Course Descriptions

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

 

Recitation/ Problem Session: Tuesdays, 8:10pm-9:30pm

Prerequisites:

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

Description:

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

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

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

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

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

If you want to brush up your MATLAB:

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

Prerequisites: Graduate-level proficiency in linear algebra, undergraduate-level proficiency in numerical methods and PDEs.

Description: The numerical solution of elliptic PDEs is a key topic in applied and computational mathematics, having applications in classic fields such as electromagnetics and fluid dynamics. Furthermore, elliptic solves are often the most time-consuming piece of numerical time marching schemes for parabolic problems. Numerical discretization of the differential formulation of these problems using finite elements or finite differences lead to sparse linear systems, while discretization of equivalent integral formulations of the same problem lead to dense linear systems. Despite these differences, many of the same hierarchical linear algebraic ideas can lead to fast solvers ( O(N) or O(N log N) ) in both situations. This course will discuss the necessary tools from numerical linear algebra and hierarchical matrix compression and inversion, as well as cover the mathematical underpinnings of why these algorithms are even possible in the first place.

Text: P.-G. Martinsson, Fast Direct Solvers for Elliptic PDEs, SIAM Books, Philadelphia, PA, (c) 2020.

Prerequisites: Procedural programming, some knowledge of Java recommended. 

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

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

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

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

Recommended Texts:

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

Prerequisites: Continuous Time Finance or permission of instructor.

Description:  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 numerical methods required to tackle these issues, and illustrate the methods with practical case studies like American option pricing, uncertain volatility, uncertain mortality, different rates for borrowing and lending, calibration of models to market smiles, credit valuation adjustment (CVA), 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.

PDE methods suffer from the curse of dimensionality. Since most quantitative finance problems are highdimensional,

we will mostly focus on simulation-based methods (a.k.a. Monte Carlo algorithms). This course exposes the students with a wide variety of Machine Learning techniques, old and new, including parametric regression, nonparametric regression, neural networks, kernel trick, etc. These techniques allow us to compute some quantities that are key ingredients of the nonlinear Monte Carlo algorithms.

The Python programming language will be used to provide simple numerical simulations illustrating the methods presented in the course. Homeworks will allow the students to check their understanding of the course by solving exercises inspired by our experience as quantitative analysts, and will involve some coding in Python.

Recommended text: 

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

Prerequisites: Financial Securities and Markets; Risk & Portfolio Management; and Computing in Finance, or equivalent programming experience.

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

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

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

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

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

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

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

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

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

Prerequisites: Risk and Portfolio Management; and Computing in Finance. In addition, students should have a working knowledge of statistics, finance, and basic machine learning. Students should have working experience with the Python stack (numpy/pandas/scikit-learn).

Description:  This half-semester elective course examines techniques dealing with the challenges of the alternative data ecosystem in quantitative and fundamental investment processes. We will address the quantitative tools and technique for alternative data including identifier mapping, stable panel creation, dataset evaluation and sensitive information extraction. We will go through the quantitative process of transferring raw data into investment data and tradable signals using text mining, time series analysis and machine learning. It is important that students taking this course have working experience with Python Stack. We will analyze real-world datasets and model them in Python using techniques from statistics, quantitative finance and machine learning. 

This is a half-semester course covering practical aspects of econometrics/statistics and data science/machine learning in an integrated and unified way as they are applied in the financial industry. We examine statistical inference for linear models, supervised learning (Lasso, ridge and elastic-net), and unsupervised learning (PCA- and SVD-based) machine learning techniques, applying these to solve common problems in finance. In addition, we cover model selection via cross-validation; manipulating, merging and cleaning large datasets in Python; and web-scraping of publicly available data.

Prerequisites: Undergraduate linear algebra.

 

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

 

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

 

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

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

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

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

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

Recommended Texts:

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

Description: Geometrical introduction first to numerical invariants (e.g., signature and index, Euler characteristic, arithmetic genus, etc.) and then their generalizations to characteristic classes (e.g., Stiefel-Whitney, Chern, Pontryjagin,) classes of manifolds, of vector bundles and of singular varieties. Sample applications from topology, geometry, algebraic geometry, analysis, combinatorics.

Prerequisites: Some familiarity with homology and cohomology. (Some supplementary sessions may be run concurrently with this course to offer further algebraic topology background as needed.)

Grading: This course won't have exams but students will do work to demonstrate or apply some of the methods.

Prerequisites: Multivariable calculus and linear algebra.

 

Description: 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.

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

 

The meetings will be Thursdays from 12:30-2:00, in room 1314; the weekly schedule is posted here.

Description: The Atmosphere Ocean Science Student Seminar focuses on research and presentation skills. The course is spread across two semesters, and participants are expected to participate in both to earn the full 3 credits. Participants will prepare and present a full length (45-50 minute) talk on their research each semester, for a total of two over the duration of the course. In addition, short “elevator talks” are developed and given in the second semester, the goal being to encapsulate the key points of your research in under 5 minutes. A main goal of the course is learning to present your research to different audiences. We consider overview talks, appropriate for a department wide colloquium, specialty talks, as would be given in a focused seminar, and a broad pitch you would give when meeting people and entering the job market. When not presenting, students are expected to engage with the speaker, asking questions and providing feedback at the end of the talk.

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

 

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

 

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

 

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

 

Secondary Text: Bass' Real Analysis for Graduate Students

Prerequisites: Advanced calculus (or equivalent).

 

Description: Complex numbers; analytic functions; Cauchy-Riemann equations; Cauchy's theorem; Laurent expansion; analytic continuation; calculus of residues; conformal mappings.

 

Text: Marsden and Hoffman, Basic Complex Analysis, 3d edition

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

 

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

 

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

 

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

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

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

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

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

Recommended Texts:

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

Prerequisites: MATH-GA 2500 (Partial Differential Equations), or a comparable introduction to PDE using Sobolev spaces and functional analysis.

Selected PDE topics of broad importance and applicability, including:  boundary integral methods for elliptic PDE; regularity via Schauder estimates; steepest-descent and dynamical systems perspectives on some nonlinear parabolic equations; weak and strong solutions of the Navier-Stokes equations; and topics from the calculus of variations, including homogenization and Gamma convergence.

Prerequisites:

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

Description:

Fourier series and integrals, Hardy-Littlewood maximal function, interpolation theory, Hilbert transform, singular integrals and Calderon-Zygmund theory, oscillatory integrals, Littlewood-Paley theory, pseudo-differential operators and Sobolev spaces. If time allows: paradifferential calculus, T1 theorem. 

The course will follow the book

Fourier Analysis by Javier Duoandikoetxea, Graduate Studies in Mathematics, AMS, 2001, as well as the lecture notes by Terry Tao

https://www.math.ucla.edu/~tao/247a.1.06f/
https://www.math.ucla.edu/~tao/247b.1.07w/

Prerequisites :  Real Variables, Sobolev Spaces and Basic PDEs

Description:  The topics to be discussed in the first part of the course are: harmonic maps in 2D including the Riemann mappings, univalent harmonic maps and its Jacobian, and some applications to isometric embeddings and minimal surfaces.  A few basic Sobolev inequalities and facts about Sobolev maps will be proved too. 

For the second part of the course, we start with Polyakov formula in the spectral geometry as a motivation for Moser-Trudinger inequality, then it follows by Szego limit theorem on the circle.  Afterwards, we switch to more recent progress of generalization to the 2-sphere and its close relation to minimal nodes problem in numerical analysis.  We shall briefly discuss sphere covering inequality and applications to mean field equations. Various open problem will be remarked.

This course introduces the student to the first fundamental ideas of differentiable dynamical systems, focusing on hyperbolic dynamics. Hyperbolicity in dynamical systems refers to the fast separation of nearby orbits, from unstable fixed points to chaotic behavior. No prior knowledge of the subject is assumed. The three main topics I plan to cover in some detail are (1) stable, unstable and center manifolds at fixed points and periodic orbits, (2) geometric theory of chaotic systems: horseshoes, homoclinic orbits, and attractors; and (3) hyperbolic dynamics from a probabilistic or ergodic theory point of view, including the ergodic theorem and Lyapunov exponents. I intend also to set aside a few lectures at the end of the term to discuss examples and for brief excursions into topics in other parts of dynamical systems, to present a broader view of the subject.

Prerequisite: For topics (1) and (2), analysis of several variables is a must, basic knowledge of manifolds helpful; measure theory is assumed for topic (3).

Recommended text: I will not be following any text. Class notes will be made available. The closest text is Introduction to Dynamical systems by Brin and Stuck.

Prerequisites: Elementary linear algebra and differential equations.

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

Recommended Texts:

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

Prerequisites: Introductory complex variable and partial differential equations.

 

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

 

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

 

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

Prerequisites: Financial Securities and Markets; Scientific Computing in Finance (or Scientific Computing); and familiarity with basic probability.

Description:  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: Multivariate calculus, linear algebra, and calculus-based probability.

Description:  Risk management is arguably one of the most important tools for managing investment portfolios and trading books and quantifying the effects of leverage and diversification (or lack thereof).

This course is an introduction to portfolio and risk management techniques for portfolios of (i) equities, delta-1 securities, and futures and (ii) basic fixed income 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 portfolios (e.g. portfolio optimization and risk). As part of the course, we review current risk models and practices used by large financial institutions.

It is important that students taking this course have good working knowledge of multivariate calculus, linear algebra and calculus-based probability.

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: Multivariate calculus, linear algebra, and calculus-based probability.

This course provides a quantitative introduction to financial securities for students who are aspiring to careers in the financial industry. We study how securities traded, priced and hedged in the financial markets. 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; clearing; valuation adjustment and capital requirements. It is important that students taking this course have good working knowledge of multivariate calculus, linear algebra and calculus-based probability.

Prerequisites: Financial Securities and Markets; and Stochastic Calculus, or equivalent.

Description: 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 programming skills; and Financial Securities and Markets, or equivalent  familiarity with Black-Scholes interest rate models.

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

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

Prerequisites: Advanced Risk Management; Financial Securities and Markets, or equivalent familiarity with market and credit risk models; and Computing in Finance, or equivalent programming experience.

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

Complex systems are ubiquitous in science and engineering and are characterized by a large dimensional phase space and a large dimension of strong instabilities which transfer energy throughout the system.

Typical examples can be found in the atmosphere and ocean science, plasma physics, neural and material sciences. Modeling and statistical prediction of typical phenomena such as extreme events and anomalous transport have important scientific significance and large societal impacts.

Key mathematical issues in the complex systems 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 and a lack of physical understanding. Efficient and accurate algorithms for parameter estimation, data assimilation, and prediction are required using cheap stochastic parameterizations, judicious linear feedback control, and a new nonlinear modeling framework, which provide accurate predictions of both the observed and hidden extreme events, as well as the strongly non-Gaussian statistics in highly intermittent nonlinear models.

This is a research expository course on the development of new mathematical framework of building suitable nonlinear approximate models, which aim at predicting both the observed and hidden statistics in complex nonlinear dynamical systems for short, medium, and long range forecasting using only short and partially observed training time series. The applied mathematics of various complex dynamical systems are studied through the paradigm of modern applied mathematics involving the blending of rigorous mathematical theory, qualitative and quantitative modeling, 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 and plasma physics as well as laboratory water wave experiments. Also recent mathematical strategies for turbulent dynamical systems as well as rigorous results are briefly surveyed. Accessible open problems are often mentioned.

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

Prerequisite: Some familiarity with applied differential equations (seek consent of instructor if in doubt).

Description:  This course will focus on neurophysiology and biophysics at the cellular level - the mechanistic and mathematical descriptions of neuronal dynamics and input/output properties – some linear and many quite nonlinear. How do neurons integrate synaptic inputs over their dendrites, generate spikes and transmit spikes to targets?  With a range of different intrinsic properties neurons can temporally organize their distinctive spiking signatures to perform as integrators, as differentiators, as fast or slow pacemakers, or as bursters.  Neuronal functional architecture varies;  different branching structures and ionic channel distributions allow for local processing in dendrites or more feedforward transmission to the soma; coupling between soma and axon can shape spike train output.  Excitability and propagation will be described with Hodgkin-Huxley-like models and reductions; synaptic transduction will feature ionic channel kinetics, dynamics of depression/facilitation/plasticity and control at dendritic spines.  Both numerical simulation and analytical techniques (perturbation and bifurcation methods) will be described and used, serving as an applied introduction to these methodologies. Students will undertake computing projects related to the course material.

Prerequisites:

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

 

Description:

The course introduces the basic concepts and methods of probability.

 

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

 

Recommeded text:

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

Prerequisite: Multivariate calculus, linear algebra, and calculus-based probability.

Description: The goal of this half-semester course is for students to develop an understanding of the techniques of stochastic processes and stochastic calculus as it is applied in financial applications. We begin by constructing the Brownian motion (BM) and the Ito integral, studying their properties. Then we turn to Ito’s lemma and Girsanov’s theorem, covering several practical applications. Towards the end of the course, we study the linkage between SDEs and PDEs through the Feynman-Kac equation. It is important that students taking this course have good working knowledge of calculus-based probability.

Prerequisites:

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

 

Description:

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: Graduate course in Probability

Description: We will discuss the construction and approximations of Gaussian (and non-Gaussian) multiplicative chaos, and their role in the analysis of logarithmicaly correlated Gaussian fields, random matrices, and polymers.

Texts: There will be no textbook. I will rely on articles from the current literature, and probably will also distribute notes.

Description:

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

Recommended Texts:

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

Prerequisites: Geophysical Fluid Dynamics, or equivalent knowledge of rotating/stratified fluid dynamics

The dynamical structure and circulation of planetary atmospheres and oceans are shaped by turbulent motions occurring at scales ranging from the scale of the planet itself, to the scale where viscosity absorbs its energy (millimeters in Earth’s atmosphere). These motions are largely characterized by high rotation and stratification, but also by their ubiquity near boundaries. This course will survey a wide range of observed turbulent processes, with a focus on their phenomenology, interpretation through simplified models, and the tools necessary to analyze them. Topics to be covered include: scaling, power spectra and structure functions, Reynolds averaging and eddy diffusivity, generating instabilities, inertial-range theory, rotating and stratified limits, geostrophic turbulence, coherent structure formation, boundary layer turbulence, convection, passive scalar advection by turbulent flows, closure theories and parameterization.

Reading: A reading list will be provided, drawing from a few basic texts and articles

Assessment: Grades will be calculated based on the completion of a project (chosen in consultation with the instructor midway through the course), a presentation on the project, as well as on a few problem sets.

Prerequisites: Undergraduate calculus, linear algebra, introduction to fluid dynamics.

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.