Course Descriptions

Prerequisites:

MA-UY 2122 or permission of adviser.

Description:

This course and its sequel MA-GY 6223 rigorously treat the basic concepts and results in real analysis. Course topics include limits of sequences, topological concepts of sets for real numbers, properties of continuous functions and differentiable functions. Important concepts and theorems include supremum and infimum, Bolzano-Weierstrass theorem, Cauchy sequences, open sets, closed sets, compact sets, topological characterization of continuity, intermediate value theorem, uniform continuity, mean value theorems and inverse function theorem.

Prerequisites:

MA-UY 2034  and MA-UY 2114 or Graduate Standing.

Description:

This course covers basic ideas of linear algebra: Groups, rings, fields, vector spaces, basis, dependence, independence, dimension. Relation to solving systems of linear equations and matrices. Homomorphisms, duality, inner products, adjoints and similarity.

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

Recommended Text:

• Real Analysis: A First Course, 2nd Edition, Russell Gordon. https://www.amazon.com/Real-Analysis-First-Course-2nd/dp/0201437279

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

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

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

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

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

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

If you want to brush up your MATLAB:

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

Cross-listing: CSCI-GA 2420.001.

This six-week course will be structured in an unusual way. Each of our six meetings will be independent. At each meeting, the first hour will be a lecture aimed at anyone interested in numerical analysis at a high level, organized around a well-known topic and mixing historical perspectives, recent developments, and always some new mathematics. The second hour will be for enrolled students only, a hands-on work session making use of Chebfun.

• Lecture 1. Gaussian elimination
• Lecture 2. Random functions and random differential equations
• Lecture 3. Chebyshev series
• Lecture 4. Rational functions
• Lecture 5. Quadrature
• Lecture 6. ODEs

Course homepage

This course will cover many of the standard PDEs from classical mathematical physics, including the Laplace equation, the Helmholtz equation, Maxwell’s equations, Stokes’ equation, and their alternative integral equation formulation. Numerically solving these associated integral equations requires a special set of tools, in particular, quadrature for singular functions and fast algorithms for the resulting dense matrices. Numerical methods for each of these aspects will be presented (e.g. singular quadrature, fast multipole methods, iterative solvers, and fast direct solvers), and open problems in the field will be discussed.

Course Website

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

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

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

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

Recommended Texts: 

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

Cross-listing: CSCI-GA 2112.001.

Prerequisites: Scientific Computing or Numerical Methods II, Continuous Time Finance, or permission of instructor.

The classical curriculum of mathematical finance programs generally covers the link between linear parabolic partial differential equations (PDEs) and stochastic differential equations (SDEs), resulting from Feynmam-Kac's formula. However, the challenges faced by today's practitioners mostly involve nonlinear PDEs. The aim of this course is to provide the students with the mathematical tools and computational methods required to tackle these issues, and illustrate the methods with practical case studies such as American option pricing, uncertain volatility, uncertain mortality, different rates for borrowing and lending, calibration of models to market smiles, credit valuation adjustment (CVA), portfolio optimization, transaction costs, illiquid markets, super-replication under delta and gamma constraints, etc. 



We will strive to make this course reasonably comprehensive, and to find the right balance between ideas, mathematical theory, and numerical implementations. We will spend some time on the theory: optimal stopping, stochastic control, backward stochastic differential equations (BSDEs), McKean SDEs, branching diffusions. But the main focus will deliberately be on ideas and numerical examples, which we believe help a lot in understanding the tools and building intuition.



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

Prerequisites: Derivative Securities, Risk & Portfolio Management with Econometrics, and Computing in Finance (or equivalent programming experience).

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

Prereqisites:  Risk & Portfolio Management with Econometrics, Scientific Computing in Finance (or Scientific Computing) and Computing in Finance (or equivalent programming experience.

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

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

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

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

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

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

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

Prerequisites: Undergraduate linear algebra or permission of the instructor.

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

Recommended Text: Lipschutz, S., & Lipson, M. (2012). Schaum’s Outlines [Series]. Schaum’s Outline of Linear Algebra (5th ed.). New York, NY: McGraw-Hill.

Note: Extensive lecture notes keyed to these texts will be issued by the instructor.

Prerequisites: Undergraduate linear algebra.

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

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

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

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

Basic concepts of groups, rings and fields. Symmetry groups, linear groups, Sylow theorems; quotient rings, polynomial rings, ideals, unique factorization, Nullstellensatz; field extensions, finite fields.

Recommended Texts:

• Artin, M. (2010). Featured Titles for Abstract Algebra [Series]. Algebra (2nd ed.). Upper Saddle River, NJ: Pearson
• Chambert-Loir, A. (2004). Undergraduate Texts in Mathematics [Series]. A Field Guide to Algebra (2005 ed.). New York, NY: Springer-Verlag
• Serre, J-P. (1996). Graduate Texts in Mathematics [Series, Vol. 7]. A Course in Arithmetic (Corr. 3rd printing 1996 ed.). New York, NY: Springer-Verlag.

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

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

Recommended Texts:

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

This will be an introduction to topics in the study, construction and classification of manifolds and submanifolds. These will include smooth structures, classification theories, and new applications to the construction of manifolds with prescribed L² Betti numbers.

Prerequisites: Multivariable calculus and linear algebra.

Differentiable manifolds, tangent bundle, embedding theorems, vector fields and differential forms. Introduction to Riemannian metrics, connections and geodesics.

Text: Lee, J.M. (2009).Graduate Studies in Mathematics [Series, Vol. 107]. Manifolds and Differential Geometry. Providence, RI: American Mathematical Society.

 

I start with definition and basic properties of the scalar curvature and present a list of relevant examples. Then I will explain basic constructions: gluing and surgery. With this I will prove some classification results for simply connected manifolds with positive scalar curvature.

Then I will explain the Schoen-Yau method combined with symmetrisation and prove basic inequalities for manifolds with boundaries.

After that I will expose the basics on spinors and on the index theory for the Dirac operators followed by application to scalar curvature.

I plan to discuss problems of birational algebraic geometry focusing on the related questions in group theory.

The main topics will be:

1. basic algebraic geometry
2. basic birational geometry
3. Invariant theory
4. Galois groups of functional fields
5. Projective geometry related to the structure of galois groups above

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

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

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

Text:   

• Main Text: Folland's "Real Analysis: Modern Techniques and Their Applications''
• Secondary Text: Bass' "Real Analysis for Graduate Students''

Prerequisites: Advanced calculus (or equivalent).

Complex numbers; analytic functions, Cauchy-Riemann equations; linear fractional transformations; construction and geometry of the elementary functions; Green's theorem, Cauchy's theorem; Jordan curve theorem, Cauchy's formula; Taylor's theorem, Laurent expansion; analytic continuation; isolated singularities, Liouville's theorem; Abel's convergence theorem and the Poisson integral formula.

Text: Brown, J., & Churchill, R. (2008). Complex Variables and Applications (8th ed.). New York, NY: McGraw-Hill.

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

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

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

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

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

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

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

See the syllabus at http://math.nyu.edu/faculty/kohn/teaching.html for more information (including a tentative semester plan).

Recommended Texts:

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

Elliptic regularity: Harmonic functions and Harnack Inequality: Liouville's theorem, removable singularity, Harnack convergence theorems. Cacciopolli inequality and some of its consequences. De Giorgi-Nash Theory. Hyperbolic equations: Local existence and regularity of nonlinear problems, conserved quantities, vector fields method. Further topics, such as variational methods, homogenization, dispersive PDEs.

Prerequisites: Linear algebra, real variables (including measure theory), and basic complex analysis.

Topics: Banach spaces. Functionals and operators. Principle of uniform boundedness, open mapping and closed graph theorems. Duality and weak topologies. Alaoglu's theorem. Invariant subspaces. Spectral theorem for compact operators on Banach spaces and self-adjoint operators on Hilbert spaces. Hilbert-Schmidt operators. Semigroups. Fixed-point theorem. Applications to differential equations and harmonic analysis.

The course will concentrate on both general theory and concrete examples. Working knowledge of measure and integral is expected.

• Lax, P.D. (2002). Functional Analysis, Wiley

• Reed, M., & Simon, B. (1972). Functional Analysis, Academic Press

• Conway, J.B. (1985). A Course in Functional Analysis, Springer

The course will cover both the basic theory of Riemann-Hilbert problems and also present some of the many applications of the theory to the analysis of integrable PDEs such as KdV and NLS, to random matrix theory, to combinatorial problems such as Ulam's increasing subsequence problem, and also to orthogonal polynomials.

The prerequisites for the course are a graduate course in complex analysis and a graduate course in functional analysis.

Prerequisites: Basic Probability; Applied Stochastic Analysis.

One of the greatest scientific achievements of the 19th century was the development by Boltzmann and Gibbs  of the theory of statistical mechanics that relates the microscopic properties of a system to the macroscopic ones using a probabilistic framework. In its original formulation statistical mechanics applies to thermal systems whose dynamics satisfy detailed-balance (microscopic reversibility) — this is appropriate if one wants to understand the collective behavior of the molecules in a gas or a liquid and relate them to standard thermodynamic quantities, such as temperature or pressure. However, many systems of interest are intrinsically out-of-equilibrium, e.g. living systems that consume energy from their environment or exert work on it, or systems that are being driven by external forces, like the atmosphere or the ocean. How would one generalize the framework of statistical mechanics to such situations remains an open question. To this end, in recent years a collection of new theoretical tools has emerged, sometimes referred to collectively as macroscopic fluctuation theory, with results such as  the Jarzinsky equality or Crooks fluctuation theorem.

In a nutshell, these results are based on a reformulation of the problem that includes both space and time in the statistical description, and study the fluctuations not only in the system's variables but also in its current. This typically requires one to sample trajectories solution of a stochastic differential equation or a Markov jump process conditioned on them doing something specific, which, from a mathematical viewpoint, amounts to sampling probability distributions defined on some Hilbert space. These problems require one to design specific Monte Carlo sampling methods whose efficiency do not deteriorate with the dimensionality of the system. They also typically require one to use importance sampling strategies to lower the variance of the estimators. The aim of this course is to give an introduction to the type of methods that can be used in this context, and illustrate them in applications from non-equlibrum statistical mechanics. The class will use examples from material science (dynamic phase transition), active matter (self-assembly), fluid dynamics (regime changes, rogue waves), chemical-physics (metastability), and biology (motility-induced phase transition).

Topics: Transition Paths Sampling; Umbrella Sampling and Stratification in Trajectory Space; Importance Sampling by Reweighting via Girsanov Transform; Diffusion Monte Carlo; Optimal Control.

Prerequisites: Elementary linear algebra and differential equations.

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

Recommended Texts:

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

Prerequisites: Introductory complex variable and partial differential equations.

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

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

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

Prerequisites: Derivative Securities, Scientific Computing, Computing for Finance, and Stochastic Calculus.

The term "statistical arbitrage" covers any trading strategy that uses statistical tools and time series analysis to identify approximate arbitrage opportunities while evaluating the risks inherent in the trades (considering the transaction costs and other practical aspects). This course starts with a review of Time Series models and addresses econometric aspects of financial markets such as volatility and correlation models. We will review several stochastic volatility models and their estimation and calibration techniques as well as their applications in volatility based trading strategies. We will then focus on statistical arbitrage trading strategies based on co-integration, and review pairs trading strategies. We will present several key concepts of market microstructure, including models of market impact, which will be discussed in the context of developing strategies for optimal execution. We will also present practical constraints in trading strategies and further practical issues in simulation techniques. Finally, we will review several algorithmic trading strategies frequently used by practitioners.

Prerequisites: Univariate statistics, multivariate calculus, linear algebra, and basic computing (e.g. familiarity with MATLAB or co-registration in Computing in Finance).

A comprehensive introduction to the theory and practice of portfolio management, the central component of which is risk management. Econometric techniques are surveyed and applied to these disciplines. Topics covered include: factor and principal-component models, CAPM, dynamic asset pricing models, Black-Litterman, forecasting techniques and pitfalls, volatility modeling, regime-switching models, and many facets of risk management, both theory and practice.

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: Risk Management, Derivative Securities (or equivalent familiarity with market and credit risk models).

The course is divided into two parts. The first addresses the institutional structure surrounding capital markets regulation. It will cover Basel (1, MRA, 2, 2.5, 3), Dodd-Frank, CCAR and model review. The second part covers the actual models used for the calculation of regulatory capital. These models include the Gaussian copula used for market risk, specific risk models, the Incremental Risk Calculation (single factor Vasicek), the Internal Models Method for credit, and the Comprehensive Risk Measure.

An introduction to arbitrage-based pricing of derivative securities. Topics include: arbitrage; risk-neutral valuation; the log-normal hypothesis; binomial trees; the Black-Scholes formula and applications; the Black-Scholes partial differential equation; American options; one-factor interest rate models; swaps, caps, floors, swaptions, and other interest-based derivatives; credit risk and credit derivatives.

Prerequisites: Derivative Securities and Stochastic Calculus, or equivalent.

A second course in arbitrage-based pricing of derivative securities. The Black-Scholes model and its generalizations: equivalent martingale measures; the martingale representation theorem; the market price of risk; applications including change of numeraire and the analysis of quantos. Interest rate models: the Heath-Jarrow-Morton approach and its relation to short-rate models; applications including mortgage- backed securities. The volatility smile/skew and approaches to accounting for it: underlyings with jumps, local volatility models, and stochastic volatility models.

Prerequisites:  Computing in Finance (or equivalent programming skills) and Derivative Securities (familiarity with Black-Scholes interest rate models)

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

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

Prerequisites:  Derivate Securities and Computing in Finance (or equivalent familiarity with financial models and computing skills)

This half-semester course introduces the institutional market for bonds and loans subject to default risk and develops concepts and quantitative frameworks useful for modeling the valuation and risk management of such fixed income instruments and their associated derivatives. Emphasis will be put on theoretical arbitrage restrictions on the relative value between related instruments and practical applications in hedging, especially with credit derivatives. Some attention will be paid to market convention and related terminology, both to ensure proper interpretation of market data and to prepare students for careers in the field.

We will draw on the fundamental theory of derivatives valuation in complete markets and the probabilistic representation of the associated valuation operator. As required, this will be extended to incomplete markets in the context of doubly stochastic jump-diffusion processes. Specific models will be introduced, both as examples of the underlying theory and as tools that can be (and are) used to make trading and portfolio management decisions in real world markets.

Prerequisites: undergraduate courses in ODEs and PDEs.

The course should be interesting for undergraduate students, graduate students, and postdocs in pure and applied mathematics, physics, engineering, and climate, atmosphere, ocean science interested in modeling and simulating climate systems and other turbulent dynamical systems.

The El Niño-Southern Oscillation (ENSO) is the strongest signal on yearly time scale beyond the seasonal cycle. It has significant impact on global climate and relevance for seasonal forecasts. Roughly speaking, ENSO is a periodic phenomenon that people are familiar with. However, surprisingly the ENSO in the past few decades is very different from the traditional picture with large impacts on global warming and weather and climate around the world. This course will focus on a new perspective on ENSO modeling, which is consistent with the recent observational record. Rigorous theories, simple nonlinear dynamics, simple stochastic models and numerical methods will be combined with physical reasoning to produce new simple nonlinear dynamics and the comparison of the model results with real observations will be emphasized. We will show how this can be done in a simple modeling framework that is consistent with complex observations of ENSO over the last a few decades. This course is a self-contained introduction to these topics and elementary material from nonlinear dynamics, stochastic modeling and numerical algorithms will be developed in a self-contained way. Students and young researchers can certainly follow the lectures with elementary background.

The lectures will cover the basic physics of ENSO, real observational records and mathematical tools with interesting new nonlinear dynamical stochastic models capturing the observational phenomena. Open problems and future work will also be discussed.

There will be no final exam and homework. The enrolled students will participate actively in some lectures being mentored by Professor Majda and his two postdocs Nan Chen and Sulian Thual.

Information theory is a branch of applied probability theory which has seen application in fields as diverse as computer science, mathematical statistics, dynamical systems, financial mathematics, large deviation theory and complexity theory. In this seminar course we shall provide a comprehensive introduction to the theory and explain the many applications.  In the second part of the course we shall apply the theory to the study of predictability in realistic dynamical systems.  The relationship between predictability and statistical disequilibrium will also be studied using tools from information theory developed in the first part of the course. A full set of lecture notes will be available.

Prerequisite:  Working knowledge of differential equations and probability as used in applications.

This course is about the role of feedback in physiological systems. The course begins with an introduction to feedback, with emphasis on the subtle role of dynamics in determining whether a feedback system is stable or unstable.  The rest of the course will focus on specific biological examples, drawn both from the whole-organism level and also from the cellular level.  The whole-organism topics are control of the circulation (blood pressure, cardiac output and its distribution), the control of body salt and water, and the control of ovulation number (which is a whole-organism mechanism, since it involves circulating hormones, and of particular mathematical interest since it involves symmetry breaking and the control of an integer-valued quantity). Cell-level topics are the control of cell volume (which has as a by-product the electrical membrane potential that allows neurons to function), the molecular circadian clock (a deliberately unstable biochemical feedback system that involves gene expression), telomere length regulation, and the control of copy number of bacterial plasmids.  Particularly at the cellular level, stochastic models and their relationship to deterministic models will be emphasized.  There will be homework assignments, some of which will involve computing, and a final computing project, but no exam. Students are encouraged to work in teams on homework and also on the final project.

Text (recommended for background reading): Guyton and Hall, Textbook of Medical Physiology, 13th Edition, Elsevier 2015.

Prerequisite: familiarity with applied differential equations; most neurobiological background will be provided.

Lecture course on the formulation and analysis of differential equation models for neuronal ensembles and neuronal computations. Topics include neuronal rhythms, motor pattern generators, perceptual dynamics, decision-making, etc.

This course will involve the formulation and analysis of differential equation models for neuronal ensembles and neuronal computations.  Spiking and firing rate mechanistic treatments of network dynamics as well as probabilistic behavioral descriptions will be covered.  We will consider mechanisms of coupling, synaptic dynamics, rhythmogenesis, synchronization, bistability, adaptation,…  Applications will likely include:  central pattern generators and frequency control, perceptual bistability, working memory, decision-making and neuro-economics, feature detection in sensory systems, cortical dynamics (gamma and other oscillations, up-down states, balanced states,…).  Students will undertake computing projects related to the course material: some in homework format and a term project with report and oral presentation.

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

The course introduces the basic concepts and methods of probability.

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

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

The link to the Springer page for this book is: http://link.springer.com/book/10.1007/978-3-642-55682-1

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

First semester in an annual sequence of Probability Theory, aimed primarily for Ph.D. students. Topics include laws of large numbers, weak convergence, central limit theorems, conditional expectation, martingales and Markov chains.

Recommended Text: S.R.S. Varadhan, Probability Theory (2001)

Prerequisites: Graduate level (measure theoretic) probability, but no prior knowledge of percolation or statistical mechanics will be assumed.

The first half of the course will introduce Ising models, independent (Bernoulli) percolation, Fortuin-Kasteleyn (FK) dependent percolation (or random cluster) models, and how they're related to each other. We'll discuss correlation inequalities, infinite volume limits and phase transitions.

The second half will be more seminar-like and will touch on a number of topics related to critical and near-critical scaling limits for the classical Ising model on the two-dimensional plane. These may include conformal loop and measure ensembles, novel couplings between FK and Ising variables and how to prove exponential decay of near-critical correlations. The aim is to explain some of the techniques used in recent papers of Camia-Garban-Newman and Camia-Jiang-Newman.

Prerequisites: basic knowledge of probability and Markov chains.

Isomorphism theorems link properties of local time of (symmetric) Markov processes to certain Gaussian fields. A particular case is the generalized Ray-Knight theorem, that has found applications in the study of cover times and of interlacements. The course will review these results and recent developments.

The applications of probability to many areas in mathematics and in other fields have multiplied dramatically in recent years. Rich interactions with classical analysis have been found in the study of random fractals; researchers in statistics, combinatorics, theoretical computer science, and high-dimensional geometry (to name a few) increasingly need sophisticated probabilistic tools. The aim of this course is to provide such tools, focusing on martingales and their applications. These will include: optional stopping; maximal inequalities; concentration inequalities with special emphasis on Hilbert spaces; martingale convergence; L^p martingales; uniform integrability and backward martingales; applications to reversible Markov chains; continuous-time martingales; and more.

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

Recommended Texts:

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

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

This course introduces geophysical fluid dynamics from an experimental, laboratory point of view. Laboratory instrumentation centers on a turntable equipped with velocity field (PIV, particle imaging velocimetry) and density field (LIF, laser induced florescence) measurement equipment. The course intertwines the laboratory observations with geophysical fluid dynamics theory, while numerical models are used as a platform to reconcile the observations with theoretical understanding.

Recommended Texts:

• Atmosphere, Ocean and Climate Dynamics: An Introductory Text. Plumb & Marshall
• Introduction to Geophysical Fluid Dynamics. Benoit Cushman-Roisin