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.

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: 
Graduate-level proficiency in multivariate calculus, probability theory, linear algebra. Programming experience recommended, but not required.

Description: 
Numerical linear algebra has been the bedrock of computational science for the past several decades. The ability to solve linear systems, find eigenvalues/eigenvectors, and compute matrix factorizations has allowed many problems in the physical sciences to be solved quite accurately (even nonlinear ones, as often the nonlinear term can be iterated on in order to solve the overall system). However, as the dimension of problems grows, standard dense numerical linear algebra algorithms will fail to keep up due to their inherent computational complexity, often quadratic or cubic in the dimension of the matrix. Recently, methods of randomization have been introduced which can partially overcome this computational complexity growth, oftentimes even avoiding a significant loss in precision. This course will serve as an overview of this class of randomized techniques, how they work in practice, the underlying analysis of accuracy guarantees, and modern applications of such techniques to problems in math, physics, chemistry, and statistics.

This course will cover fundamental methods that are essential for the numerical solution of differential equations. It is intended for students familiar with ODE and PDE and interested in numerical computing; computer programming assignments in MATLAB will form an essential part of the course. The course will introduce students to numerical methods for (1) ordinary differential equations (explicit and implicit Runge-Kutta and multistep methods, convergence and stability); (2) elliptic partial differential equations such as the Poisson eq. (finite difference, finite element and integral equation methods); (3) parabolic  and hyperbolic equations such as the heat or wave equation (finite difference and finite volume methods). We will also discuss spectral methods and the FFT, exponential temporal integrators, and multigrid iterative solvers.

Prerequisites:
Risk and Portfolio Management, Financial Securities and Markets, and Computing in Finance.

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

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

Prerequisites: 
Multivariate calculus, linear algebra, and calculus-based probability. Students should also have working knowledge of basic statistics and machine learning (such as what is covered in Data Science & Data-Driven Modeling).

Course Description:   
This half-semester course (a natural sequel to the course “Data Science & Data-Driven Modeling”) examines techniques in machine learning and computational statistics in a unified way as they are used in the financial industry. We cover supervised learning (regression and classification using linear and nonlinear models), specifically examining splines and kernel smoothers, bagging and boosting approaches; and how to evaluate and compare the performance of these machine learning models. Cross-validation and bootstrapping are important techniques from the standard machine learning toolkit, but these need to be modified when used on many financial and alternative datasets. In addition, we discuss random forests and provide an introduction to neural networks. Hands-on homework form an integral part of the course, where we analyze real-world datasets and model them in Python using the machine learning techniques discussed in the lectures.

Prerequisites:
Undergraduate linear algebra or permission of the instructor.

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

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

Prerequisites:
Linear Algebra I or permission of the instructor.

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

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

Plus: Extensive instructor’s class notes.

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

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

Recommended Texts:

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

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

 Description:
This graduate course will cover several analytic techniques in number theory, as well as properties of number fields and their rings of integers. Topics include: primes in arithmetic progressions, zeta-function, prime number theorem, number fields, rings of integers, Dedekind zeta-function, introduction to analytic techniques: circle method, sieves.

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.

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).

Prerequisite:
Knowledge of PDE and probability theory.

Description:
This course will be concerned with recent developments in the derivation of mean-field evolution PDEs from discrete systems of particles with pair interaction potentials, with or without noise terms. Motivations are numerous and come from physics, biology and social sciences, convergence of particle methods and stochastic gradient descent, neural networks, etc.

We will review the classical theory in the regular case, discuss the relative entropy based methods, the modulated energy approach for singular interactions, and the characterization of fluctuations around the mean-field limit. It will be a cross between a course and a reading group in the sense that part of the material will be learnt through student presentations. 

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Description:

The term “directed polymers” refers to a class of Gibbs distributions on polymer chains with local self-interaction and interaction with the random environment. We will mostly work with polymers with discrete steps in continuous environments and study localization issues, thermodynamic limits, the dynamic polymer approach, applications to the ergodic theory of stochastic PDEs.

Prerequisites:
Some familiarity with manifolds and differential geometry.

Description:
This course is an introduction to quantitative and asymptotic techniques in geometry and topology. Quantitative geometry studies how the size or complexity of a map or space affects its geometry and topology, and how maps and spaces behave at large scales or under various asymptotics. We will introduce some of the ideas and methods of quantitative geometry and apply them to questions in areas like geometric group theory and systolic geometry.

Description:
This course will be devoted to the study of elliptic curves over various fields: finite fields, fields of rational or complex numbers. For elliptic curves defined over finite fields we will also discuss applications to cryptography. 

Description:
In this seven lectures we shall tour over various results concerning the unique continuation: Analytic unique continuation, uniqueness of Cauchy problems, 3 spheres theorem, monotonicity of frequency and doubling estimates, and some recent results including that by Logunov and Malinnikova. 

References:
A list of papers would be presented in lectures. 

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Description:
This will be a seminar style course concerning the approach to the Riemann Hypothesisbased on whether the zeros of the Laplace transform of a certain function on the real line have only pure imaginary zeros. We will review results of Polya (1920's), de Bruijn (1950), Newman (1976) and Rodgers-Tao (2020).  We will also discuss possible connections with a theorem by Lee and Yang (1952) about Ising models. A recent review paper covering many of these topics is by Newman and Wu (Bull. Amer. Math. Soc. 57 (2020), 595-614).

Course Description:

This is an intermediate level PDE course. Students who would like to register for this class should have taken the graduate level PDE class (MATH-GA.2500).

This course will cover basic material that may not be covered in a regular PDE sequence,

1. Waves and rays. This will be a refresher class on ray tracing, dispersion, characteristics and bi-characteristics.
2. Energy method. This will include, Noether’s theorem, energy estimates for symmetric hyperbolic equations.
3. Scattering (including modified scattering) of solutions to semilinear equations. Prototype problems are the Schrodinger Equation and the Wave Equation. We will explore the effects of multi waves resonances on the asymptotic behavior of solutions.
4. WKB method and Geometric optics. We will explore these methods to construct good approximate solutions to linear non homogenous equations and to semi linear equations.
5. Wigner transform approach to high frequency asymptotics (if time permits)

The level of the material that will be presented in class is that of the books Hyperbolic Partial Differential Equations and Geometric Optics, by Jeffrey Rauch. Hyperbolic Partial Differential Equations, by Peter Lax.

Prerequisites:
Working knowledge of Fourier analysis and some elementary functional analysis.

Description:
Theory of almost periodic functions (classical theories of Bohr, Bochner, Besicovitch, and von Neumann) and generalized harmonic analysis of Wiener. Applications to differential equations and dynamical systems, number theory, and signal processing.

Prerequisites:
Graduate-level Linear Algebra, Basic Probability, Numerical Methods. Having taken the Inverse Problems course, which mainly focused on deterministic problems and PDE-constrained optimization in the fall 2021 is an advantage, but not necessary.

Description:
This half-course will cover theoretical and computational aspects of Bayesian inverse problems. We discuss how to define probability distributions that model uncertainty given observational data and physical or statistical models. In particular, we plan to cover Bayes’ formula for linear and nonlinear problems, priors, Gaussian random fields, Matern kernels, and Karhunen-Loeve expansions. We focus on problems where the likelihood function is expensive to evaluate and on approximations of the posterior distribution such as MAP estimation, Laplace approximation, and sampling methods. While the basics of Markov chain Monte Carlo sampling will be covered, this is not a main focus of the course. Assignments will be a mixture of practical and theoretical problems.

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

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

Prerequisites:
Complex Variables I (or equivalent).

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

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

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

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

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

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

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

Main Texts:

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

Description:
The course will present various ideas and methods in the theory of elliptic free boundary problems with particular emphasis on the regularity of free boundaries. Topics to be discussed include: variational inequalities, classical obstacle problems, regularity of solutions and regularity of free boundaries, structure of singularities and their generalizations.  

References:

1. D. Kinderlehrer and G. Stampacchia, . An Introduction to Variational Inequalities and Their Applications (Classics in Applied Mathematics 31). Society for Industrial and Applied Mathematics, 2000.　
2. Various research and survey papers by L.Caffarelli and others.

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 and Portfolio Management, or equivalent.

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

Description:
This course provides 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.

Prerequisites:
Multivariate calculus, linear algebra, and calculus-based probability. 

Description:
Risk management is arguably one of the most important tools for managing investment portfolios and trading books and quantifying the effects of leverage and diversification (or lack thereof). This course is an introduction to portfolio and risk management techniques for portfolios of (i) equities, delta-1 securities, and futures and (ii) basic fixed income securities. A systematic approach to the subject is adopted, based on selection of risk factors, econometric analysis, extreme-value theory for tail estimation, correlation analysis, and copulas to estimate joint factor distributions. We will cover the construction of risk measures (e.g. VaR and Expected Shortfall) and portfolios (e.g. portfolio optimization and risk). As part of the course, we review current risk models and practices used by large financial institutions. It is important that students taking this course have good working knowledge of multivariate calculus, linear algebra and calculus-based probability.

Prerequisites:
Risk & Portfolio Management and Computing in Finance.

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

Prerequisites:
Financial Securities and Markets, and Computing in Finance or equivalent programming experience.

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

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

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

Prerequisites:   
Multivariate calculus, linear algebra, and calculus-based probability.

 Description:    
This course provides a quantitative introduction to financial securities for students who are aspiring to careers in the financial industry. We study how securities traded, priced and hedged in the financial markets. Topics include: arbitrage; risk-neutral valuation; the log-normal hypothesis; binomial trees; the BlackScholes formula and applications; the Black-Scholes partial differential equation; American options; one factor interest rate models; swaps, caps, floors, swaptions, and other interest-based derivatives; credit risk and credit derivatives; clearing; valuation adjustment and capital requirements. It is important that students taking this course have good working knowledge of multivariate calculus, linear algebra and calculus-based probability.

Prerequisites:  
Calculus-based probability, Stochastic Calculus, and a one semester course on derivative pricing (such as what is covered in Financial Securities and Markets).

Description:   
This is an advanced course on asset pricing and trading of derivative securities. Using tools and techniques from stochastic calculus, we cover (1) Black-Scholes-Merton option pricing; (2) the martingale approach to arbitrage pricing; (3) incomplete markets; and (4) the general option pricing formula using the change of numeraire technique. As an important example of incomplete markets, we discuss bond markets, interest rates and basic term-structure models such as Vasicek and Hull-White. It is important that students taking this course have good working knowledge of calculus-based probability and stochastic calculus. Students should also have taken the course “Financial Securities and Markets” previously. In addition, we recommend an intermediate course on mathematical statistics or engineering statistics as an optional prerequisite for this class.

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

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

Prerequisites:
Stochastic Calculus, and Financial Securities and Markets or equivalent knowledge of basic bond mathematics and bond risk measures (duration and convexity).

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

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

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

Prerequisites:
Financial Securities and Markets, and Stochastic Calculus.

Description:
The course provides a comprehensive overview of most commonly traded quantitative strategies in energy markets. The class bridges quantitative finance and energy economics covering theories of storage, net hedging pressure, optimal risk transfer, and derivatives pricing models. Throughout the course, the emphasis is placed on understanding the behavior of various market participants and trading strategies designed to monetize inefficiencies resulting from their activities and hedging needs. We discuss in detail recent structural changes related to financialization of energy commodities, crossmarket spillovers, and linkages to other financial asset classes. Trading strategies include traditional risk premia, volatility, correlation, and higher-order options Greeks. Examples and case studies are based on actual market episodes using real market data.

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

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

Prerequisites:
Financial Securities and Markets, Risk and Portfolio Management, and Computing in Finance or equivalent programming experience.

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

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

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

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

Prerequisites:
Students should have taken PDE, Methods of Applied Math, and hopefully Fluid Dynamics and Computational PDE.

Description:
I will introduce the area of active matter where large-scale dynamics of a material, synthetic or biological, arises from its active microscopic constituents performing work on the system. Bacterial suspensions and cytoskeletal networks are two examples arising in biology. I will discuss methods for coarse-graining the microscopic dynamics to derive a macroscopic model, typically partial differential equations that reflect conservation laws for mass and momentum, plus evolution equations for so-called conformation variables. Many of these PDE models have
surprising behaviors with novel instabilities and dynamics, and we will spend time studying such features. I will also discuss variants of such models arising in describing the dynamics of transport in living cells.  Final projects will be required of registered students.

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

Description: 
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 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), a genetic switch (in which positive feedback is used to make the expression of a gene bistable) and the control of the 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, 14th Edition, Elsevier 2020

Prerequisites.
Math: familiarity with applied differential equations or permission of instructor; Neurobiology: most background will be provided.

Description:
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, feature detection in sensory systems, cortical dynamics (gamma and other oscillations, up-down states, balanced states,…), time estimation and learning a rhythm.  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

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

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

Texts:

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

Prerequisites:
MATH-GA 2901 Basic Probability or equivalent.

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

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

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

Prerequisites:
Probability Limits Theorems 1

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

Recommended Text:

• Stochastic Processes by Bass
• Stochastic Processes by Varadhan (Courant Lecture Series in Mathematics, volume 16)
• Theory of Probability and Random Processes by Koralov and Sinai
• Brownian Motion and Stochastic Calculus by Karatzas and Shreve

Description:
Concentration of measure is a principle that informally states that Lipschitz functions that depend on many parameters are almost constant. We will cover many manifestations of this principle, for example for product and log-concave measures, with proofs based inductions and dynamics (martingales, Markov chains, PDEs). We will then consider extensions and modern applications in probability and mathematical physics, such as the analysis of the sine-Gordon model.