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: Thursdays, 7:10-9:00PM (following the course)

 

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

Convex optimization problems have many important properties, including a powerful duality theory and the property that any local minimum is also a global minimum. Nonsmooth optimization refers to minimization of functions that are not necessarily convex, usually locally Lipschitz, and typically not differentiable at their minimizers. Topics in convex optimization that will be covered include duality, linear and semidefinite programming, CVX ("disciplined convex programming"), gradient and Newton methods, Nesterov's complexity bound, the alternating direction method of multipliers, the nuclear norm and matrix completion, the primal barrier method, primal-dual interior-point methods for linear and semidefinite programs. Topics in nonsmooth optimization that will be covered include subgradients and subdifferentials, Clarke regularity, and algorithms, including gradient sampling and BFGS, for nonsmooth, nonconvex optimization. Homework will be assigned, both mathematical and computational. Students may submit a final project on a pre-approved topic or take a written final exam.

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.

Prerequisites:

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

Description:

his 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.

 

Text:

"Finite Difference Methods for Ordinary and Partial Differential Equations" by Randy LeVeque. This textbook is now available freely to you in PDF format.

Cross-listing:

CSCI-GA 2421.001.

 

Prerequisites:

Risk and Portfolio Management with Econometrics, Derivative Securities, and Computing in Finance

Description:

This is a version of the course Scientific Computing (MATH-GA 2043.001) designed for applications in quantitative finance.  It covers software and algorithmic tools necessary to practical numerical calculation for modern quantitative finance.  Specific material includes IEEE arithmetic, sources of error in scientific computing, numerical linear algebra (emphasizing PCA/SVD and conditioning), interpolation and curve building with application to bootstrapping, optimization methods, Monte Carlo methods, and the solution of differential equations.

Please Note: Students may not receive credit for both MATH-GA 2043.001 and MATH-GA 2048.001

 

 

Prerequisites: Undergraduate linear algebra or permission of the instructor.

Description: Linear spaces, subspaces. Linear dependence, linear independence; span, basis, dimension, isomorphism. Quotient spaces. Linear functionals, dual spaces. Linear mappings, null space, range, fundamental theorem of linear algebra. Underdetermined systems of linear equations. Composition, inverse, transpose of linear maps, algebra of linear maps. Similarity transformations. Matrices, matrix Multiplication, Matrix Inverse, Matrix Representation of Linear Maps determinant, Laplace expansion, Cramer's rule. Eigenvalue problem, eigenvalues and eigenvectors, characteristic polynomial, Cayley-Hamilton theorem. Diagonalization.

Text:

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

 

 

Prerequisites: Linear Algebra I or permission of the instructor.

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

Text:

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

Plus: Extensive instructor’s class notes.

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

Text:

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

Recommended Texts:

• Lang, S. (2005). Graduate Texts in Mathematics [Series, Bk. 211]. Algebra (3^rded.). New York, NY: Springer-Verlag.
• Serre, J.P. (1977). Graduate Texts in Mathematics [Series, Bk. 42]. Linear Representations of Finite Groups. New York, NY: Springer-Verlag.
• Reid, M. (1989). London Mathematical Society Student Texts [Series]. Undergraduate Algebraic Geometry. New York, NY: Cambridge University Press.
• James, G., & Lieback, M. (1993). Cambridge Mathematical Textbooks [Series]. Representations and Characters of Groups. New York, NY: Cambridge University Press.
• Fulton, W., Harris, J. (2008). Graduate Texts in Mathematics/Readings in Mathematics [Series, Bk. 129]. Representation Theory: A First Course (Corrected ed.). New York, NY: Springer-Verlag.
• Sagan, B.E. (1991). Wadsworth & Brooks/Cole Mathematics Series [Series]. The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions. Pacific Grove, CA: Wadsworth & Brooks/Cole.
• Brocker, T., & Dieck, T. (2003). Graduate Texts in Mathematics [Series, Bk. 98]. Representations of Compact Lie Groups. New York, NY: Springer-Verlag.

 

Prerequisites:

Undergraduate elementary number theory, abstract algebra, including groups, rings and ideals, fields, and Galois theory (e.g. undergraduate Algebra I and II). A background in complex analysis, as well as in algebra, is required.

 

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.

 

 

 

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.

This course will introduce and develop some key invariants of manifold theory and their roles in manifold classification and other applications. The invariants to be studied include some from algebraic K-theory, e.g., Whitehead torsion and Reidemeister torsion, and some from bundle theory, e.g., characteristic classes. 

 

This class will meet once a week but will also have some supplementary opportunities to give students more familiarity with background algebraic topology. 

 

Description:

Differential Geometry II will focus on Riemannian geometry. Topics to be covered may include: second variation of arc length, Rauch comparison theorem and applications, Toponogov's theorem, invariant metrics on Lie groups, Morse theory, cut locus, the sphere theorem, complete manifolds of nonnegative curvature.

Recommended Texts:

• John Milnor, Morse Theory (Princeton University Press, 1963).
• John M. Lee, Riemannian Manifolds: An Introduction to Curvature (Springer, 1997).

 

Description:

We will sketch a sequence of developments in riemannian geometry which have taken place over roughly the last 50 years. These concern structure theories for manifolds satisfying bounds on sectional or Ricci curvature, and related theories of geometric convergence and the structure of limit spaces. The emphasis in the lectures will be on high level ideas and on providing a guide to the technical aspects of the proofs.  The foundational results on Ricci curvature are proved in detail my short book "Degeneration of riemannian metrics on Ricci curvature bounds". A broader and less technical overview is given in my expository paper "Structure theory and convergence in riemannian geometry".  I will make the pdf files of both of these available to people who are attending the lectures. If time permits, we will discuss recent results concerning quantitative behavior of singular sets and the proof of the codimension 4 conjecture.

Prerequisites:  Familiarity with real analysis, complex analysis, differential geometry

 

Description:

Differentiability and rectifiability describe how well functions and sets can be linearly approximated at infinitesimal scales.  For many problems, knowing the behavior of a set at small scales isn't enough; one needs quantitative versions of differentiability and rectifiability that bound how well functions and sets can be approximated at many different scales.  Examples include singular integrals; covering sets in the plane by a curve; and the distortion of maps between metric spaces.  In this course, we will study applications of quantitative differentiability and rectifiability in geometric measure theory, quantitative geometry, and harmonic analysis.  Topics covered may include:

 

• Differentiability of Lipschitz functions
• The Analyst's Traveling Salesman Problem
• Uniform rectifiability and singular integrals
• Quasi-isometries and quasi-isometric rigidity 
• Metric embeddings

Large ensembles of points with Coulomb type interactions arise in various settings of condensed matter physics, classical and quantum mechanics, statistical mechanics, random matrices and even approximation theory. We focus on the Gibbs measure associated to such systems in general dimension and on describing the microscopic behavior of the system as the number of points tends to infinity.  Results include Large Deviations Principles and Central Limit Theorems for fluctuations. This allows for instance to observe the effect of the temperature and to connect with crystallization questions. 

 

We will introduce the basic statistical mechanics notions needed and describe the tools and methods used to prove the results (electric formulation of the energy, screening procedure, transport method...).

As part of our new NSF research training group (RTG) in Modeling & Simulation, we will be organizing a lunchtime group meeting for students, postdocs, and faculty working in applied mathematics who do modeling & simulation. The aim is to create a space to discuss applied mathematics research in an informal setting: to (a) give students and postdocs a chance to present their research (or a topic of common interest) and get feedback from the group, (b) learn about other ongoing and future research activities in applied math at the Institute, and (c) discuss important open problems and research challenges.

 

The meetings will be Thursdays from 12:30-2:00, in room 1314; the weekly schedule is posted here.

Pre-requisties: Basic Measure Theory and Linear Elliptic PDEs.

Topics to be discussed:

1. Quantitative unique continuation and doubling estimates.
2. Hausdorff dimension and measure bounds on nodal sets.
3. Structure and Geometric measure estimates for critical sets.
4. Other geometrical and topological properties of nodal sets.

 

 

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 variables.

Description:

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

Recommended Text:

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

 

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

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

Main texts:

1.      L.C. Evans, Partial Differential Equations, American Mathematical Society

2.      M. Renardy and R. Rogers, An Introduction to Partial Differential Equations, Springer-Verlag

 

Prerequisites:

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

Description:

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. and 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.

Prerequisites:

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

Description:

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

Text:

• 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.

Description:

In this course we develop a quantitative investment and trading framework. In the first part of the course, we study the mechanics of trading in the financial markets, some typical trading strategies, and how to work with and model high frequency data. Then we turn to transaction costs and market impact models, portfolio construction and robust optimization, and optimal betting and execution strategies. In the last part of the course, we focus on simulation techniques, back-testing strategies, and performance measurement. We use advanced econometric tools and model risk mitigation techniques throughout the course. Handouts and/or references will be provided on each topic.

 

 

 

Description:

This course provides 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.

 

 

 

 

Description:

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.

Description:

The first part of the course will cover the theoretical aspects of portfolio construction and optimization. The focus will be on advanced techniques in portfolio construction, addressing the extensions to traditional mean-variance optimization including robust optimization, dynamical programming and Bayesian choice. The second part of the course will focus on the econometric issues associated with portfolio optimization. Issues such as estimation of returns, covariance structure, predictability, and the necessary econometric techniques to succeed in portfolio management will be covered. Readings will be drawn from the literature and extensive class notes.

 

 

Prerequisites:

Derivative Securities, Computing in Finance or equivalent programming.

Description:

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

Text:

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

 

 

Description:

Students in the Mathematics in Finance program conduct research projects individually or in small groups under the supervision of finance professionals. The course culminates in oral and written presentations of the research results.

 

 

Description:

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.

Description:

This is a second course in arbitrage-based pricing of derivative securities. Concerning equity and FX models: We 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 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 give an introduction to inflation models.

We cover a few special topics: We provide an introduction to stochastic optimal control with applications, as well as optimal stopping time theory and its application to American options pricing. We introduce Cox default processes and discuss their applications to unilateral and bilateral CVA/DVA.

 

Prerequisites:

Derivative Securities, Stochastic Calculus, and Computing in Finance (or equivalent familiarity with financial models, stochastic methods, and computing skills).

Description:

The course is divided into two parts. The first addresses the fixed-income models most frequently used in the finance industry, and their applications to the pricing and hedging of interest-based derivatives. The second part covers the foreign exchange derivatives markets, with a focus on vanilla options and first-generation (flow) exotics. Throughout both parts, the emphasis is on practical aspects of modeling, and the significance of the models for the valuation and risk management of widely-used derivative instruments.

 

 

 

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

 

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

 

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

 

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

 

Prerequisites: Derivative Securities and Stochastic Calculus.

 

The course provides 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, cross-market spillovers, and linkages to other financial asset classes.

 

Trading strategies include traditional risk premia, volatility, correlation, and higher-order options Greeks. Examples and case studies are based on actual market episodes using real market data.

 

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

 

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

 

Prerequisites:

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

Description:

This is a half-semester course covering topics of interest to both buy-side traders and sell-side execution quants. The course will provide a detailed look at how the trading process actually occurs and how to optimally interact with a continuous limit-order book market.

We begin with a review of early models, which assume competitive suppliers of liquidity whose revenues, corresponding to the spread, reflect the costs they incur. We discuss the structure of modern electronic limit order book markets and exchanges, including queue priority mechanisms, order types and hidden liquidity.  We examine technological solutions that facilitate trading such as matching engines, ECNs, dark pools, multiple venue problems and smart order routers.

The second part of the course is dedicated pre-trade market impact estimation, post-trade slippage analysis, optimal execution strategies and dynamic no-arbitrage models. We cover Almgren-Chriss model for optimal execution, Gatheral’s no-dynamic-arbitrage principle and the fundamental relationship between the average response of the market price to traded quantity,  and properties of the decay of market impact.

Homework assignments will supplement the topics discussed in lecture. Some coding in Java will be required and students will learn to write their own simple limit-order-book simulator and analyze real NYSE TAQ data.

 

 

 

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 {x_i} the variability attributable to a factor z. The existence of such attributable variability means that the conditional distribution rho(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. Tools will include conditional optimal transport and the Wasserstein barycenter problem, and their adversarial formulations based on duality and mini-maximization.

 

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

 

A variety of topics  in biology and neuroscience will be addressed, including: (1) Stochastic gene expression: analytical modeling of stochastic messenger RNA synthesis and degradation; discrete and continuous models; master equation; probability 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; maximum likelihood retrospective distribution of random generation times along  lineages of a cells selected randomly from extant cells in a population; discrete models for population growth, including birth and death. (3) Single-photon responses of retinal rods; statistical measures of variability; reproducibility of the single-photon response; explicit stochastic biochemical kinetic models; model testing with Monte Carlo simulations. (4) Optimal spatiotemporal filtering of photon noise in vision; physiological correlates in retinal light adaptation (5) Stochastic witching between bistable percepts, e.g. 2 very different auditory percepts induced by A-B-A tone sequence. (6) Stochastic behavior of neurons in the central nervous system: models for synaptic noise; spike train statistics and renewal theory. (7) Probability density methods for large-scale modeling of neural networks: partial differential-integral equations; Fokker-Plank approximation; applications to modeling orientation tuning in visual cortex.

 

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

 

Living systems exist in a thermal environment, and entropy plays a fundamental role in their dynamics.  This course begins with an introduction to entropy as defined and used in thermodynamics and in statistical mechanics, including the principle of detailed balance and the Einstein and Onsager relations.  The rest of the course will focus on specific biological topics: osmotic pressure and the control of cell volume; ion transport through membrane channels; rotary molecular motors driven by ionic elecrochemical potential differences across membranes; entropy-driven biochemical reactions; entropic spring forces, hydrophobic forces, and the mechanics of gels; the molecular mechanism and thermodynamics of muscle contraction; and multicomponent diffusion.  A theme of the course will be the interplay between macroscopic and microscopic models, and the use of computer simulation to make entropic phenomena visible.

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

 

Description:

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

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

Texts:

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

 

 

Prerequisites: MATH-GA 2901 Basic Probability or equivalent.

 

Description:

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

 

Optional Problem Session:

Thursdays, 5:30-7:00.

Text:

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

 

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

 

Description:

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

Recommended & On Reserve Text:

• 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
• Markov Processes: Characterization and Convergence by Ethier and Kurtz

Convergence of Probability Measures 

The course will cover the basic theory of invariant unitary, orthogonal and symplectic random matrix ensembles.

Some universality results and some applications of random matrix theory will also be discussed.