MATH-GA.1002-001 Multivariable Analysis
3 Points, Mondays, 7:10-9:00PM, Gilles Francfort
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.
MATH-GA.1420-001 Introduction To Math Analysis II
3 Points, Wednesdays, 8:00-9:50PM, Samuel Boury
Measure theory and Lebesgue integration on the Euclidean space. Convergence theorems. L^p spaces and Hilbert spaces. Fourier series. Introduction to abstract measure theory and integration.
MATH-GA.1420-002 Introduction To Math Analysis II
3 Points, Thursdays, 8:00-9:50PM, Jumageldi CharyyevDescription TBA
MATH-GA.2012-002 Advanced Topics In Numerical Analysis: High Performance Computing
3 Points, Mondays, 5:10-7:00PM, Georg Stadler
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.
MATH-GA.2012-003 Advanced Topics In Numerical Analysis: Immersed Boundary Method For Fluid-Structure Interaction
3 Points, Mondays, 11:00-12:50PM, Charles Peskin
Familiarity with numerical methods and fluid dynamics.
The immersed boundary (IB) method is a general framework for the computer simulation of flows with immersed elastic boundaries and/or complicated geometry. It was originally developed to study the fluid dynamics of heart valves, and it has since been applied to a wide variety of problems in biofluid dynamics, such as wave propagation in the inner ear, blood clotting, fish swimming, and insect flight.
Non-biological applications include sails, parachutes, flows of suspensions, and two-fluid or multifluid problems.
Topics to be covered include: mathematical formulation of fluid-structure interaction in Eulerian and Lagrangian variables with the Dirac delta function as the link between these two kinds of variables; discretization of the structure, fluid, and interaction equations, including energy-based discretization of the structure equations, finite-difference discretization of the fluid equations, and IB delta functions with specified mathematical properties; an IB method with an exactly divergence-free interpolated velocity field; IB methods for immersed boundaries with non-trivial mass and for fluids with non-uniform density and viscosity; IB methods for immersed filaments with bend and twist; and a stochastic IB method for thermally fluctuating hydrodynamics within biological cells.
Course requirements include homework assignments and a computing project, but no exam. Students may collaborate on the homework and on the computing project, and are encouraged to present the results of their computing projects to the class.
The Immersed Boundary Method. Lecture notes freely available at: http://www.math.nyu.edu/faculty/peskin/ib_lecture_notes/index.html These notes will be supplemented by selected publications on the topics of the course.
MATH-GA.2012-004 Advanced Topics In Numerical Analysis: Stochastic Modeling And Uncertainty Quantification In Complex Systems
3 Points, Thursdays, 1:25-3:15PM, Benjamin Peherstorfer
As physical and time domains of partial differential equations (PDEs) can be resolved with ever higher accuracy, the quality of numerical simulations of physical phenomena increasingly tends to be limited by noisy and incomplete data that insufficiently describe boundary conditions, coefficients, and other parameters of the problem setups. This course provides an introduction to the stochastic modeling of these data uncertainties as random coefficients and random forcing terms of PDEs and discusses dimensionality reduction methods for efficiently estimating moments of the solutions of the corresponding stochastic PDEs. The first part focuses on stochastic modeling and sampling (Karhunen-Loeve expansion, polynomial chaos expansion, stochastic Galerkin, (multilevel) Monte Carlo, stochastic collocation, sparse grids). The second part discusses dimensionality reduction techniques to make estimation of moments of the PDE solutions computationally tractable (proper orthogonal decomposition, a posteriori error estimation and error control of reduced models, empirical interpolation).
MATH-GA.2020-001 Numerical Methods II
3 Points, Tuesdays, 5:10-7:00PM, Leslie Greengard
Numerical Methods I or equivalent graduate course in numerical analysis (numerical linear algebra, iterative solvers, nonlinear systems, interpolation, integration), undergraduate or graduate courses in ODE and (hyperbolic, parabolic, and elliptic) PDEs
This course will cover fundamental methods that are essential for the numerical solution of differential equations. It is intended for students familiar with ODE and PDE and interested in numerical computing; computer programming assignments in MATLAB will form an essential part of the course. The course will introduce students to numerical methods for (1) ordinary differential equations, explicit and implicit Runge-Kutta and multistep methods, convergence and stability; (2) finite difference and finite element and integral equation methods for elliptic partial differential equations (Poisson eq.); (4) spectral methods and the FFT, exponential temporal integrators, and multigrid iterative solvers; and (5) finite difference and finite volume parabolic (diffusion/heat eq.) and hyperbolic (advection and wave) partial differential equations.
"Finite Difference Methods for Ordinary and Partial Differential Equations" by Randy LeVeque. This textbook is now available freely to you in PDF format.
MATH-GA.2048-001 Scientific Computing In Finance
3 Points, Wednesdays, 5:10-7:00PM, Richard Lindsey and Mehdi Sonthonnax
Note: Students may not receive credit for both MATH-GA 2043.001 and MATH-GA 2048.001
Risk and Portfolio Management, Financial Securities and Markets, and Computing in Finance.
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
MATH-GA.2071-001 Machine Learning & Computational Statistics (1st Half Of Semester)
1.5 Points, Mondays, 7:10-9:00PM, Ivailo Dimov
Note: 1st Half of SemesterPrerequisites: 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).Course Description: This half-semester course (a natural sequel to the course “Data Science & Data-Driven Modeling”) examines techniques in machine learning and computational statistics in a unified way as they are used in the financial industry. We cover supervised learning (regression and classification using linear and nonlinear models), specifically examining splines and kernel smoothers, bagging and boosting approaches; and how to evaluate and compare the performance of these machine learning models. Cross-validation and bootstrapping are important techniques from the standard machine learning toolkit, but these need to be modified when used on many financial and alternative datasets. In addition, we discuss random forests and provide an introduction to neural networks. Hands-on homework form an integral part of the course, where we analyze real-world datasets and model them in Python using the machine learning techniques discussed in the lectures.
MATH-GA.2110-001 Linear Algebra I
3 Points, Tuesdays, 5:10-7:00PM, Michael Lindsey
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.
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.
MATH-GA.2120-001 Linear Algebra II
3 Points, Wednesdays, 5:10-7:00PM, Gaoyong Zhang
Prerequisites: Linear Algebra I or permission of the instructor.
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.
Friedberg, S.H., Insel, A.J., and Spence, L.E. (2003). Linear Algebra (4th ed.). Upper Saddle River, NJ: Prentice-Hall/ Pearson Education.
Plus: Extensive instructor’s class notes.
MATH-GA.2140-001 Algebra II
3 Points, Tuesdays, 7:10-9:00PM, Fedor Bogomolov
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.
Artin, M. (2010). Algebra (2nd ed.). Upper Saddle River, NJ: Prentice-Hall/ Pearson Education.
- 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.
MATH-GA.2210-001 Introduction To Number Theory I
3 Points, Tuesdays, 3:20-5:10PM, Yuri Tschinkel
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.
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.
MATH-GA.2320-001 Topology II
3 Points, Mondays, 7:10-9:00PM, Sylvain Cappell
Homology and cohomology from simplicial, singular, cellular, axiomatic and differential form viewpoints. Axiomatic characterizations and applications to geometrical problems of embedding and fixed points. Manifolds and Poincaré duality. Products and ring structures. Vector bundles, tangent bundles, DeRham cohomology and differential forms.
MATH-GA.2360-001 Differential Geometry II
3 Points, Wednesdays, 1:25-3:15PM, Fengbo Hang
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.
- John Milnor, Morse Theory (Princeton University Press, 1963).
- John M. Lee, Riemannian Manifolds: An Introduction to Curvature (Springer, 1997).
MATH-GA.2420-001 Advanced Topics In Geometry: Topics In Minimal Surfaces
1.5 Points, Guido DePhilippis
Note: 1st Half of Semester
I will present some classical and recent results on the theory of minimal surfaces, starting from the Douglas Rado solution of the Plateau problem and arriving to the Schoen-Simon-Yau curvature estimates.
MATH-GA.2420-002 Advanced Topics: Stochastic Multi-Armed Bandits (1st Half Of Semester)
1.5 Points, Tuesdays, 1:25-3:15PM, Jean-Christophe Mourrat
Note: 1st Half of Semester
Multi-armed bandits are basic examples of sequential decision problems. Their name is derived from "one-armed bandits", a colloquial term for slot machines. An agent can choose between a number of possible actions, or "arms", repeatedly over a number of rounds. Once a choice is made, the agent receives a "reward", which is random, with a fixed distribution for each given arm. These distributions are not known in advance. The agent thus needs to resolve a trade-off between the necessity to learn the behavior of each arm, and the desire to always choose the best possible arm. We will study classical methods that aim to maximize the cumulative rewards, as well as theoretical limits to what any method can achieve.
MATH-GA.2420-003 Advanced Topics: Markov Chain Analysis (1st Half Of Semester)
3 Points, Mondays, 11:00-12:50PM, Eyal LubetzkyDescription TBA
MATH-GA.2420-004 Advanced Topics In Applied Mathematics: Working Group In Modeling And Simulation
3 Points, Thursdays, 12:30-2:00PM, Aleksandar Donev and Miranda Holmes-Cerfon and Leif RistrophDescription TBA
MATH-GA.2420-005 Advanced Topics In Statistics: Statistical Aspects Of Optimal Transport (1st Half Of Semester)
1.5 Points, Mondays, 3:20-5:10PM, Jonathan Niles-Weed
Note: 1st Half of Semester
Optimal transport (OT) studies the geometry of probability measures defined on a metric space. Tools from OT have recently become popular in machine learning for their usefulness in graphics and computer vision. This course will focus on recent research in statistical aspects of OT—these include nonparametric density estimation based on OT, limit theorems and inference for OT distances, and statistical theory of regularization for OT.
Knowledge of probability and real analysis is assumed. Statistical background is helpful but not required. No prior exposure to optimal transport is necessary.
MATH-GA.2420-006 Advaned Topics: Introduction To The Theory Of Elliptic Curves (1st Half Of Semester)
1.5 Points, Mondays, 1:25-3:15PM, Alena Pirutka
Note: 1st Half of Semester
This course will be devoted to the study of elliptic curves over various fields: finite fields, fields of rational or complex numbers. For elliptic curves defined over finite fields we will also discuss applications to cryptography.
MATH-GA.2420-007 Advanced Topics In Geometry: Tbd (2nd Half Of Semeter)
3 Points, Mondays, 1:25-3:15PM, Jeff CheegerDescription TBA
MATH-GA.2420-008 Advanced Topics: Harmonic Analysis And Geometric Discrepancy (2nd Half Of Semester)
1.5 Points, Tuesdays, 9:00-10:50AM, Sinan Gunturk
Note: 2nd Half of Semester
Geometric discrepancy is about quantifying the quality of distribution of finite sets of points relative to a given, typically continuous distribution on some space with a geometric structure, e.g. a torus. The theory has wide-ranging applications from numerical analysis and approximation theory to probability theory, dynamical systems, number theory, and computer science. In this course, we will primarily focus on harmonic analysis methods in studying problems in geometric discrepancy. We will cover the classical theory (starting with Weyl's criterion, upper bounds of Koksma-Hlawka type, Erdos-Turan inequality and other uses of exponential sums) as well as some more modern tools such as Roth's orthogonal function method.
MATH-GA.2420-009 Advaned Topics: Mathematical Models Of Development, Learning And Plasticity: Cellular-Molecular Mathematical Models For Ltp (1st Half Of Semester)
3 Points, Mondays, 9:00-10:50AM, Dave McLaughlin
In neuronal systems, development and learning occur through some form of Hebb’s postulate – neurons that fire together, wire together. That is, modifications in synaptic strength and transmission efficiency are driven by correlations between pre and post synaptic neuronal firing activity. This leads to fascinating dynamical systems models in which the parameters of the model themselves depend on 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 occurs over two distinct time scales – “Early” (over a few hours) and “Late” (over years), termed E-LTP and L-LTP respectively. In this course, we will focus on E-LTP -- specifically upon cellularmolecular descriptions of the synaptic connections located at dendritic spines of the “receiving” neuron, together with cellular-molecular descriptions of the E-LTP plasticity of these synapses. (A description of L-LTP would involve additional cellular-molecular mechanisms beyond those required for E-LTP.)
We will study in some detail mathematical models in the neuroscience literature of electro-chemical descriptions of the components of E-LTP occurring at a dendritic spine, such as models of
- the ion channels at the synapses (Ca+2 channels, NMDAR receptors, AMPA receptors, Ca+2 pumps, Ca+2 buffers);
- the chemical diffusion reactions between ions and proteins within the spine and dendritic shaft (Ca+2 , CaM, CaMKII, Ca+2 channels, NMDAR receptors, AMPA receptors);
- and the phosphorylation of these molecules.
These are the ingredients that will be required to develop a mathematical model of the cellular molecular mechanisms underlying E-LTP of Hebbian Learning -- such as the learning of “place fields” by hippocampal CA1 pyramidal neurons.
MATH-GA.2440-001 Real Variables II
3 Points, Tuesdays, 3:20-5:10PM, Sylvia Serfaty
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.
MATH-GA.2460-001 Complex Variables II
3 Points, Mondays, 5:10-7:00PM, Percy Deift
Prerequisites: Complex Variables I (or equivalent).
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.
Ahlfors, L. (1979). International Series in Pure and Applied Mathematics [Series, Bk. 7]. Complex Analysis (3rd ed.). New York, NY: McGraw-Hill.
MATH-GA.2470-001 Ordinary Differential Equations
3 Points, Wednesdays, 11:00-12:50PM, Scott Armstrong
Undergraduate background in analysis, linear algebra and complex variables.
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.
Teschl, G. (2012). Graduate Studies in Mathematics [Series, Vol. 140]. Ordinary Differential Equations and Dynamical Systems. Providence, RI: AMS Chelsea Publishing/ American Mathematical Society.
MATH-GA.2500-001 Partial Differential Equations
3 Points, Wednesdays, 3:20-5:10PM, Fanghua Lin
Note: Masters students should consult the course instructor before registering for this class
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.
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.
1. L.C. Evans, Partial Differential Equations, American Mathematical Society
2. M. Renardy and R. Rogers, An Introduction to Partial Differential Equations, Springer-Verlag
MATH-GA.2510-001 Advanced Partial Differential Equations
3 Points, Thursdays, 9:00-10:50AM, Sylvia Serfaty
Calculus of variations. Elliptic regularity theory: Cacciopoli inequality, Schauder estimates, De Giorgi-Nash theory. Basics of homogenization. Viscosity solutions. Introduction to dispersive equations.
MATH-GA.2550-001 Functional Analysis
3 Points, Mondays, 9:00-10:50AM, Guido DePhilippisDescription TBA
MATH-GA.2620-001 Advanced Topics In PDEs: Formation Of Singularities For Compressible Euler - Shocks & Implosion
3 Points, Thursdays, 1:45-3:35PM, Jalal Shatah and Vlad Vicol
PART I (First seven weeks). Title: Wave Turbulence. Lectures by professor Shatah.
Wave Turbulence is the study of statistical properties of random weakly nonlinear dispersive waves. The Wave Kinetic Equation (WKE) describes the evolution of the wave spectrum. In this class we will focus on the rigorous derivation of the WKE for the archetype dispersive models, the nonlinear Schrödinger equation. The lectures will cover the following,
- Resonances in dispersive equations.
- Continuum limit of discrete resonances.
- Introduction to counting resonate frequencies.
- Existence of solutions via tree expansion.
- Dispersive equations with Random data.
- The emergence of WKE.
PART II (Second seven weeks). Title: Formation Of Singularities For Compressible Euler Shocks & Implosion. Lectures by professor Vicol.
We discuss two recent results concerning the dynamic formation of singularities for the compressible Euler system. The proofs are constructive and give a precise description of the solution at the time of the first singularity.
The formation of shocks, from an open set of smooth initial data of finite energy and no vacuum, is discussed first. This follows works by Buckmaster-Shkoller-Vicol. Second, we discuss a different set of initial data, for which the finite time implosion of the fluid may be established. This follows works by Merle-Raphael-Rodnianski-Szeftel.
- Blowup for Burgers and very closely related problems: virial proofs vs self-similar analysis.
- Singularity formation in 2D compressible Euler with azimuthal symmetry: the role of modulated self-similar analysis.
- Shock formation in 3D Euler: the role of Lagrangian coordinates in self-similar analysis.
- Implosion and explosion: the Guderley problem at the formal level.
- Implosion in 3D Euler: the search for globally self-similar solutions via ODE methods
MATH-GA.2660-001 Advanced Topics In Analysis: Riemann-Hilbert Theory
3 Points, Percy Deift
The course will cover the basics of Riemann-Hilbert Theory and how it is sed to analyze the asymptotic behavior of a variety of physical and mathematical systems, such as the KdV equation, random matrix theory and as swell as some combinatorial problems, such as Ulam's longest increasing subsequence problem.
Complex variables, functional analysis.
MATH-GA.2704-001 Applied Stochastic Analysis
3 Points, Wednesdays, 1:25-3:15PM, Jonathan Weare
Basic Probability (or equivalent masters-level probability course), Linear Algebra (graduate course), and (beginning graduate-level) knowledge of ODEs, PDEs, and analysis.
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.
- Stochastic Processes and Applications, by G. A. Pavliotis.
- C. Gardiner, Stochastic Methods: A Handbook for the Natural and Social Sciences
MATH-GA.2708-001 Algorithmic Trading & Quantitative Strategies
3 Points, Tuesdays, 7:10-9:00PM, Petter Kolm and Lee Maclin
Computing in Finance, and Risk and Portfolio Management, or equivalent.
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.
3 Points, Tuesdays, 1:25-3:15PM, Eric Vanden Eijnden
This course provides brief mathematical introductions to elasticity, classical mechanics, and statistical mechanics -- topics at the interface where differential equations and probability meet physics and materials science. For students preparing to do research on physical applications, the class provides an introduction to crucial concepts and tools; for students of analysis the class provides valuable context by exploring some central applications. No prior exposure to mechanics or physics is assumed.
The segment on elasticity (about 6 weeks) will include: one-dimensional models (strings and rods); buckling as a bifurcation; nonlinear elasticity for 3D solids; and linear elasticity. The segment on classical mechanics (about 5 weeks) will include: basic examples; alternative formulations including action minimization and Hamilton's equations; relations to the Calculus of Variations including Hamilton-Jacobi equations, optimal control, and geodesics; stability and parametric resonance. The segment on statistical mechanics (about 3 weeks) will include basic concepts such as the microcanonical and canonical ensembles, entropy, and the equilibrium distribution; some simple examples; and the numerical method known as Metropolis sampling.
MATH-GA.2751-001 Risk & Portfolio Management
3 Points, Wednesdays, 7:10-9:00PM, Aaron Brown
Multivariate calculus, linear algebra, and calculus-based probability.
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.
MATH-GA.2752-001 Active Portfolio Management
3 Points, Mondays, 5:10-7:00PM, Jerome Benveniste
Risk & Portfolio Management and Computing in Finance.
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.
MATH-GA.2753-001 Advanced Risk Management
3 Points, Wednesdays, 7:10-9:00PM, Ken Abbott and Irena Khrebtova
Financial Securities and Markets, and Computing in Finance or equivalent programming experience.
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.
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.
MATH-GA.2755-001 Project & Presentation
3 Points, Wednesdays, 5:10-7:00PM, Petter Kolm
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.
MATH-GA.2791-001 Financial Securities And Markets
3 Points, Mondays, 7:10-9:00PM, Alireza JavaheriPrerequisites: Multivariate calculus, linear algebra, and calculus-based probability.Course 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.
MATH-GA.2793-001 Dynamic Asset Pricing (2nd Half Of Semester)
3 Points, Mondays, 7:10-9:00PM, Bruno Dupire and Montacer Essid
Prerequisites: Calculus-based probability, Stochastic Calculus, and a one semester course on derivative pricing (such as what is covered in Financial Securities and Markets).Course 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.
MATH-GA.2798-001 Interest Rate & Fx Models
3 Points, Thursdays, 5:10-7:00PM, Fabio Mercurio and Travis Fisher
Financial Securities and Markets, Stochastic Calculus, and Computing in Finance (or equivalent familiarity with financial models, stochastic methods, and computing skills).
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.
MATH-GA.2799-001 Modeling And Risk Management Of Bonds And Securitized Products (2nd Half Of Semester)
1.5 Points, Thursdays, 7:10-9:00PM, Rodney Sunada-Wong
Note: 2nd Half of Semester
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.
MATH-GA.2800-001 Trading Energy Derivatives (1st Half Of Semester)
1.5 Points, Thursdays, 7:10-9:00PM, Ilia Bouchouev
Note: 1st Half of Semester
Prerequisites: Financial Securities and Markets, and Stochastic Calculus.
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.
MATH-GA.2801-001 Advanced Topics In Equity Derivatives (2nd Half Of Semester)
1.5 Points, Wednesdays, 7:10-9:00PM, Sebastien Bossu
Note: 2nd Half of Semester
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.
MATH-GA.2802-001 Market Microstructure (1st Half Of Semester)
1.5 Points, Wednesdays, 7:10-9:00PM, Gordon Ritter
Note: 1st Half of Semester
Financial Securities and Markets, Risk and Portfolio Management, and Computing in Finance or equivalent programming experience.
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.
MATH-GA.2840-003 Advanced Topics In Applied Math: Stochastic Models In Modern Finance And Economics
3 Points, Tuesdays, 11:00-12:50PM, Marco Avellaneda
We cover selected models in finance and economics and related mathematical techniques, including:
- Correlation modeling in multi-asset markets, Random Matrix Theory, PCA, Hierarchical PCA, Statistical Clustering Applications to Modern Portfolio Theory
- Trade impact theory: Almgren-Chriss model for liquidating portfolios and macro-hedging. Stochastic control theory, linear and non-linear regulator, Hamilton-Jacobi equations and their applications to optimal liquidations.
- Option pinning at strikes and market impact due to aggregate Delta-hedging.
- Term-structure models for commodity and exotic index futures, such as the Volatility Index (VIX).
- Data-driven models for credit scoring of Wall Street firms. Forecasting the size of the hedge- fund industry, etc.
The course will be based on reading selected research papers and projects involving implementing the related models.
MATH-GA.2840-004 Advanced Topics In Applied Math: Modeling, Simulation, And Experiment In Fluid Dynamics
3 Points, Tuesdays, 3:20-5:10PM, Leif Ristroph
This course will explore how applied mathematics can productively interact with the experimental sciences and with real-world data. The course will involve projects in fluid dynamics, each of which has an experimental system in the Applied Math Lab. Students will work in small groups to gather experimental data, with an emphasis on discovery and characterization of phenomena, and they will formulate mathematical or computational models to account for these observations and make testable predictions. Assignments will include journal-style papers and conference-style talks. The projects will be drawn from modern fluid dynamics research and will explore fascinating questions relevant to biological and geophysical fluid dynamics.
MATH-GA.2840-005 Advanced Topics In Applied Math: Mathematical Tools For Data Science
3 Points, Thursdays, 3:30-5:10PM, Carlos Fernandez-GrandaDescription TBA
MATH-GA.2840-006 Advanced Topics In Applied Math: Mathematical Tools For Data Science Lab
3 Points, Mondays, 3:30-4:20PM, TBADescription TBA
MATH-GA.2840-007 Advanced Topics In Applied Math: Mathematical Tools For Data Science Lab
3 Points, Mondays, 8:00-8:50AM, TBADescription TBA
MATH-GA.2901-001 Essentials Of Probability
3 Points, Wednesdays, 7:10-9:00PM, Charles Newman
Prerequisites: Calculus through partial derivatives and multiple integrals; no previous knowledge of probability is required
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.
- Probability and Random Processes, 3rd ed., by Grimmett and Stirzaker
MATH-GA.2902-002 Stochastic Calculus Optional Problem Session
3 Points, Thursdays, 5:30-7:00PM, TBADescription TBA
MATH-GA.2903-001 Stochatic Calculus (2nd Half Of Semester)
1.5 Points, Tuesdays, 7:10-9:00PM, Alexey Kuptsov
Note: 2nd Half of Semester
Prerequisites: MATH-GA 2901 Basic Probability or equivalent.
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:
Durrett, R. (1996). Probability and Stochastics Series [Series, Bk. 6]. Stochastic Calculus: A Practical Introduction. New York, NY: CRC Press.
MATH-GA.2912-001 Probability Theory II
3 Points, Wednesdays, 9:00-10:50AM, Gerard Ben Arous
Prerequisites: Probability Limits Theorems 1
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.
- 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
MATH-GA.2931-001 Advanced Topics In Probability: Multiplicative Chaos, From Gff To Polymers (Marc 25 To May 10)
3 Points, Tuesdays, Thursdays, 11:00-12:50PM, Ofer Zeitouni
Note: 2nd Half of Semester
We will discuss the construction and approximations of Gaussian (and non-Gaussian) multiplicative chaos, and their role in the analysis of logarithmicaly correlated Gaussian fields, random matrices, and polymers.
Graduate course in probability
There will be no textbook. I will rely on articles from the current literature, and probably will also distribute notes.
MATH-GA.2962-001 Mathematical Statistics
3 Points, Wednesdays, 11:00-1:30PM, Michael O'neilDescription TBA
MATH-GA.3003-001 Ocean Dynamics
3 Points, Tuesdays, 3:20-5:10PM, Laure ZannaDescription TBA
MATH-GA.3010-001 Advanced Topics In AOS: Climate Change
3 Points, Thursdays, 9:00-10:50AM, Edwin Gerber
After starting off with the warmest January in recorded history, 2020 is on pace to be the warmest year ever observed. The last 6 years (2014-2019) are the 6 warmest years in recorded history. It is likely the hottest the Earth has ever been since the last interglacial period 125,000 years ago.
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.
A background in atmosphere or ocean science is plus, but the course will be structured so that anyone with a reasonable background in mathematics and physics can participate.
MATH-GA.3011-001 Advanced Topics In AOS: Climte Dynamics
3 Points, Shafer SmithDescription TBA