Course Descriptions

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.

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.

Recitation/Problem Session: 7:10-9:00 (following the course)

Prerequisites: (serial) programming experience with C/C++ (I will use C in class) or FORTRAN, basic knowledge of command line UNIX tools, and some familiarity with numerical methods.

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 using and programming these machines (distributed and shared memory parallelism: MPI, OpenMP, OpenCL). 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 computing homework assignments, in which you will explore material by yourself and try things out.  There will be a larger final project at the end. Students who have code they want to parallelize or speed up are encouraged to attend, and use that for their final project.

Text: Besides various online resources, we will use:

• "Parallel Programming for Multicore and Cluster Systems" by T. Rauber and G. Ruenger, Springer, 2nd edition (2013). Available online in the institute.

Cross-listing: CSCI-GA 2945.001

Prerequisites: Students must have a good upper level undergraduate math background including linear algebra, probability, and multi-variate calculus. Course work in numerical computation is desirable. Students should be able to do numerical programming in Python, C/C++, Java, Fortran, R, or Matlab. Students without experience in Python will have to put in some extra effort in the first few weeks.

First half: An introduction to practical Monte Carlo methods, with some of the basic theory, with applications to statistical physics and chemistry, and to Bayesian statistics. Pseudo random number generators. Direct sampling methods, including mappings and rejection. Markov chain Monte Carlo (MCMC), Metropolis Hastings, detailed balance, partial resampling/heat bath. Basic MCMC theory, including Perron Frobenius and the ergodic theorem, the Markov chain central limit theorem, and the Kubo formula for auto-correlation time. Methods for estimating the auto-correlation time and estimating error bars for MCMC results.

Second half: More specialized topics depending on the interests and background of students in the class. Topics may include (a) advanced samplers: Hamiltonian samplers, affine invariant ensemble samplers, multi-level methods, adaptive samplers, (b) thermodynamic integration, (c) Bayesian model selection, (d) rare event simulation, (e) stochastic differential equations, (f) more theory: spectral gap, Poincare and Cheeger inequalities.

Assignments will include significant programming in Python as well as theoretical exercises. Assignments will emphasize elements of programming methodology relevant to Monte Carlo methods, including verification protocols and visualization methods. There will be a long term project done individually or in small groups.

Course Website

Prerequisites: Numerical linear algebra, elements of ODE and PDE.

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 form an essential part of the course. The course will introduce students to numerical methods for (1) nonlinear equations, Newton's method; (2) ordinary differential equations, Runge-Kutta and multistep methods, convergence and stability; (3) finite difference, finite element and integral equation methods for elliptic partial differential equations; (4) fast solvers, multigrid methods; and (5) parabolic and hyperbolic partial differential equations.

Text: LeVeque, R. (2007). Classics in Applied Mathematics [Series]. Finite Difference Methods for Ordinary and Partial Differential Equations. Philadelphia, PA: Society for Industrial and Applied Mathematics.

Cross-listing: CSCI-GA 2421.001

Prerequisites: Risk and Portfolio Management with Econometrics, Derivative Securities, 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

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

Prerequisites: Linear Algebra I or permission of the instructor.

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 decompositions. Complexification and diagonalization over R. Matrix norms, series and the matrix exponential map, applications to ODE. Bilinear and quadratic forms and their normal forms. The classical matrix groups: unitary, orthogonal, symplectic. Implicit function theorem, smooth surfaces in Rⁿ and their tangent spaces. Intro to matrix Lie algebras and Lie groups.

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

Plus: Extensive instructor’s class notes.

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

Text:

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.

Recommended Texts:

Lang, S. (2005). Graduate Texts in Mathematics [Series, Bk. 211].Algebra (3^rd ed.). 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., & Liebeck, M. (1993). Cambridge Mathematical Textbooks [Series]. Representations and Characters of Groups. New York, NY: Cambridge University Press.

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

Sagan, B.E. (1991). Wadsworth Series in Computer Information Systems [Series]. The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions. Pacific Grove, CA: Wadsworth & Brooks/Cole.

Brocker, T., & Dieck, T. (2003). Graduate Texts in Mathematics [Series, Bk. 98]. Representations of Compact Lie Groups. New York, NY: Springer-Verlag.

Prerequisites: Undergraduate elementary number theory, abstract algebra, including groups, rings and ideals, fields, and Galois theory (e.g. undergraduate Algebra I and II).

This course is a graduate level introduction to algebraic number theory, in which we will cover fundamentals of the subject. Topics include: the theory of the valuation (p-adic numbers, completion, local fields, henselian fields, ramification theory, Galois theory of valuations) and Riemann-Roch theory.

For additional information, see the course website.

Text: Neukirch, J. (1999).Grundlehren der mathematischen Wissenschaften [Series, Book 322].Algebraic Number Theory. New York, NY: Springer-Verlag.

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 Poincar duality. Products and ring structures. Vector bundles, tangent bundles, DeRham cohomology and differential forms.

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.

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.

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

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

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 Poincar-Bendixson theorems.

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

Prerequisites: MATH-GA 2490.001 PDE I and MATH-GA 2430.001 Real Variables, or theTH-GA 2430.001 equivalent.

This course is a continuation of MATH-GA 2490 and is designed for students who are interested in analysis and PDEs. The course gives an introduction to Sobolev spaces, Holder spaces, and the theory of distributions; elliptic equations, harmonic functions, boundary value problems, regularity of solutions, Hilbert space methods, eigenvalue problems; general hyperbolic systems, initial value problems, energy methods; linear evaluation equations, Fourier methods for solving initial value problems, parabolic equations, dispersive equations; nonlinear elliptic problems, variational methods; Navier-Stokes and Euler equations.

Recommended Texts: Garabedian, P.R. (1998). Partial Differential Equations (2^nd Rev. ed.). Providence, RI: AMS Chelsea Publishing/ American Mathematical Society.

Evans, L.C. (2010). Graduate Studies in Mathematics [Series, Bk. 19]. Partial Differential Equations (2^nd ed.). Providence, RI: American Mathematical Society. 

John, F. (1995). Applied Mathematical Sciences [Series, Vol. 1]. Partial Differential Equations (4th ed.). New York, NY: Springer-Verlag.

Prerequisites: Real analysis; basic knowledge of complex variables and functional analysis.

Classical Fourier Analysis: Fourier series on the circle, Fourier transform on the Euclidean space, Introduction to Fourier transform on LCA groups. Stationary phase. Topics in real variable methods: maximal functions, Hilbert and Riesz transforms and singular integral operators. Time permitting: Introduction to Littlewood-Paley theory, time-frequency analysis, and wavelet theory.

Recommended Text: Muscalu, C., & Schlag, W. (2013). Cambridge Studies in Advanced Mathematics [Series, Bk. 137]. Classical and Multilinear Harmonic Analysis (Vol.1). New York, NY: Cambridge University Press. (Online version available to NYU users through Cambridge University Press.)

This class is intended to explain the mathematical tools used to study the effect of resonances on the long-time dynamics of nonlinear hyperbolic and dispersive PDEs. The model equations that we will consider are the nonlinear Klein-Gordon equation and the nonlinear Schrodinger equation. The lectures will proceed as follows: 1) Review of resonances in ODEs, Poincare Normal forms and Birkhoff Normal forms; 2) Normal forms analysis for PDEs and its impact on the time of existence of solutions; and 3) Asymptotic dynamics of the NLS with periodic boundary conditions.

The mathematical tools that we will present in this class are: 1) Normal forms and averaging; 2) Fourier analysis and equidistribution of lattice points; 3) The circle method. These tools will be presented to explain resonant frequencies and how to count them.  A considerable portion of the class will be devoted to explaining the Fourier analysis techniques used to derive the asymptotics of the nonlinear Schrodinger equation with periodic boundary conditions.

Prerequisites: Real analysis at the graduate level.

This is the second semester of a year course on dynamical systems. It is an introductory sequence, requiring no prior knowledge of the subject. In the fall semester, I will cover mostly ergodic theory, a probabilistic approach to dynamical systems. Topics include ergodicity, mixing properties, entropy; ergodic theory of continuous and differentiable maps, Lyapunov exponents etc. In the spring semester, the focus will be on differentiable dynamical systems. Topics include invariant manifolds, hyperbolicity, and various models of chaotic systems.

Prerequisites: Real variables (measure theory), basic elliptic partial differential equations and surfaces in Euclidian spaces.

This topics course will cover a basic theory of varifolds. In particular, first variations, monotonicicy formula, isoperimetric inequality, rectifiability, tangent cone and regularity theory. Some examples of applications and generalizations will be discussed in the later part of the course.

References: Allard, William K.
[1] On the first variation of a varifold. Ann. of Math. (2) 95 (1972), 417–491.
[2] On the first variation of a varifold: boundary behavior. Ann. of Math.
   (2) 101 (1975), 418–446.
F.H.Lin, Geometric Measure Theory--An Introduction, International Press, Boston (2002).

Prerequisites: Real Variables I and PDE I (or equivalent)

A modern introduction to the Calculus of Variations, with equal emphasis on theory and applications. Topics will include: existence of solutions and convergence of numerical schemes; convex duality; one-dimensional variational problems; multidimensional nonconvex problems; relaxation; Gamma-convergence; homogenization; and energy-driven pattern formation. Along the way, we'll discuss many applications including minimal surfaces, optimal control, nonlinear elasticity, composite materials, and pattern formation problems involving defects or walls.

There will be no required text. (The syllabus will include a list of relevant books, and those not available electronically will be placed on reserve.)

In this course we will be interested in using tools from analysis and probability to describe Coulomb-type systems.

More precisely this means large sets of points interacting via Coulomb, logarithmic or more generally Riesz (i.e. inverse powers of the distance) interactions, in general with temperature. These systems are important statistical mechanics ensembles and arise in various settings related to physics (vortices in superconductors...), approximation theory (Fekete sets...) and random matrix theory (GUE, GOE and Ginibre ensembles), and have been the object of many studies in recent years with various viewpoints.

In this course, after reviewing some of the motivations for studying such systems, we will a point of view based on the detailed expansion of the interaction energy. This allows to describe the macroscopic and microscopic behavior of the systems. In particular the goal of the course is to show a Large Deviations Principle for the empirical field and a Central Limit Theorem for fluctuations down to the mesoscopic scales. This allows to observe the effect of the temperature as it gets very large or very small, and to connect with crystallization questions.

The course will assume knowledge of real analysis, functional analysis and very basic notions of probability.

Prerequisites: 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.

Texts:

• Stochastic Processes and Applications, by G. A. Pavliotis.
• C. Gardiner, Stochastic Methods: A Handbook for the Natural and Social Sciences 

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

Prerequisites: Undergraduate ordinary and partial differential equations.

This course provides a mathematical introduction to mechanics aimed at graduate students preparing for research on physical science topics in applied mathematics or applied probability. It covers fundamental core topics at an advanced mathematical level and it also provides examples of applications drawn from recent research. 

The core topics are:

1. classical mechanics (discrete and continuous),
2. statistical mechanics, and
3. quantum mechanics.

No prior knowledge of physics is required to attend this class.

Textbook: Oliver Buhler, A Brief Introduction to Classical, Statistical, and Quantum Mechanics Courant Lecture Notes 13, American Mathematical Society, 2006.

Risk Management is arguably one of the most important tools for managing a trading book and quantifying the effects of leverage and diversification (or lack thereof).

This course is an introduction to risk-management techniques for portfolios of (i) equities and delta-1 securities and futures (ii) equity derivatives (iii) fixed income securities and derivatives, including credit derivatives, and (iv) mortgage-backed 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 historical back-testing of portfolios. We also review current risk-models and practices used by large financial institutions and clearinghouses.

If time permits, the course will also cover models for managing the liquidity risk of portfolios of financial instruments.

Prerequisites: Risk & Portfolio Management with Econometrics, 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.

Prerequisites: Derivative Securities, Computing in Finance or equivalent programming.

The importance of financial risk management has been increasingly recognized over the last several years. This course gives a broad overview of the field, from the perspective of both a risk management department and of a trading desk manager, with an emphasis on the role of financial mathematics and modeling in quantifying risk. The course will discuss how key players such as regulators, risk managers, and senior managers interact with trading. Specific techniques for measuring and managing the risk of trading and investment positions will be discussed for positions in equities, credit, interest rates, foreign exchange, commodities, vanilla options, and exotic options. Students will be trained in developing risk sensitivity reports and using them to explain income, design static and dynamic hedges, and measure value-at-risk and stress tests. Students will create Monte Carlo simulations to determine hedge effectiveness. Extensive use will be made of examples drawn from real trading experience, with a particular emphasis on lessons to be learned from trading disasters.

Text: Allen, S.L. (2003). Wiley Finance [Series, Bk. 119]. Financial Risk Management: A Practitioner’s Guide to Managing Market and Credit Risk. Hoboken, NJ: John Wiley & Sons.

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.

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.

A second course in arbitrage-based pricing of derivative securities. Concerning equity and FX models: we'll discuss numerous approaches that are used in practice in these markets, such as the local volatility model, Heston, SABR, and stochastic local volatility. The discussion will include calibration and hedging issues and the pricing of the most common structured products. Concerning interest rate models: we'll start with a thorough discussion of one-factor short-rate models (Vasicek, CIR, Hull-White) then proceed to more advanced topics such as two-factor Hull-White, forward rate models (HJM) and the LIBOR market model. Throughout, the pricing of specific payoffs will be considered and practical examples and insights will be provided. We'll conclude with a discussion of inflation models.

Prerequisites: Derivative Securities, 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.

Prerequisites: Basic bond mathematics and bond risk measures (duration and convexity); Derivative Securities and Stochastic Calculus.

This half-semester course will cover the fundamentals of Securitized Products, emphasizing Residential Mortgages and Mortgage-Backed Securities (MBS). We will build pricing models that generate cash flows taking into account interest rates and prepayments. The course will also review subprime mortgages, CDO’s, Commercial Mortgage Backed Securities (CMBS), Auto Asset Backed Securities (ABS), Credit Card ABS, CLO’s, Peer-to-peer / MarketPlace Lending, and will discuss drivers of the financial crisis and model risk.

Prerequisites: Derivative Securities and Stochastic Calculus.

This half-semester course focuses on energy commodities and derivatives, from their basic fundamentals and valuation, to practical issues in managing structured energy portfolios. We develop a risk neutral valuation framework starting from basic GBM and extend this to more sophisticated multi-factor models. These approaches are then used for the valuation of common, yet challenging, structures. Particular emphasis is placed on the potential pitfalls of modeling methods and the practical aspects of implementation in production trading platforms. We survey market mechanics and valuation of inventory options and delivery risk in the emissions markets.

Prerequisites: Derivative Securities, Stochastic Calculus, and Computing in Finance or equivalent programming experience.

This half-semester course will give a practitioner’s perspective on a variety of advanced topics with a particular focus on equity derivatives instruments, including volatility and correlation modeling and trading, and exotic options and structured products. Some meta-mathematical topics such as the practical and regulatory aspects of setting up a hedge fund will also be covered.

Prerequisites: Derivative Securities, Risk & Portfolio Management with Econometrics, 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.

This course will present an evolving methodology for the explanation of variability in data in terms of known and unknown factors, based on the mathematical theory of optimal transport. Real world observations can be highly individualized. Medical data, for instance, aggregates samples of patients having each a unique combination of age, sex, diet, prior conditions and prescribed drugs; samples that are often collected and analyzed at facilities with different equipment and personnel. In addition, each patient has an underlying health state that one would like to extract from the data. The individualized nature of data provides a door to personalized medicine and, more generally, to increased predictability, but also brings in a number of mathematical challenges. The connection to optimal transport presents itself naturally in the context of removing from observations {xi} the variability attributable to a factor z.

The existence of such attributable variability means that the conditional distribution p(x|z) depends on z. Removing the attributable variability is therefore tantamount to estimating a set of maps x  y = Y (x; z) so that none of the variability remaining in y can be attributed to z. In addition, one wants these maps to distort the data minimally, so that the variability not related to z is unaffected by the transformation from x to y. This class will explore how these ideas can be expanded to provide a general framework for the analysis of data, including broad generalizations of classical tools such as clustering by k-means and principal component analysis.

This course provides the mathematical and computational tools needed for an operational knowledge of discrete choice models, matching models, and network flow models. A number of economic applications of these concepts will be discussed. The first part of the course will introduce basic results around Optimal Transportation theory: the Monge-Kantorovich duality, the Optimal Assignment Problem, basic results in Linear Programming, and Convex Analysis. Those concepts will serve as building blocks in the sequel. The second part will cover discrete choice models, from the classical theory to more recent advances. The classical Generalized Extreme Value (GEV) specification will be recalled, as well as Maximum Likelihood estimation in the parametric case. Comparative statics results will be derived using tools from Convex Analysis, and nonparametric identification will be worked out using Optimal Transport theory. Simulation methods will be covered. A computationally intensive application will be demonstrated. The third part will be devoted to matching models with stochastic utility, starting with the Transferable Utility (TU) case which is then generalized to Imperfectly Transferable Utility (ITU) including Non-transferable Utility (NTU). Equilibrium computation in the general case will be worked out using techniques from General Equilibrium. The more specific, but empirically relevant logit case, will be efficiently addressed using more the specific techniques or Iterative Fitting. Various algorithms will be described and compared in practice. Moment Matching Estimation and Maximum Likelihood Estimation will be worked out and compared. Several applications, to Collective Models of Family Economics, and to Labor Markets with taxes, will be described. Time permitting, the fourth and last part will provide an introduction to problems on networks. The basic tools to describe the topology on a network will be described: discrete differential operators, diffusions on networks, shortest paths on networks. The Optimal Transport problem on networks will be formulated, along with its extension to stochastic
utility.

Required Texts: The first part of the course will be based on my textbook: [OTME] Optimal Transport Methods in Economics (Princeton University Press, in press), a draft of which is available here.

Recommended Texts:

• [TSM] A. Roth and M. Sotomayor, Two-Sided Matching A study in Game-Theoretic Modeling and Analysis, Monographs of the Econometrics Society, 1990.
• [DCMS] Train, K.. Discrete Choice Methods with Simulation. 2nd Edition. Cambridge University Press, 2009.
• [TOT] C. Villani, Topics in Optimal transportation, AMS, 2003.

Cross-listing: ECON-GA 1702.001.

Many problems arising e.g. in the context of statistical mechanics, material science, or data assimilation involve computing expectations over probability distributions defined on high-dimensional spaces. These problems are typically beyond analytical solutions and therefore requires one to use numerical methods such as Monte Carlo (MC) simulations. In fact, even vanilla MC simulations are inadequate for many tasks at hand: rather specific methods must be developed that take into account the specificity of the problem to reduce the computational cost: these techniques go by the generic name of importance sampling. In this class, I will discuss selected topics in this context, with applications to the calculation of free energies, the analysis of reactive events in metastable systems, the estimation of the likelihood of extreme events, the calculation of large deviation functions arising e.g. from work relations in nonequilibrium statistical mechanics, and the calculation of parameters by Bayesian estimation.

This course will explore how applied mathematics and math modeling 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. The projects will be drawn from modern fluid dynamics research and will explore fascinating questions relevant to life and earth, such as: What is the fluid dynamics of bird flocks and fish schools, and How does erosion by water or wind sculpt the landforms and landscapes around us?

Prerequisites: elementary background in ODEs, PDEs, probability theory, Fourier transforms.

A variety of topics of current interest in biology and neuroscience will be addressed. Topics include: (1) Stochastic gene expression: analytical modeling of stochastic messenger RNA synthesis and degradation; discrete and continuous models; master equation; generating function; steady-state distributions; temporal evolution of the distributions; stochastic protein product. (2) Stochastic cell divisions and population growth: mean growth rate; age distributions; optimal lineage. (3) Evolution of fitness in yeast populations in the absence of selective pressures. (4) Single-photon responses of retinal rods; statistical measures of variability; reproducibility of the single-photon response; explicit biochemical kinetic models; model testing with Monte Carlo simulations. (5) Stochastic switching between bistable percepts, e.g. 2 very different auditory percepts induced by A-B-A tone sequence. (6) Optimal filtering of photon noise in vision. (7) Stochastic behavior of neurons in the central nervous system: models for synaptic noise; spike train statistics and renewal theory. (8) Probability density methods for large-scale modeling of neural networks: partial differential-integral equations; Fokker-Plank approximation; applications to modeling visual cortex.

The course aims to demonstrate the richness and complexity of the living cell by way of introducing basic phenomena of biological processes in cells. In demonstrating underlying unifying physical principles, the course will emphasize physical intuitive pictures and mathematical tools for understanding properties of the living cell. 

The course will cover the following materials:

1. Basics of molecular biology of the cell/sequences/specificity/evolution
2. Mechanical/chemical/thermodynamic processes in living cells
   1. Mechanical properties of cytoskeletons/Polymers/ (beam theory)
   2. Energy and the life of cells/ATP hydrolysis/Entropy/hydrophobicity/depletion forces/osmotic pressure/ free energy
   3. Statistical mechanics of cells/Chemical forces/Delta_G/Transition rate theory/Kramer's theory/self-assembly
3. Random walks/Transport/Diffusion in the cell/Fluctuation-dissipation theorem
4. Hydrodynamics/Stokes flow/drag/swimming bacteria/Blood flow
5. Electrostatics/Salty solutions/Poisson-Boltzmann Equations/electrochemical equilibrium and the Nernst equation/Action Potential/Hodgkin-Huxley model
6. ** Biological membranes/Ion channels/vesicles/active membranes
7. ** Case study: Molecular motors

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

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

Texts:

• Probability Essentials by J. Jacod and P. Protter
• Probability: Theory and Examples (4th edition) by R. Durrett

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: Monday, 6:00-7:00.

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

Independent increment processes, especially Poisson processes and Brownian motion. Markov chains. Stochastic differential equations and diffusions, Markov processes in general, semi-groups, generators and connection with partial differential equations.

Recommended & On Reserve Text: McKean (Henry) Probability: The Classical Limit Theorems. Cambridge University Press, Cambridge, U.K. 2014.

Supplementary Reading: Varadhan, S.R.S. (2007). Courant Lecture Series in Mathematics [Series, Bk. 16]. Stochastic Processes. Providence, RI: American Mathematical Society/CIMS

Prerequisites: The prerequisites for this class are mainly Limit Theorems in Probability, and some differential geometry. If you know some random matrix theory, or statistical physics, it is helpful but not required.

Random functions of many variables tend to be very complex. The number of critical points can be very large, the topology of the level sets very intricate. This has important consequences for optimization problems in random landscapes in high dimensions.

We will illustrate these geometric sentences by examples coming from statistical physics of disordered media, and then from machine learning. The common mathematical tool underlying these different questions is given by Random Matrix Theory, through the classical Kac-Rice formula of random differential geometry.

We will begin by giving a brief introduction to the general framework for studying random gaussian fields, or random gaussian functions, of many variables, as given in the excellent books by Adler-Taylor and/or Azais-Wschebor, and get quickly to the Kac-Rice formula.

We will then survey quickly the needed results from Random Matrix Theory.

The next step will be to survey recent work describing this complexity phenomenon and its consequences, first from a geometric point of view, i.e random Morse theory. We will illustrate it first in the case of general Gaussian random functions on the high-dimensional sphere. These random functions happen to be the energy landscapes of important models of statistical physics of disordered media, i.e spherical spin glasses. Some recent important progress has been achieved, but many open questions remain in this field, and we will explore some of them.

We will then see how this picture could be extended to the random landscapes of deep learning algorithms, which are at the heart of some of the recent progress in Data Science.

The goal of this course is to introduce students to modern dynamical oceanography, with a focus on mathematical models for observed phenomena.  The lectures will cover the observed structure of the ocean, the thermodynamics of sea-water, the equations of motion for rotating-stratified flow, and the most useful approximations thereof: the primitive, planetary geostrophic and quasi-geostrophic equations. The lectures will demonstrate how these approximations can be used to understand boundary layers, wind-driven circulation, buoyancy-driven circulation, oceanic waves (Rossby, Kelvin and inertio-gravity), potential vorticity dynamics, theories for the observed upper-ocean stratification (the thermocline), and for the global general circulation. Additionally the course will cover relevant oceanic fluid instabilities and their resulting turbulence: mesoscale turbulence driven by baroclinic instability, convective turbulence and high-latitude sinking, and mixing across density surfaces due to shear-driven turbulence.  Finally, we will discuss tides and the observed  internal wave spectrum.  Throughout the lectures, the interplay between observational, theoretical, and modeling approaches to problems in oceanography will be highlighted.

Course activities will include a few problem sets and class presentations.

Recommended Text:

• "The Theory of Large-Scale Ocean Circulation" by R. Samelson (Cambridge 2011)
• "Atmospheric and Oceanic Fluid Dynamics" by G.K. Vallis (Cambridge 2006)
• "Lectures on Geophysical Fluid Dynamics" by R. Salmon (Oxford 1998)

In addition, will read relevant journal articles each week.