Course Descriptions

This course continues MA-GY 6213. The topics are integration, series of real numbers, sequences and series of functions and Fourier series. Important concepts and theorems include Riemann and Riemann-Stieltjes integral, fundamental theorem of calculus, the mean value theorem of integrals, Dirichlet test, absolute and conditional convergence, uniform convergence, Weierstrass test, power series, orthogonal functions and Fourier series. | Offered in the spring.

Prerequisite(s): MA-GY 6213.

This course continues MA-GY 7033. Topics covered are basic concepts of linear algebra continuing with: range, nullity, determinants and eigenvalues of matrices and linear homomorphisms, the polar decomposition and spectral properties of linear maps, orthogonality, adjointness and its applications. | Offered in the spring.

Prerequisite(s): MA-GY 7033.

Description:

Differentiation and integration for vector-valued functions of one and several variables: curves, surfaces, manifolds, inverse and implicit function theorems, integration on manifolds, Stokes' theorem, applications. 

Description:

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.

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.

Prerequisite:  Familiarity with numerical methods and fluid dynamics.

Description:

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

Topics to be covered include: mathematical formulation of fluid-structure interaction in Eulerian and Lagrangian variables, with interaction equations involving the Dirac delta function; discretization of the structure, fluid, and interaction equations, including energy-based discretization of the structure equations, finite-difference discretization of the fluid equations, and IB delta functions with specified mathematical properties; a simple but effective method for adding mass to an immersed boundary; numerical simulation of rigid immersed structures or immersed structures with rigid parts; IB methods for immersed filaments with bend and twist; and a stochastic IB method for thermally fluctuating hydrodynamics within biological cells.  Some recent developments to be discussed include stability analysis of the IB method and a Fourier-Spectral IB method with improved boundary resolution.

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

Text:
The Immersed Boundary Method.  Lecture notes freely available at: http://www.math.nyu.edu/faculty/peskin/ib_lecture_notes/index.html

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

Prerequisites:
Linear Algebra I or permission of the instructor.

Description:
Review of: Trace, determinant, characteristic and minimal polynomial. Eigenvalues, diagonalization, spectral theorem and spectral mapping theorem. When diagonalization fails: nilpotent operators and their structure, generalized eigenspace decomposition, Jordan canonical form. Polar and singular value decomposition. Complexification and smooth surfaces in R^n and their tangent spaces. Intro to matrix Lie algebras and Lie groups.

Text:
Friedberg, S.H., Insel, A.J., and Spence, L.E. (2003). Linear Algebra (4^th ed.). Upper Saddle River, NJ: Prentice-Hall/ Pearson Education.

Plus: Extensive instructor’s class notes.

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

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

  Recommended Texts:

  • Lang, S. (2005). Graduate Texts in Mathematics [Series, Bk. 211]. Algebra (3^rded.). 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.

• MATH-GA.2210-001 Introduction To Number Th

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: 

The AOS Seminar is a practicum focused on scientific presentations.  PhD students present their research as either a formal talk of the type that would be given for an invited presentation, or as an informal introduction to a topic.  Faculty and other students provide constructive feedback. 

Description:  In neuronal systems, development and learning occur through some form of Hebb’s postulate – neurons that fire together, wire together. That is, modifications in synaptic strength and efficacy are driven by correlations between pre and post synaptic neuronal firing activity. This leads to fascinating dynamical systems models in which the parameters of the model that fix the synaptic strength dependon the solution’s temporal activity.

In Neuroscience, this type of synaptic plasticity is known as Long-Term Potentiation (LTP) and Long-Term Depression (LTD). LTP/D occurs over time scales that are much slower than those at which the neurons in the network respond to stimulation. This slow temporal evolution of LTP/D is driven by local time averages of the dynamics of the neurons’ responses, which slowly modifies these neuronal responses.

In this short course, we will study mathematical models of LTP/D – some at the level of neuronal systems and others at the cellular-molecular level, setting the stage for combining the two levels of representation.

Refs at the Systems Level:

P. Dayan and L. Abbott, “Theoretical Neuroscience” MIT Press, Part III (2002)

W. Gerstner and W. Kistler, “Spiking Neuron Models”, Cambridge University Press, Ch 10 & 11, (2002).

S. Song, K. Miller and L. Abbott, Nature Neuroscience 3 no 9, pp 919 – 926, (2000).

S. Song and L. Abbott, Neuron 32, pp339-350 (2001).

J-P Pfister and W. Gerstner, J Neuroscience 26(38), pp 9673-9682, (2006)

Milstein,.., K. Bittner,...,J. Magee,...,S.Romani, eLife (2021)

 

Refs at the Cellular-Molecular Level:

M. Graupner and N. Brunel, PLoS Computational Biology, 3:11, 2299-2323, (2007).

H. Pi & J. Lisman, J. Neuro. Sci, 28(49) 13132-13138 (2008)

J. Lisman, R. Yasuda and S. Raghavachari, Nature Rev-Neuroscience 13, 169-182, (2012).

K. Bittner, A. Milstein, C. Grienberger, S. Romani & J. Magee, Science, 357, 1033-1036 (2017)

H. Ma, N. Mandelberg, S. Cohen, X. He, and R. Chen, preprint (2019).

G. Li, C. Peskin & D. McLaughlin, “A Biochemical Cellular-Molecular Description of Early Long-Term Potentiation – Toward “One-Shot” Learning”, in draft (2023)

Description:

This is a lunchtime group meeting aimed at graduate students, postdocs, and faculty working in applied mathematics who do modeling & simulation.  The goal is to create a space to discuss applied mathematics research in an informal setting: to (a) give current graduate 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.

Description:

The course will cover the basic theory of elliptic functions, with applications, in particular, to integrable systems.

Description:

On a finite time interval, small noisy perturbations do not change the behavior of a dynamical system much. But the long-term dynamics of the perturbed system can be drastically different, and the behavior depends on the geometry of the problem. First, I will discuss the classical metastability results of Freidlin and Wentzell describing the behavior of a system with multiple stable attractors and transitions between them. Then I will discuss newer results concerning the behavior of systems perturbed by noise near saddle points and heteroclinic networks. Familiarity with stochastic calculus will be assumed.

Description:

A concise introduction to the fundamental hydrodynamic PDEs -- the Euler equations (incompressible and compressible) and the Navier-Stokes equations (incompressible). We focus on the well-posedness theory (weak and strong solutions), long-time dynamics, the formation of singularities, and other topics. The results will be presented in the least possible generality, the goal being to provide a sampler of some of the main ideas in the field. 

Prerequisites: Mostly working knowledge of real, complex, functional, and harmonic analysis.

Description:

Frames and Riesz bases of exponentials, Paley-Wiener theorems, nonuniform sampling and interpolation in Paley-Wiener space, connections to other areas of mathematics and applications. 

Prerequisites: familiarity with Riemannian geometry and basic algebraic topology

Description:
This course is an introduction to subriemannian geometry and the geometry of nilpotent groups. Nilpotent groups are the simplest noncommutative groups. Their simplicity means that they appear in many areas of mathematics, especially geometry and geometric group theory. Their noncommutativity leads to distinctive and unusual geometry that make them a productive source of examples and a useful tool in a mathematician's toolbox.

In this course, we will study the geometry and analysis of nilpotent groups, possibly including topics such as:

• subriemannian metrics and manifolds
• lattices, large-scale geometry and asymptotic cones
• embeddings and metric geometry
• geodesics, surfaces, and geometric measure theory

Prerequisites: 

Multivariate calculus and calculus-based probability. Students should have completed Computing in Finance (MATH-GA-2401) or equivalent, have strong coding skills in Python, and working experience with the Python stack (numpy/pandas/scikit-learn).

Description: 

This half-semester course examines the building technologies and concepts in distributed ledger technologies and the workings of crypto financial markets.

We begin by an overview of the traditional central banking system and the mechanics of central bank money and commercial bank lending as the two dominant mechanisms of money creation. We explore the current network of banking in traditional finance (TradFi) and its hierarchy of commercial banks, central banks, correspondent banks, settlement and clearing mechanism, and the instruments used to create and transmit money.

We cover the principles of private and public key cryptography and its usage in encryption, digital signature, and message authentication. Hash functions serve as one-way functions that play a prominent role in creating message digests and solving the cryptographic puzzle in proof-of-work-based blockchains. We cover the main challenges of secure communication and typical attacks such as replay, man-in-the-middle, Sybil attacks and the cryptographic techniques used to tackle them.

Next, we take a deep-dive in the original Bitcoin whitepaper and show how the integration of cryptographic digital signatures, recursive blockchains, hash-based proof-of-work consensus mechanism to solve the 51% attack, and double-spend problem gave rise to the pioneering Bitcoin blockchain.

The Ethereum blockchain and its smart contracts have given rise to a variety of distributed apps (dApps), prominent among them decentralized exchanges (DEX) using constant function demand curves for creating automatic market-making. We cover the mechanics of these markets and concepts of swapping, liquidity pairs, yield farming and the general landscape of decentralized finance (DeFi).

Blockchain data is public by design and there is a wealth of real-time and historical data. We discuss some of the data analysis and machine learning methods utilized to analyze this type of data.

Given that blockchain is a software protocol, it is important that students taking this course have strong coding skills in Python and working experience with the Python stack (numpy/pandas/scikit-learn).

Description:

Elliptic curves, group law, algebraic varieties, Counting points over finite fields, Applications to cryptography, Mordell-Weil theorem over number fields.

Description:

Communication, both oral and written, is essential in academic careers and beyond. This course aims to help graduate students in mathematics develop skills to more effectively communicate their discipline and their research through writing and oral presentation. This half course (1.5 credits) will focus on academic writing and presentations, and will help students understand the “logic” of writing so as to construct clearer prose both at the sentence, paragraph, and article level. Throughout, the course will pay attention to how skills from both of these areas transfer to creating clearer, more engaging research presentations.

This seminar-style course will be highly interactive, with much of the learning occurring through feedback from other students. Students are expected to actively participate during the course time, and to complete several assignments including observing a class, teaching a short class, writing a research report and completing shorter writing exercises.

Description:
Basics of Functional Analysis. Rearrangement Inequalities. Basics of Fourier Analysis. Distributions. Sobolev Spaces. BV Functions. Interpolation. Maximal Function.

Suggested Texts:
Analysis, Lieb and Loss. Fourier Analysis, an introduction, Stein and Shakarchi. Functional Analysis, Sobolev spaces and PDE, Brezis.

Prerequisites:
Complex Variables I (or equivalent).

Description:
The fundamental theorem of algebra, the argument principle; calculus of residues, Fourier transform; the Gamma and Zeta functions, product expansions; Schwarz principle of reflection and Schwarz-Christoffel transformation; elliptic functions, Riemann surfaces; conformal mapping and univalent functions; maximum principle and Schwarz's lemma; the Riemann mapping theorem.

Text:
Ahlfors, L. (1979). International Series in Pure and Applied Mathematics [Series, Bk. 7]. Complex Analysis (3^rd ed.). New York, NY: McGraw-Hill.

Prerequisites:
Undergraduate background in analysis, linear algebra and complex variables.

Description:
Existence and uniqueness of initial value problems. Linear equations. Complex analytic equations. Boundary value problems and Sturm-Liouville theory. Comparison theorem for second order equations. Asymptotic behavior of nonlinear systems. Perturbation theory and Poincaré-Bendixson theorems.

Recommended Text:
Teschl, G. (2012). Graduate Studies in Mathematics [Series, Vol. 140]. Ordinary Differential Equations and Dynamical Systems. Providence, RI: AMS Chelsea Publishing/ American Mathematical Society.

Prerequisites: 
MATH-GA 2490 (Introduction to Partial Differential Equations) and MATH-GA 2430 (Real Variables), or equivalent background.  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:

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

Prerequisites:
Basic Probability (or equivalent masters-level probability course), Linear Algebra (graduate course), and (beginning graduate-level) knowledge of ODEs, PDEs, and analysis.

Description:
This is a graduate class that will introduce the major topics in stochastic analysis from an applied mathematics perspective.  Topics to be covered include Markov chains, stochastic processes, stochastic differential equations, numerical algorithms, and asymptotics. It will pay particular attention to the connection between stochastic processes and PDEs, as well as to physical principles and applications. The class will attempt to strike a balance between rigour and heuristic arguments: it will assume that students have some familiarity with measure theory and analysis and will make occasional reference to these, but many results will be derived through other arguments. The target audience is PhD students in applied mathematics, who need to become familiar with the tools or use them in their research.

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.

Description:
This course provides mathematical introductions to classical mechanics, elasticity, statistical mechanics and soft matter -- topics at the interface where ODEs, PDEs and stochastic processes meet physics, biology and engineering. For students preparing to do research on physical/biological applications, the class provides an introduction to crucial concepts and tools of mechanics; for students of analysis/PDE the class provides context by exploring some central applications. No prior exposure to mechanics or physics is assumed.

The segment on classical mechanics (about 5 weeks) will include: equations (and their integration) of motion for a system of particles, conservation laws, oscillations, motion of a rigid body, Hamilton-Jacobi equations, systems' control, stability and resonance.  The segment on elasticity (about 5 weeks) will include: basics of linear elasticity theory, equilibrium of rods and plates, elements of nonlinear elasticity, buckling and bifurcations.

The segment on statistical mechanics (about 2 weeks) will include: basic concepts such as Gibbs and Boltzmann distributions, energy and entropy, thermodynamic inequalities; phase equilibrium, numerical Monte-Carlo method.  The segment on soft matter (about 2 weeks) will include: theory of active polar gels, applications in cell biophysics.

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:
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:   
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 BlackScholes 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:  
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.

Description:

We will take a tour of modern theoretical physics, emphasizing the interplay between physical and mathematical ideas. Topics will include, on the physical side, electromagnetism, special relativity, quantum mechanics, general relativity and quantum field theory. On the mathematical side, elements of differential geometry, partial differential equations, dispersive waves, Lagrangian and Hamiltonian dynamics and calculus of variations

Prerequisites: Linear Algebra, Multi-dimensional analysis (e.g. gradients, Hessians), Probability Theory. Recommended but not required: experience training DNNs, high dimensional probability.

Description:

An overview of the many distinct training dynamics of Deep Neural Networks (DNNs) and their impact on the function learned by the network. The focus will mostly be on mathematical analysis of deep networks with a large number of neurons. Topics covered: the Neural Tangent Kernel and related dynamics; the spectral bias of DNNs; the transition between lazy and active regimes; the implicit bias of DNNs under the cross-entropy loss or with L2-regularization; Saddle-to-Saddle dynamics; appearance of sparsity for different DNN architectures. The grade will be based on a group project.

Prerequisite: DS-GA 1002 Probability and Statistics or similar

 

Description:

This course is a continuation of Probability and Statistics for Data Science. We will cover more advanced concepts from probability and statistics emphasizing their application to real-world data. Evaluation will be based on weekly homeworks and a project.

 
Syllabus: Correlation, the law of large numbers, the central limit theorem, confidence intervals, the bootstrap, hypothesis testing, the covariance matrix, principal-component analysis, low-rank models, regression and classification. 

This is the lab course associated with MATH-GA 2840.005. 

Prerequisite: DS-GA 1002 Probability and Statistics or similar

Description:

This course is a continuation of Probability and Statistics for Data Science. We will cover more advanced concepts from probability and statistics emphasizing their application to real-world data. Evaluation will be based on weekly homeworks and a project.

Syllabus: Correlation, the law of large numbers, the central limit theorem, confidence intervals, the bootstrap, hypothesis testing, the covariance matrix, principal-component analysis, low-rank models, regression and classification. 

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

 Description:
The one-semester course introduces the basic concepts and methods of probability.

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

Texts:

• Probability and Random Processes, 3rd ed., by Grimmett and Stirzaker

Prerequisites:
MATH-GA 2901 Basic Probability or equivalent.

Description:
Review of basic probability and useful tools. Bernoulli trials and random walk. Law of large numbers and central limit theorem. Conditional expectation and martingales. Brownian motion and its simplest properties. Diffusion in general: forward and backward Kolmogorov equations, stochastic differential equations and the Ito calculus. Feynman-Kac and Cameron-Martin Formulas. Applications as time permits.

Optional Problem Session:
Thursdays, 5:30-7:00.

Text:
Durrett, R. (1996). Probability and Stochastics Series [Series, Bk. 6]. Stochastic Calculus: A Practical Introduction. New York, NY: CRC Press.

Prerequisites:  Probability Limits Theorems 1

Description:

Stochastic processes in continuous time. Brownian motion. Poisson process. Processes with independent increments. Stationary processes. Semi-martingales. Markov processes and the associated semi-groups. Connections with PDEs. Stochastic differential equations. Convergence of processes.

Recommended Text:

• Stochastic Processes by Bass
• Stochastic Processes by Varadhan (Courant Lecture Series in Mathematics, volume 16)
• Theory of Probability and Random Processes by Koralov and Sinai
• Brownian Motion and Stochastic Calculus by Karatzas and Shreve
• D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, Springer, 1999.
• J.-F. Le Gall, Brownian Motion, Martingales, and Stochastic Calculus. Springer, 2016.

Prerequisite(s): Undergraduate-level proficiency in linear algebra and multivariable calculus; graduate-level proficiency in probability, at the level of MATH-GA 2901, MA-GY 6813, or ECE-GY 6303.

Description:

Topics to be covered in this course include various statistical models (sampling model, randomization methods), estimation and margins of error (MLE, confidence intervals, asymptotic theory, efficiency and sufficiency, robustness), likelihood theory (score functions and ratio tests), Bayes theory, decision theory (hypothesis testing, goodness of fit, shrinkage), and finally, an introduction to some common computational methods (bootstrap, Markov Chain-Monte Carlo).

Prerequisite:  Geophysical Fluid Dynamics MATH-GA.3001-001

Description: This is a course on modern dynamical oceanography, with a focus on mathematical models for observed phenomena. The lectures will cover the observed structure of the ocean, the thermodynamics of sea-water, the equations of motion for rotating-stratified flow, and the most useful approximations thereof: the primitive, planetary geostrophic and quasigeostrophic equations. Lectures will demonstrate how these approximations can be used to understand boundary layers, wind-driven circulation, buoyancy-driven circulation, oceanic waves (Rossby, Kelvin and Internal), potential vorticity dynamics, theories for the observed upper-ocean stratification (the thermocline), and for the global general circulation. Additional topics include baroclinic instability and mesoscale eddies; Ekman and boundary layer dynamics, including submesoscale dynamics; and the observed internal wave spectrum.

Assessment: Problem sets, class presentations, final project

Recommended Texts:

• Gill: Atmosphere-Ocean Dynamics, Academic Press
• Olbers, Willebrand, Eden: Ocean Dynamics, Springer
• Salmon: Lectures on Geophysical Fluid Dynamics, Oxford
• Samelson: The Theory of Large-Scale Ocean Circulation, Cambridge