

Risk & Portfolio Management with Econometrics, Computing in Finance.

Description:

The first part of the course will cover the theoretical aspects of portfolio construction and optimization. The focus will be on advanced techniques in portfolio construction, addressing the extensions to traditional mean-variance optimization including robust optimization, dynamical programming and Bayesian choice. The second part of the course will focus on the econometric issues associated with portfolio optimization. Issues such as estimation of returns, covariance structure, predictability, and the necessary econometric techniques to succeed in portfolio management will be covered. Readings will be drawn from the literature and extensive class notes.

Prerequisites:

Derivative Securities, Computing in Finance or equivalent programming.

Description:

The importance of financial risk management has been increasingly recognized over the last several years. This course gives a broad overview of the field, from the perspective of both a risk management department and of a trading desk manager, with an emphasis on the role of financial mathematics and modeling in quantifying risk. The course will discuss how key players such as regulators, risk managers, and senior managers interact with trading. Specific techniques for measuring and managing the risk of trading and investment positions will be discussed for positions in equities, credit, interest rates, foreign exchange, commodities, vanilla options, and exotic options. Students will be trained in developing risk sensitivity reports and using them to explain income, design static and dynamic hedges, and measure value-at-risk and stress tests. Students will create Monte Carlo simulations to determine hedge effectiveness. Extensive use will be made of examples drawn from real trading experience, with a particular emphasis on lessons to be learned from trading disasters.

Text:

Allen, S.L. (2003). Wiley Finance [Series, Bk. 119]. Financial Risk Management: A Practitioner’s Guide to Managing Market and Credit Risk. Hoboken, NJ: John Wiley & Sons.

Description:

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.

Description:

An introduction to arbitrage-based pricing of derivative securities. Topics include: arbitrage; risk-neutral valuation; the log-normal hypothesis; binomial trees; the Black-Scholes formula and applications; the Black-Scholes partial differential equation; American options; one-factor interest rate models; swaps, caps, floors, swaptions, and other interest-based derivatives; credit risk and credit 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:

Derivative Securities, Stochastic Calculus, and Computing in Finance (or equivalent familiarity with financial models, stochastic methods, and computing skills).

Description:

The course is divided into two parts. The first addresses the fixed-income models most frequently used in the finance industry, and their applications to the pricing and hedging of interest-based derivatives. The second part covers the foreign exchange derivatives markets, with a focus on vanilla options and first-generation (flow) exotics. Throughout both parts, the emphasis is on practical aspects of modeling, and the significance of the models for the valuation and risk management of widely-used derivative instruments.

Prerequisites:

Basic bond mathematics and bond risk measures (duration and convexity); Derivative Securities and Stochastic Calculus.

Description:

This half-semester course will cover the fundamentals of Securitized Products, emphasizing Residential Mortgages and Mortgage-Backed Securities (MBS). We will build pricing models that generate cash flows taking into account interest rates and prepayments. The course will also review subprime mortgages, CDO’s, Commercial Mortgage Backed Securities (CMBS), Auto Asset Backed Securities (ABS), Credit Card ABS, CLO’s, Peer-to-peer / MarketPlace Lending, and will discuss drivers of the financial crisis and model risk.

Prerequisites:

Derivative Securities and Stochastic Calculus.

Description:

This half-semester course focuses on energy commodities and derivatives, from their basic fundamentals and valuation, to practical issues in managing structured energy portfolios. We develop a risk neutral valuation framework starting from basic GBM and extend this to more sophisticated multi-factor models. These approaches are then used for the valuation of common, yet challenging, structures. Particular emphasis is placed on the potential pitfalls of modeling methods and the practical aspects of implementation in production trading platforms. We survey market mechanics and valuation of inventory options and delivery risk in the emissions markets.

Prerequisites:

Derivative Securities, Stochastic Calculus, and Computing in Finance or equivalent programming experience.

Description:

This half-semester course will give a practitioner’s perspective on a variety of advanced topics with a particular focus on equity derivatives instruments, including volatility and correlation modeling and trading, and exotic options and structured products.  Some meta-mathematical topics such as the practical and regulatory aspects of setting up a hedge fund will also be covered.

Prerequisites:

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

Description:

This is a half-semester course covering topics of interest to both buy-side traders and sell-side execution quants. The course will provide a detailed look at how the trading process actually occurs and how to optimally interact with a continuous limit-order book market.

We begin with a review of early models, which assume competitive suppliers of liquidity whose revenues, corresponding to the spread, reflect the costs they incur. We discuss the structure of modern electronic limit order book markets and exchanges, including queue priority mechanisms, order types and hidden liquidity.  We examine technological solutions that facilitate trading such as matching engines, ECNs, dark pools, multiple venue problems and smart order routers.

The second part of the course is dedicated pre-trade market impact estimation, post-trade slippage analysis, optimal execution strategies and dynamic no-arbitrage models. We cover Almgren-Chriss model for optimal execution, Gatheral’s no-dynamic-arbitrage principle and the fundamental relationship between the average response of the market price to traded quantity,  and properties of the decay of market impact.

Homework assignments will supplement the topics discussed in lecture. Some coding in Java will be required and students will learn to write their own simple limit-order-book simulator and analyze real NYSE TAQ data.

Description:

Data-driven methods for pattern extraction and prediction in dynamical systems

This seminar-style course will survey methods combining aspects of ergodic theory and machine learning for identifying and predicting coherent patterns generated by dynamical systems. We will first review some of the classical constructions in the operator-theoretic description of dynamical systems, which allow one to represent a dynamical system by means of linear evolution operators acting on spaces of observables (functions of the state). We will then discuss how such operators provide a natural way of decomposing observables into characteristic coherent patterns, whose evolution can be approximated in a consistent manner from empirical measurements. Planned topics include Koopman operators, unitary evolution groups, Stone's theorem, delay-coordinate maps, approximation in reproducing kernel Hilbert spaces, and spatiotemporal pattern formation. These topics will be illustrated with applications to various ODE and PDE models and atmosphere ocean science.

Prerequisites:

The course has no formal prerequisites, but a basic background in dynamical systems, functional analysis, or data science would be helpful.

Assessment:

Lecture attendance and presentation of a relevant paper from the literature at the end of the semester.

Description:

This course will provide graduate students preparing for teaching and research careers with several skills and tools for more effective professional oral and written presentation. It will also provide a platform for supervised teaching practice. Students from all fields of mathematics are welcome, both pure and applied. The first part of the course, taught primarily by Prof. Mutiara Sondjaja, will focus on teaching pedagogy and effective class management. The second part of the course, co-taught with Prof. Aleks Donev, will focus on scientific writing, from abstracts to complete papers. Students will practice both by writing a review article or lecture notes on a topic from their field of study, aimed at their peers and not at specialists. They will deliver lectures to the class on the chosen topic and get feedback from the instructors and other students. The use of LaTex or tools based on LaTex such as LyX or sharelatex/Overleaf will be strongly encouraged. We will also have some guest lectures from professional writers and career service professionals, and will provide, as time permits, help with basic job search skills like writing CVs, teaching and research statements, and cover letters. Students will be encouraged to help each other and learn from peers.

Text:

"Handbook of writing for the mathematical sciences" by Nicholas J Higham, published by SIAM, any edition, strongly recommended. We will also use other optional more general sources such as Strunk and White's "The Elements of Style."

Prerequisites:

Familiarity with differential equations as used in applications.

Description:

Processes occurring in space and time, or in more than one space dimension, are governed by partial differential equations. In biology, these PDE typically have special features, such as inhomogeneity or nonlinearity, that give their solutions remarkable and unexpected behaviors. Two such examples are (1) wave propagation in the inner ear, in which an exponential stiffness gradient of a membrane immersed in fluid makes it possible for each temporal frequency of the incident sound to excite primarily a particular spatial location, thus laying the foundation for the perception of music and helping to separate signals from noise; and (2) the propagation of electrical signals in nerve and heart, in which nonlinearity conspires with diffusion to produce pulse-shaped traveling waves known as action potentials. Other biological processes described by PDE that we shall consider are: (3) the random walks of the biomolecular motors myosin and kinesin; (4) diffusion in the labyrinth of the cell nucleus (how transcription factors find their targets); (5) the fiber architecture of the heart and its valves, in which structure emerges as a mathematical consequence of function; and (6) the role of photon noise in the optimal processing of visual information by the retina.

This course can serve as an applied introduction to PDE. Asymptotic and numerical methods will be introduced as needed. There will be homework, some of which will involve computing, and a final computing project, but no exam. Both the homework and the final project may be done by students working individually or in groups.

Texts:

An online collection of reprints and lecture notes, available here.

Prerequisites:

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

Description:

The one-semester 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 martingales in discrete time, and if time permits Markov chains and Brownian motion.

Texts:

• Probability Essentials by J.Jacod and P.Protter.
• Probability: Theory and Examples (4^th edition) by R. Durrett.

Prerequisites:

MATH-GA 2901 Basic Probability or equivalent.

Description:

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:

Monday, 6:00-7:00.

Text:

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

Description:

Stochastic processes in continuous time. Brownian motion. Poisson process. Processes with independent increments. Stationary processes. Semi-martingales. Markov processes and the associated semi-groups. Connections with PDEs. Stochastic differential equations. Convergence of processes.

Recommended & On Reserve Text:

• Stochastic Processes by Varadhan (Courant Lecture Series in Mathematics, volume 16)
• Theory of Probability and Random Processes by Koralov and Sinai
• Brownian Motion and Stochastic Calculus by Karatzas and Shreve
• Markov Processes: Characterization and Convergence by Ethier and Kurtz
• Convergence of Probability Measures by Billingsley

Description:

What effects the large scale circulation of the atmosphere? Like the antiquated heating system of a New York apartment, solar radiation unevenly warms the Earth, leading to gradients in energy in both altitude and latitude. But unlike the simple convection of air in your drafty home, the effects of rotation, stratification, and moisture lead to exotic variations in weather and climate, giving us something to chat about over morning coffee... and occasionally bringing modern life to a standstill.

The goals of this course are to describe and understand the processes that govern atmospheric fluid flow, from the Hadley cells of the tropical troposphere to the polar night jet of the extratropical stratosphere, and to prepare you for research in the climate sciences. Building on your foundation in Geophysical Fluid Dynamics, we will explore how stratification and rotation regulate the atmosphere's response to gradients in heat and moisture. Much of our work will be to explain the zonal mean circulation of the atmosphere, but in order to accomplish this we’ll need to learn a great deal about deviations from the zonal mean: eddies and waves. It turns out that eddies and waves, planetary, synoptic (weather system size) and smaller in scale, are the primary drivers of the zonal mean circulation throughout much, if not all, of the atmosphere.

There will also be a significant numerical modeling component to the course. You will learn how to run an atmospheric model on NYU's High Performance Computing facility, and then design and conduct experiments to test the theory developed in class for a final course project.

Recommended Texts:

• Vallis, G.K. (2006). Atmospheric and Oceanic Fluid Dynamics: Fundamentals and Large-scale Circulation. New York, NY: Cambridge University Press.
• Lorenz, E.N. (1967). The Nature and Theory of the General Circulation of the Atmosphere. World Meteorological Org.
• Walker, G. (2007). An Ocean of Air: Why the Wind Blows and Other Mysteries of the Atmosphere. Orlando, FL: Houghton Mifflin Harcourt.

Prerequisites:

Fluid dynamics and some elementary probability are essential. Geophysical fluid dynamics and a class on stochastic processes are desirable.

Syllabus:

Kolmogorov initiated the study of statistical fluid dynamics and turbulence as a subject in probability theory by treating the velocity field as a random function. This quickly led to celebrated results such as the -5/3 power law and the 4/5 law for third-order structure functions, one of the few exact results in turbulence theory. Since then much work has been done to adapt the statistical theory to more general fluid flows, including the rotating stratified flows that are typical for atmosphere and ocean fluid dynamics. This class will review the classic theory and then work towards contemporary research applications in AOFD.

Planned topics include:

kinematics of homogenous random functions, correlation theory and power spectra, structure functions, classical theory of isotropic turbulence, derivation of the exact 4/5 law for third-order structure functions. Rotation and stratification, linear waves and vortices, statistical Helmholtz decomposition of horizontal flows, random advection of tracers, third-order structure functions and spectral energy fluxes in rotating stratified flows.

Fluid models of interest are the 1d Burger’s equation, 2d incompressible and quasi-geostrophic flow, 3-d quasi-geographic flow and the 3d Boussinesq equations. Numerical simulations will illustrate the theory.

Selected reading will be assigned during the course. This course is a seminar course with a final project that consists of presenting a relevant paper.

Description:

The goals of this course are to explore the foundations of weather prediction and synoptic meteorology. We will take a historical perspective, reading the classic papers that propelled the field forward, and reviews looking backward to capture the full context of these pivotal efforts. We'll start at the turn of the previous century, when scientists first started to think about applying physics to predict the weather (as opposed to relying on the behavior of large marmots, the length of bands in fuzzy caterpillars, etc.). This will lead us to the "Bergen school of meteorology" and the first numerical weather predictions efforts in the UK and Princeton. The direction from there on will depend part on the interests of the class, but I hope to explore the "air mass" view of meteorology, frontegenesis (and the more explosive elements of extratropical cyclones, such as we witnessed earlier this month), the fundamentals of tropical cyclones, the mesoscale meteorology associated with extreme convection and thunderstorms, and the limits of predictability.