Course Descriptions

Description:
Linear and multilinear algebra. Differentiation and integration in several variables. Introduction to manifolds, tangent and cotangent bundle. Vector fields, differential forms, exterior derivative. curves, Integration of differential forms on a manifold, Stokes' Theorem.

Textbook:
Spivak, Calculus on Manifolds

Description: 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.

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 (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.

Prerequisites: Completion of a Python summer assignment and relevant self-study.

Description:  The purpose of this course is threefold: It will teach students the popular Python programming language. Students will learn the five most important concepts of modern, object-oriented software development: testing, data structures, design, working with data, and distributed computing. All of the examples used in class will have a financial context. Projects we will work on include developing a toy exchange, building a framework for managing live price data, and tools for preparing high frequency data for simulations and backtests. Additional topics include Google’s Firebase realtime database and using Python to work with SQL. Students will make extensive use of the Anki study system to gauge their own progress and prepare for tests. Please note that students who sign up for this course should prepare by completing a summer assignment.

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

Description: 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 (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

Prerequisites: The following four courses, or equivalent: (1) Data Science and Data-Driven Modeling, (2) Financial Securities and Markets, (3) Machine Learning & Computational Statistics, and (4) Risk and Portfolio Management. It is important you have experience with the Python stack. 

Course description:  This is a full semester course covering recent and relevant topics in alternative data, machine learning and data science relevant to financial modeling and quantitative finance.  This is an advanced course that is suitable for students who have taken the more basic graduate machine learning and finance courses Data Science and Data-Driven Modeling, and Machine Learning & Computational Statistics, Financial Securities and Markets, and Risk and Portfolio Management. 

For the syllabus for the course, click HERE.

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 (2nd ed.). Hoboken, NJ: John Wiley & Sons/ Wiley-Interscience.

Prerequisites: Undergraduate linear algebra.

Description: Linear algebra is two things in one: a general methodology for solving linear systems, and an abstract structure underlying much of mathematics and the sciences. The course will assume that students have already had a course on linear algebra, and will be more advanced, focusing on analytical issues such as the behavior of eigenvalues and eigenfunctions.

Recommended Text: Strang, G. (2005). Linear Algebra and Its Applications (4^th ed.). Stamford, CT: Cengage Learning. Lax, P.D. (2007). Pure and Applied Mathematics: A Wiley Series of Texts, Monographs and Tracts [Series, Bk. 78]. 

• 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:   Undergraduate analysis and algebra at the level of MATH-UA 325 Analysis and MATH-UA 343 Algebra are strongly recommended.  Undergraduate students planning to take this course must have MATH-UA 343 Algebra and MATH-UA 325 Analysis (or the respective Honors versions) or permission of the Department.

Course Description:  After introducing metric spaces and 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, and homotopy.  Additional material may be covered at the discretion of the instructor, such as degree theory, transversality and intersection theory, and examples from knot theory.

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.

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:  'Real Analysis' by Stein and Shakarchi 

Secondary Text:  “Real Analysis” by Royden and Fitzpatrick

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.

Overview of the course: The plan is to cover the transport equation, the method of char-
acteristics, and the fundamental second order PDEs: the wave, Laplace and heat equations. Time permitting we will discuss: transform methods, Sobolev spaces, weak solutions, and some nonlinear PDEs.

Textbook: Evans, L.C. Partial Differential Equations (2nd ed), 2010. Graduate Studies in
Mathematics. Providence, RI: American Mathematical Society.

Prerequisites: Undergraduate Linear Algebra and ODE. Also,
PDE strongly recommended.
 

There is no assigned textbook for the course, but this book contains a
fair cross-section of topics: MH Holmes, Introduction to Perturbation Methods, Springer, 2nd edition 2013. Free download
from NYU via SpringerLink

Syllabus:
Regular and singular perturbations of algebraic equations, asymptotic expansions,
integral asymptotics. Dimensional analysis, scaling. Method of multiple scales for ODEs, averaging, WKB
solution, Kapitza’s pendulum. Similarity solutions for PDEs. Matched asymptotic expansions, boundary layers,
matching rules.
Green’s function asymptotics, near-field, far-field, and multipole expansions.
Fourier methods for dispersive PDEs, group velocity, stationary phase asymptotics.
Geometric wave theory, eikonal and transport equations, ray tracing for inhomogeneous
media, caustics. Possible additional topics: homogenization theory, Gaussian random functions, stochastic processes.

Asymptotic and exact solution to a dispersive PDE
Prerequisites: elementary linear algebra and differential
equations.

This is a first-year graduate course for all incoming PhD and
Master students interested in pursuing research in Applied
Mathematics.

This course 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, Fourier transform methods, geometric wave theory, and calculus
of variations.

Grading: this course will be graded as a regular course with a grad

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.

Course Description:

This full-semester course integrates key elements of econometrics, data science, and machine learning in a financial context. Students will learn to model financial data using statistical techniques and computational methods while gaining hands-on experience with Python-based tools. The course emphasizes:
Financial Econometrics & Statistical Inference: Understanding linear regression frameworks, hypothesis testing, and model selection.
Supervised Learning: Implementing regression and classification models, optimizing performance through cross-validation and regularization.
Unsupervised Learning: Applying dimensionality reduction techniques such as PCA and SVD.
Machine Learning in Finance: Exploring algorithmic trading, risk modeling, and portfolio optimization using advanced methods like boosting, bagging, and deep learning.
Data Manipulation & Web Scraping: Handling real-world financial data, including structured and alternative datasets, using Python.
Hands-on assignments will reinforce theoretical concepts, ensuring students gain practical experience in implementing machine learning techniques for financial applications.
Prerequisites:
It is essential that students enrolling in this course have a solid understanding of multivariate calculus, linear algebra, calculus-based probability, and statistics. Familiarity with the standard Python stack is recommended but not required.  

Course Description:

The first half of the course provides a hands-on exploration of fixed-income markets, equipping students with the quantitative and strategic skills needed for roles in trading, risk management, and quantitative modeling. Students will develop pricing models for bonds, Residential Mortgages, and Mortgage-Backed Securities (MBS), analyzing market reactions, risk positioning, and hedging strategies. Key topics include interest rates, prepayments, credit spreads (OAS), and model risk. The course also covers structured credit products such as CLOs, CMBS, ABS, and CDOs, along with credit derivatives like CDX and CMBX, emphasizing modeling risks and lessons from the 2008 Financial Crisis.
The second half of the course focuses on real-world applications in fixed-income and rate-derivatives markets, bridging the gap between theory and practice. Students will examine bonds, swaps, flow options, and structured products, gaining insight into how economic trade ideas translate into trading and risk management strategies. A problem-oriented approach reinforces intuition about product structuring, market dynamics, and the practical constraints faced by sell-side practitioners in an evolving financial landscape.
 

Prerequisites:
Financial Securities and Markets (MATH-GA.2791) and Stochastic Calculus (MATH-GA.2903). Familiarity with the standard Python stack and basic proficiency in Excel /Google Sheets is required. While prior exposure to interest-rate products is beneficial, it is not a requirement.

Course Description:

The goal of the first half of the semester of the 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.
In the second half of the semester, we turn to asset pricing and the trading of derivative securities using stochastic calculus techniques. Using tools and techniques from stochastic calculus, we cover (a) Black-Scholes-Merton option pricing; (b) the martingale approach to arbitrage pricing; (c) incomplete markets; and (d) the general option pricing formula using the change of numeraire technique. As an important example of incomplete markets, we discuss bond markets, interest rates and basic term-structure models such as Vasicek and Hull-White.
It is essential that students enrolling in this course have a solid understanding of multivariate calculus, linear algebra, and calculus-based probability. While not required, a one-semester course on derivative pricing, such as "Financial Securities and Markets," is recommended. Additionally, we suggest an intermediate course in mathematical statistics or engineering statistics as an optional prerequisite for this class.

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.

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

Description: 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: 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.

Description:

The interplay between information theory, statistics, and machine learning is a constant theme in the development of all fields. This course will discuss how techniques rooted in information theory play a key role in understanding the fundamental limits of statistical and machine learning problems in terms of minimax risk and sample complexity, and develop procedures that attain the statistical optimality. The primary focus is on information-theoretic applications to statistics and machine learning, rather than the classical “IEEE-style information theory”. 

This course will cover the following topics: 1) entropy, mutual information, KL divergence, f-divergences, and their many applications (source coding, channel coding, adaptive data analysis, PAC Bayes, binary hypothesis testing, large deviation, strong converse, I-MMSE formula, area theorem, functional inequalities, etc); 2) techniques for minimax lower bounds (mutual information method; Le Cam, Assouad, and Fano; methods of second moment and orthogonal polynomials; metric entropy and global Fano; etc); 3) constructive procedures (entropic upper bounds of statistical estimation; redundancy, aggregation, prediction via universal compression; sampling via strong data-processing inequality; etc). Detailed examples from many areas will be provided for each topic.

Prerequisite: 

• Maturity with probability theory on a graduate level
• Prior exposure to information theory is NOT required

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.

Required text:

Probability and Random Processes, 3rd edition by G.Grimmett and D. Stirzaker, Oxford Press 2001 (Note: this is NOT the newer 4th edition).

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.

Description:

The Earth’s atmosphere and oceans have been warming for the last 50 years, and are expected to do so until at least the end of this Century. Human activities, primarily the emission of carbon dioxide due to the burning of fossil fuel, are the main driving forces behind these changes. We will review the physical basis for anthropogenic climate change, study the evidence for climate changes over the last few decades, and analyze some of its potential future impacts.

As an advanced topics course, the class will also invite students to be active participants. About one third of the course will be dedicated to more formal lectures, one third to class discussion of recent scientific papers, and one third to student-led research projects. Among potential research topics are assessing climate sensitivity from recent warming, investigating recent trends in weather extremes, and analyzing the response of the atmospheric circulation.