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.

Recommended Text: Introduction to Real Analysis, William F. Trench.

Recitation/Problem Session: 7:15-8:30 (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.

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 Texts (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 (2nd 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: TBA.

 

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, compressible flow (conservation laws, limiters, shock-capturing methods, boundary layers, turbulence), 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).

 

Texts: TBA.

Cross-listing: CSCI-GA 2945.001

Prerequisites: TBA.

Texts: TBA.

Cross-listing: CSCI-GA 2945.002

Prerequisites: TBA.

The course is intended for Courant Institute graduate students who are either mathematics students on the applied half of the spectrum or computer science students with an interest in numerical computation. ODEs, the starting point of analysis and dynamics, are one of the two foundation stones of applied mathematics (the other is linear algebra). This course will deepen your appreciation of them fundamentally.

This course will be based on a new textbook that I am 2/3 of the way through writing, Exploring ODEs, with coauthors Toby Driscoll and Asgeir Birkisson. The book aims to be interesting and informative for every applied or numerical mathematician. Its unusual missions are to be numerically-enabled, but not a numerical book and mathematically mature, but not technically advanced. To get an idea, including the table of contents, take a look at the talk I gave about the book project posted at https://people.maths.ox.ac.uk/trefethen/talks.html.

The course will be assessed by homework assignments. These will be computational (working in Chebfun), but will be about behavior of ODEs, not about algorithms or numerical analysis.

Cross-listing: CSCI-GA 2945.003

This course will provide an introduction to the finite element method and its theoretical foundation.

Self-adjoint elliptic problem and calculus of variation. Sobolev spaces, Poincare's and Friedrichs' inequalities. Triangulations of bounded domains in two and three dimensions. Lagrange and Hermite finite elements, which are conforming in H^1. The biharmonic problem and H^2 conforming elements.

Error bounds for basic finite element methods: Cea's lemma and the result of Aubin and Nitsche. Nonconforming finite elements: the lemmas due to Strang. Isoparametric elements and spectral elements. Compressible and almost incompressible elasticity. Mixed methods and the inf-sup condition due to Babuska and Brezzi. Incompressible Stokes. Raviart-Thomas elements and other elements conforming in H(div). Nedelec elements and other conforming elements in H(curl). Solving the large linear and non-linear systems of algebraic equations resulting from finite element approximations. An introduction to domain decomposition and multigrid algorithms.

Cross-listing: CSCI-GA 2945.004

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.

This course is intended to provide a practical introduction to 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 (2nd 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: Scientific Computing or Numerical Methods II, 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: 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 (4th 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 (5th 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 (2nd ed.). Hoboken, NJ: John Wiley & Sons/ Wiley-Interscience.

Recommended Text: Strang, G. (2005). Linear Algebra and Its Applications (4th 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(2nd 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. 3rd 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.

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.

This course will cover 3 topics:

1. Smale-Anosov dynamical systems: This will include basic stability and ergodicity theorems for Anosov and more genral Smale systems, including geodesic flows on manifolds with negative curvatures. The course will also prove basic results on Markov partitions and Markov codings.

2. Spaces with negative curvatures: The course will start with the classical material on Riemannian manifolds, then will prove basic properties of singular spaces of Busemann and Alexandrov with negative curvatures and explain the main constructions of these spaces.

3. Hyperbolic groups: The course will define these groups, prove their basic properties and discuss various classes of examples.

Texts: TBA.

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.

Text: Main Text: Folland's "Real Analysis: Modern Techniques and Their Applications"
Secondary Text: Bass' "Real Analysis for Graduate Students"

Prerequisites (Complex Variables I): 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 (8th ed.). New York, NY: McGraw-Hill.

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 (3rd ed.). New York, NY: McGraw-Hill.

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. For more information (including a tentative semester plan) see the syllabus here.

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 (2nd ed.). Providence, RI: American Mathematical Society.

Prerequisites: Linear algebra, real variables or the equivalent, and some complex function-theory would be helpful.

The course will concentrate on concrete aspects of the subject and on the spaces most commonly used in practice such as Lp (1? p ? ?), C, C?, and their duals. Working knowledge of Lebesgue measure and integral is expected. Special attention to Hilbert space (L2, Hardy spaces, Sobolev spaces, etc.), to the general spectral theorem there, and to its application to ordinary and partial differential equations. Fourier series and integrals in that setting. Compact operators and Fredholm determinants with an application or two. Introduction to measure/volume in infinite-dimensional spaces (Brownian motion). Some indications about non-linear analysis in an infinite-dimensional setting. General theme: How does ordinary linear algebra and calculus extend to d=? dimensions?

Recommended Texts: Lax, P.D. (2002). Pure and Applied Mathematics: A Wiley Series of Texts, Monographs and Tracts [Series, Bk. 55]. Functional Analysis (1st ed.). New York, NY: John Wiley & Sons/ Wiley-Interscience.

Reed, M., & Simon, B. (1972). Methods of Modern Mathematical Physics [Series, Vol. 1]. Functional Analysis (1st ed.). New York, NY: Academic Press.

The purpose of the course is to give an overview of some recent results in elliptic homogenization theory which focus on the random setting and are concerned with quantitative results. We will begin with an introduction to qualitative stochastic homogenization results, based on variational methods. Next we will discuss quantitative results based on the same variational methods which have a convex duality flavor, give explicit error estimates for the Dirichlet and Neumann problems, discuss variational methods for "non-variational" equations, develop a higher regularity theory and, time permitting, prove optimal error estimates and scaling limits for correctors. The course has few requisites apart from a basic knowledge of Sobolev spaces.

Prerequisites: TBA.

Texts: TBA.

I plan to discuss differentiable dynamics, i.e., systems generated by differentiable maps of flows on manifolds, focusing on the theory of chaotic dynamical systems, called hyperbolic theory in the field.

The following topics will be covered:

(1) Stable, center and unstable manifolds at fixed points of saddle type. This is where the simplest instance of chaotic behavior occurs.

(2) Uniform hyperbolic theory, including horseshoes and Anosov diffeomorphisms. The local picture is a direct generalization of (1), and a global theory will be added.

After these 2 topics, we have the following choices, of which there will be enough time to cover only one or two:

(3a) Ergodic theory of uniformly hyperbolic attractors
(3b) Nonuniform hyperbolic theory
(3c) A brief introduction to infinite dimensional theory

Prerequisites: Real analysis (with measure theory), basic knowledge of functional and harmonic analysis.

This course will cover a selection of classical and modern topics in applied harmonic analysis related to representation and processing of signals, including sampling theory, frames, time-frequency analysis, and wavelets.

Texts: None required. Relevant texts which are electronically available to the NYU community will be used as needed.

Prerequisites: Elementary linear algebra and differential equations.

This is a first-year course for all incoming PhD and Master 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). Camridge 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, irrotationall 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 cointegration, 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.

Prerequisites: Risk Management, Derivative Securities (or equivalent familiarity with market and credit risk models).

The course is divided into two parts. The first addresses the institutional structure surrounding capital markets regulation. It will cover Basel (1, MRA, 2, 2.5, 3), Dodd-Frank, CCAR and model review.  The second part covers the actual models used for the calculation of regulatory capital. These models include the Gaussian copula used for market risk, specific risk models, the Incremental Risk Calculation (single factor Vasicek), the Internal Models Method for credit, and the Comprehensive Risk Measure.

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.

A second course in arbitrage-based pricing of derivative securities. The Black-Scholes model and its generalizations: equivalent martingale measures; the martingale representation theorem; the market price of risk; applications including change of numeraire and the analysis of quantos. Interest rate models: the Heath-Jarrow-Morton approach and its relation to short-rate models; applications including mortgage- backed securities. The volatility smile/skew and approaches to accounting for it: underlyings with jumps, local volatility models, and stochastic volatility models.

Prerequisites: Computing in Finance (or equivalent), Derivative Securities (or equivalent), familiarity with analytical methods applied to interest rate derivatives.

When a corporation borrows money there is a risk that it will not fulfill its obligation to repay the lenders in the future; this is credit risk. This course develops mathematical tools and models that are useful in analyzing, valuing and managing credit risk, both in its original form as found embedded in bonds and loans, and in derived forms as it exists in derivatives like asset swaps, credit default swaps (CDS), CDS indices and options, and tranched portfolio products like synthetic CDOs. We will discuss alternative notions of credit spread, and their dynamics and relationship to fundamental quantities like probability of default and loss given default. The consideration of portfolio products will require the introduction of notions of credit correlation including, but not limited to, the Gaussian default time copula.

Required Text: O’Kane, D. (2008). Wiley Finance [Series, Bk. 545]. Modeling Single-name and Multi-name Credit Derivatives. John Wiley & Sons, Hoboken, NJ.

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

Turbulent dynamical systems are ubiquitous complex systems in geoscience and engineering and are characterized by a large dimensional phase space and a large dimension of strong instabilities which transfer energy throughout the system. They also occur in neural and material sciences. Key mathematical issues are their basic mathematical structural properties and qualitative features, their statistical prediction and uncertainty quantification (UQ), their data assimilation, and coping with the inevitable model errors that arise in approximating such complex systems. These model errors arise through both the curse of small ensemble size for large systems and the lack of physical understanding. This is a research expository course on the applied mathematics of turbulent dynamical systems through the paradigm of modern applied mathematics involving the blending of rigorous mathematical theory, qualitative and quantitative modelling, and novel numerical procedures driven by the goal of understanding physical phenomena which are of central importance. The contents include the general mathematical framework and theory, instructive qualitative models, and concrete models from climate atmosphere ocean science. New statistical energy principles for general turbulent dynamical systems are discussed with applications, linear statistical response theory combined with information theory to cope with model errors, reduced low order models, and recent mathematical strategies for UQ in turbulent dynamical systems. Also recent mathematical strategies for online data assimilation of turbulent dynamical systems as well as rigorous results are briefly surveyed. Accessible open problems are often mentioned. This research expository book is the first of its kind to discuss these important issues from a modern applied mathematics perspective.

Texts: Chapter 1, 2, 4, 6, 7, 8 from Nonlinear Dynamics and Statistical Theories for Basic Geophysical Flow, by A. J. Majda and X. Wang, Cambridge Press 2006;
Introduction to Turbulent Dynamical Systems for Complex Systems, Frontiers in Applied Dynamical Systems, by A. J. Majda, 85 pages, Springer 2016.

A mostly self-contained research-oriented and fast-paced course designed for PhD students with an interest in doing research in theoretical aspects of algorithms that aim to extract information from data. These often lie in overlaps of (Applied) Mathematics with: Computer Science, Electrical Engineering, Statistics, and/or Operations Research. Each lecture will feature either one or two Mathematical Open Problem(s) with relevance in Data Science. The main mathematical tools used will be Probability and Linear Algebra, and a basic familiarity with these subjects is required. There will also be some (although knowledge of these tools is not assumed) Graph Theory, Representation Theory, Applied Harmonic Analysis, among others. The topics treated will include Dimension reduction, Manifold learning, Sparse recovery, Random Matrices, Approximation Algorithms, Community detection in graphs, and several others.

This course covers the mathematical representations of neuronal systems, and the mathematical methods used to analyze these representations. Many of these methods are the same as those used to analyze experimental data in neural science.

The course begins with the fundamental physiological properties of neurons, including neuronal and synaptic dynamics. The Hodgkin-Huxley equations for single neuronal dynamics, with synaptic couplings, will be briefly summarized -- as well as their reductions to simpler representations such as integrate and fire neurons and “exponential” integrate and fire neurons. The course then turns to its focus – representations of systems of excitatory and inhibitory neurons. It will cover, in detail, population dynamics of such neuronal networks, including coarse-grained representations (mean-field firing rate models, models which incorporate fluctuations, and models which incorporate multi-neuron effects), and functional and structural connectivity. The course strives to bring students from a background in either the physical/mathematical sciences or neural science quickly to research topics in theoretical neural science.

Topics to be covered will be taken from the following longer list:

1. Basic Neurophysiology
   1. Anatomy of a Neuron
   2. Synapses and synaptic coupling
   3. Hodgkin-Huxley Equations
   4. Integrate and Fire Representations, including “exponential integrate and fire”
2. Neuronal Coding – rate vs spike coding
3. Signal Detection Theory
   1. Discrimination
   2. ROC
   3. Nieman-Pearsen Lemma
4. Spike train analysis
   1. Renewal processes
   2. Wiener-Khinchin Theorm
   3. Power spectrum
5. Coarse-grained representations for populations of neurons
   1. Mean-field firing rate models
   2. Mean-field representations with fluctuations
   3. Other methods to model fluctuations
      1. Kinetic Theory
   4. Multi-neuron effects
6. Fokker-Planck equation for a single neuron
   1. Kramer-Moyal Expansion
   2. Pawula Theorem
7. Compressed Sensing
8. System analysis
   1. Weiner Expansion
   2. Bussgang Theorem
   3. Receptive Fields
9. Functional vs anatomical connectivity
   1. Granger causality
10. Early visual system (Retina → LGN → V1)
    1. Receptive Fields
    2. Simple and complex cells

Texts: TBA.

This course will focus on neurophysiology and biophysics at the cellular level; the mechanistic and mathematical descriptions of neuronal dynamics and input/output properties. Topics will include: ionic channels (current-voltage relations, gating kinetics, different types), Hodgkin-Huxley equations and reductions (the action potential, repetitive firing, bursting, propagation), dendrites (branching cable theory, passive and active membrane, spines), synapses (transmitter release, depression/facilitation, plasticity). Both analytical (perturbation and bifurcation methods) and numerical techniques will be described and used, serving as an applied introduction to these methodologies. Students will undertake computing projects related to the course material.

Texts: TBA.

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

Fall Term

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.

Recommended 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: Wednesday, 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)

Prerequisites: A graduate course in probability. Prerequisites on Gaussian processes will be introduced as needed.

We will discuss the extremes of logarithmically correlated fields, starting with the classical case of branching random walks and moving to the general Gaussian case and beyond.

Texts: Notes will be made available, there is no textbook.

Prerequisites: Recommended: Probability Theory at the level of Probability Limit Theorems I and II

An introduction to some of the stochastic processes where the states, evolving in time, are assignments of, say, +1 or -1 to points in d-dimensional lattices or other graphs, which are motivated by nonequilibrium statistical physics at zero (or low) temperature. A typical transition/update rule is to agree with a strict majority of nearest neighbors or else, in the case of a tie, to toss a fair coin. We will discuss both known results and many open problems.

One old and one recent reference:

Nanda-Newman-Stein: Dynamics of Ising spin systems at zero temperature, A. M. S. Transl. (2) 198 (2000) 183-194.

Damron-Kogan-Newman-Sidoravicius: Coarsening with a frozen vertex, Elec. Commun. Prob. 21 (2016) 1-4.

Prerequisites: TBA.

This will be an introduction to entropy and its many roles in different branches of mathematics, especially information theory, probability, ergodic theory and statistical mechanics. The aim is to give a quick overview of many topics, emphasizing a few basic combinatorial problems that they have in common and which are responsible for the ubiquity of entropy.

The course divides roughly into five parts:

1. Shannon Entropy and Information Theory;

2. Large Deviations and Measure Concentration;

3. Statistical Mechanics and Thermodynamics;

4. Ergodic Theory;

5. The Thermodynamic Formalism.

Parts 1, 2 and 3 will introduce entropy in settings that can be described in terms of independent random variables. Parts 4 and 5 will generalize much of that story to stochastic processes having an invariance under the passage of time or spatial translation. The natural setting for those generalizations is provided by ergodic theory. In order to cover a wide range of subjects, I will have to sacrifice a lot of generality, and sometimes omit technical details. To make up for this, students will be expected to support the lectures with reading from a variety of sources.

Texts: TBA.

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: Cambrdige University Press.

Salmon, R. (1998). Lectures on Geophysical Fluid Dynamics. New York, NY: Oxford University Press.

Pedlosky, J. (1992). Geophysical Fluid Dynamics (2nd ed.). New York, NY: Springer-Verlag.

Prerequisites: Undergraduate calculus, linear algebra, introduction to physics.

Description: This course introduces geophysical fluid dynamics from an experimental laboratory point of view. Laboratory instrumentation centers on a turntable equipped with velocity field (PIV, particle imaging velocimetry) and density field (LIF, laser induced florescence) measurement equipment. The course intertwines the laboratory observations with geophysical fluid dynamics theory, while numerical models are used as a platform to reconcile the observations with theoretical understanding.

Grading: Graded as a seminar course.

Recommended Texts: Atmosphere, Ocean and Climate Dynamics: an Introductory Text. Plumb & Marshall

Introduction to Geophysical Fluid Dynamics. Benoit Cushman-Roisin

Laboratory Experiments from the Following:

Details: http://paoc.mit.edu/labguide/

Atmosphere:

• Solid-body Rotation: The parabolic shape taken up by the free surface of water in solid body rotation is studied. Similarly, we study why rotating planets take on the shape of an oblate spheroid.
• Dye Stirring: Interleaving patterns are created by stirring colored dyes into a rotating fluid. Examples from atmospheric and oceanic flows are studied.
• Balanced Vortex: The balance of forces in the momentum equation in a rotating frame of reference is investigated by studying the flow of water down a drain-hole in a rotating system. The ideas are applied to intense atmospheric vortices such as hurricanes.
• Fronts: We create fronts in the laboratory by bringing together two bodies of water of differing densities in a rotating system. We study atmospheric fronts using meteorological data.
• Ekman Layers: Ageostrophic flow in a bottom Ekman layer is investigated in high and low pressure surfaces created in the laboratory. The same phenomenon is studied in analyzed surface fields of atmospheric highs and lows.
• General Circulation: The Hadley circulation and middle-latitude weather systems are studied in a rotating annulus experiment, using an ice bucket to represent the pole, and in atmospheric data.
• Convection: The evolution of convective boundary layers and convective plumes are studied as an analogue of dry atmospheric and oceanic convection.
• Flow over a Barrier: Investigates Rossby waves generated by flow over a barrier on a beta plane.

Ocean:

• Taylor Columns: The rigidity imparted to a fluid by rotation, as encapsulated in the Taylor-Proudman theorem, is investigated by studying flow over a submerged obstacle.
• Density Currents: The role of density differences in driving fluid motion is studied using an experiment first carried out by Marsigli in 1695.
• Ekman Pumping/Suction: cyclonic and anticylonic circulations are set up in the laboratory through the use of fans blowing air over the surface of the water to study the role of Ekman layers in inducing vertical motion.
• Ocean Gyres: western intensification of the wind-driven ocean circulation is studied by setting up a gyre in the presence of topographic beta.
• Thermohaline Circulation: The thermohaline circulation of the ocean is studied in a rendition of the classic Stommel-Arons experiment.
• Source/Sink Flow: flow from source to sink on a topographic beta plane is studied as an analogue of mid-depth/abyssal circulation in the ocean.
• Rossby Waves: Rossby waves are studied by investigating the 'westward' drift of a vortex induced by a melting ice cube in a rotating fluid of variable depth. Rossby waves in the ocean, as seen from satellite altimetry, are also presented.

Prerequisites: This course is aimed at graduate students who want to learn more about the scientific basis for climate change. It will require to read a substantial number of complex scientific papers, to evaluate them critically, and to discuss them constructively among colleagues..

Long before being recognized as a fundamental environmental challenge for to our society, climate change has been an topic of very active scientific research, dating back to the work of Fourier in the 19th Century. Climate change is fundamentally a multidisciplinary endeavors, touching on complex issues such a radiative transfer, mathematical modeling, remote sensing and biochemestry. In this course, we will focus on various scientific papers published overall the decades that have lead to the emergence of a scientific consensus that human activities are directly affecting the Earth’s climate and will continue to do so through the course of the 21st Century. More specifically, we will address the following questions:

How do greenhouse gases impact the Earth’s energy budgets?

Where does carbon come and go in the Earth system?

How do we observe and attribute ongoing climate change?

How do we assess future climate changes?

To do so, we will analyze and discuss a dozen of important scientific papers published over the last 60 years.

The students will be expected to critically analyze a dozen scientific papers, and actively participate to their discussion during class. Each student will lead several of these discussions. The final grade will be based on a project paper reporting on recent research articles in climate science.

Texts: TBA.