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:

This course studies measure theory and integration in Euclidean and abstract spaces.  The topics of the course include Lebesgue measure, Lebesgue measurable functions, Lebesgue integration, Lebesgue-Stieltjes measure and integral,  general measure theory, Caratheodory's theorem, measurable functions, convergence in measure, Egoroff's and Lusin's theorems, integration in general measure spaces, monotone and dominated convergence theorems, Fubini-Tonelli theorem, Lebesgue-Radon-Nikodym theorem, and L^p spaces.

Textbooks:  

1. Real Analysis,  by H.L. Royden and P.M. Fitzpatrick.

2. Real Analysis, by G.B. Folland.

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.

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) elliptic partial differential equations such as the Poisson eq. (finite difference, finite element and integral equation methods); (3) parabolic  and hyperbolic equations such as the heat or wave equation (finite difference and finite volume methods). We will also discuss spectral methods and the FFT, exponential temporal integrators, and multigrid iterative solvers.

Prerequisites:
Risk and Portfolio Management, Financial Securities and Markets, 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: 
Multivariate calculus, linear algebra, and calculus-based probability. Students should also have working knowledge of basic statistics and machine learning (such as what is covered in Data Science & Data-Driven Modeling).

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:
Linear Algebra I or permission of the instructor.

Description:
Review of vector spaces, linear maps, determinant, spectral theory. Inner product spaces, Schur representation, polar and singular value decomositions, Moore-Penrose pseudoinverse. Dual vector spaces and tensors.

Text:
Treil, Linear Algebra Done Wrong

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

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

  Recommended Texts:

  • Lang, S. (2005). Graduate Texts in Mathematics [Series, Bk. 211]. Algebra (3rded.). 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, including fixed points. Manifolds and Poincaré duality. Introduction to products and ring structures,vector bundles and, in particular, tangent bundles.

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: We will discuss analytical problems related to fourth order equations in conformal geometry. In particular, we will study Q curvature equations and sharp inequalities.

Description:  This is an introductory course in Algebraic Geometry

Description: The goal of the course is to present M. Viazovska's recent proof of optical packing in eight dimensions.  The prerequisite for the course is complex analysis, particularly the theory of elliptic functions. The course will follow the presentation of the proof of Viazovska's result in the recent book by D.Romik, Topics in Complex Analysis.

The Mumford Shah functional has been introduced in the 80`s as tool for image denoising. Its rigorous study has lead to beautiful mathematical theories and several interesting problems are still unsolved. In this class I will review the classical theory and present some recent results

Prerequisites: real analysis and introductory functional analysis, working knowledge of Fourier analysis.

Description: This course will present the theory of wavelets from the viewpoint of approximation theory. After covering the fundamentals of wavelet analysis and various wavelet constructions, we will study how smoothness spaces can be characterized by means of the wavelet series of their elements. We will then study the approximation-theoretical implications of these characterizations, specifically, how wavelets offer superior approximation guarantees via non-linear approximation methods.

Prerequisites:
MATH-GA 2490 (Introduction to Partial Differential Equations) and MATH-GA 2430 (Real Variables).  Masters students should consult the course instructor before registering for this class.

Description:
Undergraduate and MS-level classes in PDE usually emphasize examples, involving solutions that are more or less explicit. This course does the opposite: it emphasizes more general methods, applicable to broad classes of PDE's. Topics to be covered include: tools from analysis (Fourier  transform, distributions, and Sobolev spaces, including embedding and  trace theorems); linear elliptic pde (weak solutions, regularity, Fredholm  alternative, symmetry and self-adjointness, completeness of eigenfunctions; maximum principles and Perron's method; boundary integral methods); selected methods for solving nonlinear elliptic pde (fixed point theorems, variational methods); parabolic and hyperbolic pde (energy methods, semigroup methods, steepest-descent pde's); viscosity solutions of first-order equations.

Main Texts:

L.C. Evans, Partial Differential Equations, American Mathematical Society
M. Renardy and R. Rogers, An Introduction to Partial Differential Equations, Springer-Verlag
 

Prerequisites:
Computing in Finance, and Risk and Portfolio Management, 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.

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

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.

Prerequisites:  
Calculus-based probability, Stochastic Calculus, and a one semester course on derivative pricing (such as what is covered in Financial Securities and Markets).

Description:   
This is an advanced course on asset pricing and trading of derivative securities. Using tools and techniques from stochastic calculus, we cover (1) Black-Scholes-Merton option pricing; (2) the martingale approach to arbitrage pricing; (3) incomplete markets; and (4) 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 important that students taking this course have good working knowledge of calculus-based probability and stochastic calculus. Students should also have taken the course “Financial Securities and Markets” previously. In addition, we recommend an intermediate course on mathematical statistics or engineering statistics as an optional prerequisite for this class.

Prerequisites:
Financial Securities and Markets, 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:
Stochastic Calculus, and Financial Securities and Markets or equivalent knowledge of basic bond mathematics and bond risk measures (duration and convexity).

Description:
This half-semester course is designed for students interested in Fixed Income roles in front-office trading, market risk management, model development (“Quants”, “Strats”), or model validation.

We begin by modeling the cash flows of a generic bond, emphasizing how the bond reacts to changes in markets, how traders may position themselves given their views on the markets, and how risk managers think about the risks of a bond. We then focus on Mortgages, covering the fundamentals of Residential Mortgages, and Mortgage-Backed Securities. Students will build pricing models for mortgages, pass-throughs, sequentials and CMO’s that generate cash flows and that take into account interest rates, prepayments and credit spreads (OAS). The goals are for students to develop: (1) an understanding of how to build these models and how assumptions create “model risk”, and (2) a trader’s and risk manager’s intuition for how these instruments behave as markets change, and (3) a knowledge how to hedge these products.  We will graph cash flows and changes in market values to enhance our intuition (e.g. in Excel, Python or by using another graphing tool).

In the course we also review the structures of CLO’s, Commercial Mortgage Backed Securities (CMBS), Auto Asset Backed Securities (ABS), Credit Card ABS, subprime mortgages and CDO’s and credit derivatives such as CDX, CMBX and ABX. We discuss the modeling risks of these products and the drivers of the Financial Crisis of 2008.  As time permits, we touch briefly on Peer-to-peer / MarketPlace Lending.

Prerequisites:
Financial Securities and Markets, and Stochastic Calculus.

Description:
The course provides a comprehensive overview of most commonly traded quantitative strategies in energy markets. The class bridges quantitative finance and energy economics covering theories of storage, net hedging pressure, optimal risk transfer, and derivatives pricing models. Throughout the course, the emphasis is placed on understanding the behavior of various market participants and trading strategies designed to monetize inefficiencies resulting from their activities and hedging needs. We discuss in detail recent structural changes related to financialization of energy commodities, crossmarket spillovers, and linkages to other financial asset classes. Trading strategies include traditional risk premia, volatility, correlation, and higher-order options Greeks. Examples and case studies are based on actual market episodes using real market data.

Prerequisites:
Financial Securities and Markets, 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:
Financial Securities and Markets, Risk and Portfolio Management, 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.

Prerequisites: Probability Limits Theorems 1

Description: Stochastic processes in continuous time. Brownian motion. Poisson process. Martingales and semimartingales. Stochastic integral. Stochastic differential equations. Markov processes. Connections with PDEs. Convergence of stochastic processes.

 

Main texts:

- Stochastic Processes by Bass

- J.-F. Le Gall, Brownian Motion, Martingales, and Stochastic Calculus. Springer, 2016

- Stochastic Processes by Varadhan, Courant Lecture Series in Mathematics, volume 16

Other recommended texts:

- Theory of Probability and Random Processes by Koralov and Sinai

- Brownian Motion and Stochastic Calculus by Karatzas and Shreve

- D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, Springer, 1999

 

Description:

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

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

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

Recommended Texts:

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

Description: We will cover data-driven analysis techniques for geophysical fluid dynamics from a theoretical and applied perspective for atmosphere and oceanic phenomena. 

Description: The first predictions of human induced global warming were made over a century ago, but the topic remains controversial despite the fact that the world has warmed 1 degree Celsius over the intervening years. In this course, we will investigate observational evidence as well as the physical and mathematical foundations upon which forecasts of future climate are based. What are the key uncertainties in the predictions, and what steps are required to reduce them?  We will find that it is not the science of global warming that is controversial, but rather, what to do about it.
By reading through a mixture of historic and current studies, investigating key processes that affect the sensitivity of our planet to greenhouse gases, and exploring a hierarchy of climate models, this course will get you up to speed on the science of climate change.  Grades will be based on a course project using a climate model to predict the impact of anthropogenic forcing on the Earth's climate.  Particularly attention will be paid to establishing reproducible science and quantifying the uncertainty in your prediction.