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:

Thursdays, 7:10-9:00PM (following the course)

Description:

This class will be an introduction to the fundamentals of parallel scientific computing. We will establish a basic understanding of modern computer architectures (CPUs and accelerators, memory hierarchies, interconnects) and of parallel approaches to programming these machines (distributed vs. shared memory parallelism: MPI, OpenMP, OpenCL/CUDA). Issues such as load balancing, communication, and synchronization will be covered and illustrated in the context of parallel numerical algorithms. Since a prerequisite for good parallel performance is good serial performance, this aspect will also be addressed. Along the way you will be exposed to important tools for high performance computing such as debuggers, schedulers, visualization, and version control systems. This will be a hands-on class, with several parallel (and serial) computing assignments, in which you will explore material by yourself and try things out. There will be a larger final project at the end. You will learn some Unix in this course, if you don't know it already. Prerequisites for the course are (serial) programming experience with C/C++ (I will use C in class) or FORTRAN, and some familiarity with numerical methods.

Cross Listing:

CSCI-GA.2945-001

Prerequisites:

Familiarity with numerical methods and fluid dynamics.

Description:

The immersed boundary (IB) method is a general framework for the computer simulation of flows with immersed elastic boundaries and/or complicated geometry. It was originally developed to study the fluid dynamics of heart valves, and it has since been applied to a wide variety of problems in biofluid dynamics, such as wave propagation in the inner ear, blood clotting, fish swimming, and insect flight. Non-biological applications include sails, parachutes, flows of suspensions, and two-fluid or multifluid problems.

Topics to be covered include: mathematical formulation of fluid-structure interaction in Eulerian and Lagrangian variables with the Dirac delta function as the link between these two kinds of variables; discretization of the structure, fluid, and interaction equations, including energy-based discretization of the structure equations, finite difference discretization of the fluid equations, and IB delta functions with specified mathematical properties; an IB method with an exactly divergence-free interpolated velocity field; IB methods for immersed boundaries with non-trivial mass and for fluids with non-uniform density and viscosity; IB methods for immersed filaments with bend and twist; and a stochastic IB method for thermally fluctuating hydrodynamics within biological cells.

Course requirements include homework assignments and a computing project, but no exam. Students may collaborate on the homework and on the computing project, and are encouraged to present the results of their computing projects to the class.

Text:

The Immersed Boundary Method, lecture notes by Charles S. Peskin. These notes will be supplemented by selected publications on the topics of the course.

Prerequisites:

Students are expected to be comfortable with numerical linear algebra and multivariate calculus, and to have programming experience (preferably in Matlab). Students without all elements of this background are likely to have difficulty with the course material.

Description:

Many problems in science, engineering, medicine, and business involve optimization, in which we seek to optimize a mathematical measure of goodness subject to constraints. This course will survey widely used methods for continuous optimization, focusing on both theoretical foundations and implementation as software.

The course will cover: linear programming (including the simplex method and interior-point methods); line search and trust region methods for unconstrained optimization; methods for convex and nonconvex quadratic programming; and penalty, barrier, augmented Lagrangian, and interior-point methods for convex and nonconvex constrained optimization. In all cases, attention will be paid to numerical issues.

Description:

As physical and time domains of partial differential equations (PDEs) can be resolved with ever higher accuracy, the quality of numerical simulations of physical phenomena increasingly tends to be limited by noisy and incomplete data that insufficiently describe boundary conditions, coefficients, and other parameters of the problem setups. This course provides an introduction to the stochastic modeling of these data uncertainties as random coefficients and random forcing terms of PDEs and discusses dimensionality reduction methods for efficiently estimating moments of the solutions of the corresponding stochastic PDEs. The first part focuses on stochastic modeling and sampling (Karhunen-Loeve expansion, polynomial chaos expansion, stochastic Galerkin, (multilevel) Monte Carlo, stochastic collocation, sparse grids). The second part discusses dimensionality reduction techniques to make estimation of moments of the PDE solutions computationally tractable (proper orthogonal decomposition, a posteriori error estimation and error control of reduced models, empirical interpolation).

This course primarily concerns the design and analysis of Monte Carlo sampling techniques for the estimation of averages with respect to high dimensional probability distributions. Standard simulation tools such as importance sampling, Gibbs and Metropolis-Hastings sampling, Langevin dynamics, and hybrid Monte Carlo will be introduced along with basic theoretical concepts regarding their convergence to equilibrium. The focus is on methods that are broadly useful for problems in science, engineering, and statistics. Particular attention will be paid to the major complicating issues like conditioning and rare events and methods to address them (e.g. tempering, interacting particle methods, and free-energy methods).

Prerequisites:

Numerical Methods I or equivalent graduate course in numerical analysis (numerical linear algebra, iterative solvers, nonlinear systems, interpolation, integration), undergraduate or graduate courses in ODE and (hyperbolic, parabolic, and elliptic) PDEs

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 in MATLAB will form an essential part of the course. The course will introduce students to numerical methods for (1) ordinary differential equations, explicit and implicit Runge-Kutta and multistep methods, convergence and stability; (2) finite difference and finite element and integral equation methods for elliptic partial differential equations (Poisson eq.); (4) spectral methods and the FFT, exponential temporal integrators, and multigrid iterative solvers; and (5) finite difference and finite volume parabolic (diffusion/heat eq.) and hyperbolic (advection and wave) partial differential equations.

Text:

“Finite Difference Methods for Ordinary and Partial Differential Equations” by Randy LeVeque.

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 (2^nd 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:

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).

Description:

Recent progress on Ricci curvature including an overview of: quantitative behavior of singular sets and their rectifiability, for noncollapsed Gromov Hausdorff limit space with Ricci curvature bounded below, the proof of the codimension 4 conjecture on the singular set of noncollapsed limits of Einstein manifolds.

Description:

In neuronal systems, development and learning occur through some form of Hebb’s postulate – neurons that fire together, wire together. That is, modifications in synaptic strength and transmission efficiency are driven by correlations between pre and post synaptic neuronal firing activity. This leads to fascinating dynamical model systems in which the parameters of the model themselves depend on the solution’s temporal activity. In this short course, we will survey the mathematical theories of synaptic plasticity that are currently used to model Hebbian learning in neuronal networks. We will explore correlation plasticity based upon mean firing rates, as well as that based upon spike-timing-dependent-plasticity (STDP) based upon the spikes times of pre- and post- synaptic neurons. We will develop learning rules for updating synapses between excitatory neurons, as well as synapses from inhibitory neurons (where pre-synaptic spiking is not relevant). We will also discuss STDP when triplets of spikes (two pre, one post; or one pre, two post) replace pairs of spikes in the learning rule. We will then explore the consequences of these synaptic plasticity rules for unsupervised learning in large-scale neuronal networks, again with rate-coding paradigms and with STDP paradigms. For example, this unsupervised learning in large-scale neuronal networks provides a model for the development of cortical maps of feature preferences such as orientation in visual cortex and whisking direction in barrel cortex of rats. We will discuss how ordered feature maps result from Turing Instabilities in the presence of symmetries – translation, rotation, shift and permutations; and how disordered or “salt and pepper” feature maps result when heterogeneities break these symmetries. Finally, if time permits, we will discuss possible roles of electrical coupling through gap junctions in the development of these cortical maps.

Texts:

• W. Gerstner and W. Kistler, “Spiking Neuron Models”, Cambridge University Press, Ch 10 & 11, (2002).
• S. Song, K. Miller and L. Abbott, “Competitive Hebbian Learning through Spike-Timing-Dependent Synaptic Plasticity”, Nature Neuroscience 3 no 9, pp 919 – 926, (2000).
• S. Song and L. Abbott, “Cortical Development and Remapping through Spike Timing – Dependent Plasticity”, Neuron 32, pp339-350 (2001).
• J-P Pfister and W. Gerstner, “Triplets of Spikes in a Model of Spike Timing-Dependent Plasticity”, J Neuroscience 26(38), pp 9673-9682, (2006).
• M. Kaschube, “Neural Maps Versus Salt-and-Pepper Organization in Visual Cortex”, Current Opinion in Biology 24, 95-102 (2014).
• P. Thomas & J. Cowan, “Symmetry Induces Coupling of Cortical Feature Maps”, Phys. Rev. Lett. 92, (2004).
• F. Wolf, “Symmetry, Multistability, and Long-Range Interactions in Brain Development”, Phys. Rev. Lett. 95 (2005).
• G. Pernelle, W. Nicola and C. Clopath, “Gap Junction Plasticity as a Mechanism to Regulate Network-Wide Oscillations”, PLOS Computational Biology, (2018).

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.)

Description:

Basic introduction to Hamiltonian mechanics and integrable systems. Computation of eigenvalues of symmetric matrices using Hamiltonian integrable mechanics. Analysis of eigenvalue computation for random matrices. Universality.

This course will be run as a working group or seminar. The goal is to study the literature on invariant distributions for the stochastic Navier--Stokes system in 2D and other dissipative SPDEs. These results span approximately last two decades and although we do not aim at complete coverage of the area, we plan to get to some fairly recent developments. It will be assumed that participants are familiar with probability theory/stochastic processes and analysis. In the first several classes, the instructors will provide a minimal introduction into: the basics of the deterministic Navier--Stokes system; stochastic calculus in finite and infinite dimensions; the Navier--Stokes with random forcing; ergodic theory of Markov processes including the coupling method. Meanwhile, research papers will be distributed among the participants, and in the remaining part of the semester, we expect the participants to present these papers to the seminar.

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.

Description:

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:

The above faculty, along with graduate students and some post-docs, will meet once per week and cover a range of topics including: integral equation formulations of PDEs, numerical methods for singular quadrature, fast algorithms, and optimization methods. The material will range from standard/introductory level to recent research results, and will be presented via paper discussion, lectures by the above faculty, and reports on recent research progress by graduate students (roughly 1/3 time spend on each). Application areas will include fluid mechanics, magnetofluid dynamics (plasma physics), and electromagnetics. The working group will be advertised and open to all graduate students, pending approval of the faculty organizers.

Description:

This course provides a rigorous introduction to mathematical tools for data science drawn from linear algebra, harmonic analysis, probability theory, and convex analysis. The main topics are the singular-value decomposition (SVD), the Fourier series, randomized projections, the randomized SVD, convex optimization, duality theory and nonconvex optimization. The material is motivated by multiple data-analysis applications including dimensionality reduction, collaborative filtering, sound and image processing, magnetic-resonance imaging, sparse regression, compressed sensing, and topic modeling.

Cross Listing:

DS-GA 1013.001

Description:

This course will explore how applied mathematics, math modeling and simulations can productively interact with the experimental sciences and with real world observations and data. The course will involve projects in fluid and solid dynamics, each of which has an experimental system in the Applied Math Lab. Students will work in small groups to gather experimental data, with an emphasis on discovery and characterization of phenomena, and they will formulate mathematical models and/or computational simulations to account for these observations and make testable predictions. Assignments will include journal-style papers and conference-style talks. The projects will be drawn from research and will explore a broad range of questions relevant to the physical, biological and engineering sciences.

Strongly suggested background:

Graduate level Fluid Dynamics, PDEs, Methods of Applied Mathematics.

Description:

Fluids and materials whose large-scale motions are driven by an active microstructure (say, microswimmers or motor-proteins) are a new frontier area of biophysics and mathematical modeling. These systems are central to self-organizing processes within organisms, and are increasingly studied in-vivo using purified cellular components. In the cell, the microstructural elements of biopolymers and molecular motors are also involved in basic transport processes such as spindle positioning prior to cell division. I'll cover modeling techniques and models of how these structural elements interact with each other collectively to yield large-scale dynamics, as well as their role in basic biomechanical processes in the cell.

The course offers an introduction to the mathematical theory of the incompressible Euler and Navier-Stokes equations. Initially, we discuss the existence of weak/strong solutions and conditional regularity theorems. In the second half of the course, we discuss topics which reflect the audiences' interests. These include Prandtl theory (and boundary layers), hydrodynamic stability, or Onsager's conjecture

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:

The goal of this course is to introduce students to modern dynamical oceanography, with a focus on mathematical models for observed phenomena. The lectures will cover the observed structure of the ocean, the thermodynamics of sea-water, the equations of motion for rotating-stratified flow, and the most useful approximations thereof: the primitive, planetary geostrophic and quasi-geostrophic equations. The lectures will demonstrate how these approximations can be used to understand boundary layers, wind-driven circulation, buoyancy-driven circulation, oceanic waves (Rossby, Kelvin and inertio-gravity), potential vorticity dynamics, theories for the observed upper-ocean stratification (the thermocline), and for the global general circulation. Additionally the course will cover relevant oceanic fluid instabilities and their resulting turbulence: mesoscale turbulence driven by baroclinic instability, convective turbulence and high-latitude sinking, and mixing across density surfaces due to shear-driven turbulence. Finally, we will discuss tides and the observed internal wave spectrum. Throughout the lectures, the interplay between observational, theoretical, and modeling approaches to problems in oceanography will be highlighted.

Course activities will include a few problem sets and class presentations.

Recommended Text:

• The Theory of Large-Scale Ocean Circulation by R. Samelson (Cambridge 2011)
• Atmospheric and Oceanic Fluid Dynamics by G.K. Vallis (Cambridge 2006)
• Lectures on Geophysical Fluid Dynamics by R. Salmon (Oxford 1998)
• In addition, will read relevant journal articles each week.