Course Descriptions

Metric spaces, elements of Hilbert and Banach spaces. Lebesgue measure and integral and Lebesgue's differentiation theorem. Classical Lp spaces. Fourier series.

This class will be an introduction to the fundamentals of

high-performance numerical computation. We will establish a basic

understanding of modern computer architectures (CPUs and accelerators,

memory hierarchies, interconnects) and of approaches to programming

these machines: single instruction-multiple data (SIMD), shared and

distributed memory parallelism. Issues such as load balancing,

communication, memory locality and synchronization will be covered and

illustrated in the context of numerical algorithms. Along the way you

will be exposed to important tools for high performance computing such

as profilers, schedulers, and version control systems. This will be a

hands-on class, with several computing assignments and a larger

project.

This course is focused on numerical methods for solving ordinary and partial differential equations, and will include topics such as: numerical approximation theory, orthogonal polynomials, the Fast Fourier Transform, finite differences, spectral methods, 2-point boundary value problems, elliptic PDEs and integral equations, high-order quadrature techniques, and fast structured matrix computations.

Prerequisites:
Linear algebra I, MATH-GA.2110.

Description:
Jordan form theorem, inner product spaces, spectral theory for self adjoint operators, matrix inequalities, convexity, duality, normed linear space, positive matrices.
 

Basic concepts including groups, rings, modules, polynomial rings, field theory, and Galois theory.

Introduction to the elementary methods of number theory. Topics: arithmetic functions, congruences, the prime number theorem, primes in arithmetic progression, quadratic reciprocity, the arithmetic of quadratic fields.

Survey of point-set topology. Funda-mental groups, homotopy, covering spaces. Singular homology, calculation of homSurvey of point-set topology. Funda-mental groups, homotopy, covering spaces. Singular homology, calculation of homology groups, applications. Homology and cohomology of manifolds. Poincar? duality. Vector bundles. De Rham 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.

Recommended Texts:

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

In this course, we will go over several landmark research papers
on mathematical modeling of biological phenomena. The papers will be
selected by using the following criteria:

1) they examine "big" quantitative data resulting from imaging, genetic
and biochemical approaches;
2) they use data science (including, but no limited to, statistics and ML)
to make sense of the data and inform models;
3) traditional applied math tools, like ODEs/PDEs, stochastic processes, etc,
are used to formulate and explore models;
4) modeling results have important implications for understanding biological
mechanisms.
Background mathematical tools and biological basics will be introduced.
The main goal of the course will be learn the art of mathematical modeling,
especially the millennial change to this art brought by big volumes of
quantitative data.

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

Analytic functions. Cauchy?s theorem and its many consequences. Fractional linear transformations and conformal mappings. Introduction to Riemann surfaces. The Riemann mapping theorems. Entire functions. Special functions.

Existence, uniqueness, and continuous dependence. Linear ODE. Stability of equilibria. Floquet theory. Poincar?-Bendixson theorem. Additional topics may include bifurcation theory, Hamiltonian mechanics, and singular ODE in the complex plane.

Local existence theory: Cauchy-Kowalewsky theorem. Laplaces equation, harmonic functions, maximum principle, single and double layer potential. Fourier transform and distributions. Sobolev spaces. Elliptic boundary value problems. The Cauchy problem for the heat equation, wave equation. Local well posedness for semilinear Cauchy problems.

This course provides a rigorous introduction to classical models and tools in applied statistics. About 50% of the lectures use exponential family structure to motivate generalized linear models and other useful applied techniques, and the other half cover models and methods in nonparametric statistics (~30%) and robust statistics (~20%). The main topics include exponential family, generalized linear model, bootstrap, survival analysis, missing data analysis, empirical Bayes, model selection, density estimation, nonparametric regression, local polynomials and splines, wavelets methods, isotonic regression, robust estimation, robust regression, and robust hypothesis testing.

 

This course provides a rigorous introduction to classical models and tools in applied statistics. About 50% of the lectures use exponential family structure to motivate generalized linear models and other useful applied techniques, and the other half cover models and methods in nonparametric statistics (~30%) and robust statistics (~20%). The main topics include exponential family, generalized linear model, bootstrap, survival analysis, missing data analysis, empirical Bayes, model selection, density estimation, nonparametric regression, local polynomials and splines, wavelets methods, isotonic regression, robust estimation, robust regression, and robust hypothesis testing.

This full-semester course integrates key elements of econometrics, data science, and machine learning in a financial context. Students will learn to model financial data using statistical techniques and computational methods while gaining hands-on experience with Python-based tools. The course emphasizes:
Financial Econometrics & Statistical Inference: Understanding linear regression frameworks, hypothesis testing, and model selection.
Supervised Learning: Implementing regression and classification models, optimizing performance through cross-validation and regularization.
Unsupervised Learning: Applying dimensionality reduction techniques such as PCA and SVD.
Machine Learning in Finance: Exploring algorithmic trading, risk modeling, and portfolio optimization using advanced methods like boosting, bagging, and deep learning.
Data Manipulation & Web Scraping: Handling real-world financial data, including structured and alternative datasets, using Python.
Hands-on assignments will reinforce theoretical concepts, ensuring students gain practical experience in implementing machine learning techniques for financial applications.
 

Prerequisites:
It is essential that students enrolling in this course have a solid understanding of multivariate calculus, linear algebra, calculus-based probability, and statistics. Familiarity with the standard Python stack is recommended but not required.

Course Description:

The goal of the first half of the semester of the course is for students to develop an understanding of the techniques of stochastic processes and stochastic calculus as it is applied in financial applications.
We begin by constructing the Brownian motion (BM) and the Ito integral, studying their properties. Then we turn to Ito’s lemma and Girsanov’s theorem, covering several practical applications. Towards the end of the course, we study the linkage between SDEs and PDEs through the Feynman-Kac equation.
In the second half of the semester, we turn to asset pricing and the trading of derivative securities using stochastic calculus techniques. Using tools and techniques from stochastic calculus, we cover (a) Black-Scholes-Merton option pricing; (b) the martingale approach to arbitrage pricing; (c) incomplete markets; and (d) 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 essential that students enrolling in this course have a solid understanding of multivariate calculus, linear algebra, and calculus-based probability. While not required, a one-semester course on derivative pricing, such as "Financial Securities and Markets," is recommended. Additionally, we suggest an intermediate course in mathematical statistics or engineering statistics as an optional prerequisite for this class.

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.

Theoretical aspects of portfolio construction and optimization, focusing on advanced techniques in portfolio construction, addressing the extensions to traditional mean-variance optimization including robust optimization, dynamical programming and Bayesian choice. Econometric issues associated with portfolio optimization, including estimation of returns, covariance structure, predictability, and the necessary econometric techniques to succeed in portfolio management will be covered.

Measuring and managing the risk of trading and investment positions: interest rate positions, vanilla options positions, and exotic options positions. The portfolio risk management technique of Value-at-Risk, stress testing, and credit risk modeling.

This full-semester course covers the theory and practice of financial market microstructure, with a focus on topics relevant to both buy-side trading and sell-side execution strategies. The course provides a detailed look at how modern electronic markets operate, how trading mechanisms shape liquidity and price formation, and how participants interact strategically in continuous limit order book environments.

 

We begin with an overview of trading mechanisms, order types, and market structures, followed by a detailed treatment of liquidity measurement and transaction cost estimation using trade and quote data. Early models of informed trading in dealer and batch markets are presented, including Glosten-Milgrom and Kyle-type models, with applications to empirical data.

The second half of the course focuses on limit order book models - both static and dynamic - with an emphasis on recent theoretical and empirical developments. Topics include propagator models of market impact, intraday volume and volatility modeling, execution in fragmented markets, empirical work on auction mechanisms, and the structure of futures and ETF markets. Emerging topics such as the role of AI, regulatory developments, and crypto and decentralized finance (DeFi) microstructure are also discussed.

Throughout the course, students gain hands-on experience with market data (including TAQ and limit order book feeds) through homework assignments that reinforce theoretical and empirical material covered in the lectures. Using Python, students perform statistical and computational analyses such as simulating trading environments, estimating price impact models, fitting limit order book models, and exploring auction dynamics.

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 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; 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.

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

Prerequisite: 

Familiarity with numerical methods and fluid dynamics.

 

The immersed boundary (IB) method is a general framework for the

computer simulation of flows with immersed elastic boundaries and/or

complicated geometry.  It was originally developed to study the fluid

dynamics of heart valves, and it has since been applied to a wide

variety of problems in biofluid dynamics, such as wave propagation in

the inner ear, blood clotting, swimming of creatures large and small,

and the flight of insects.  Non-biological applications include sails,

parachutes, flows of suspensions, and two-fluid or multifluid

problems.

 

Topics to be covered include: mathematical formulation of

fluid-structure interaction in Eulerian and Lagrangian variables, with

interaction equations involving the Dirac delta function;

discretization of the structure, fluid, and interaction equations,

including energy-based finite-element or finite-difference

discretization of the structure equations, finite-difference

discretization of the fluid equations, and IB delta functions with

specified mathematical properties; a simple but effective method for

adding mass to an immersed boundary; numerical simulation of rigid

immersed structures or immersed structures with rigid parts; IB

methods for immersed filaments with bend and twist; and a stochastic

IB method for thermally fluctuating hydrodynamics within biological

cells, and for the direct numerical simulation of osmotic phenomena.

 

Numerical issues to be covered include stability analysis of the IB

method, and the challenge of making the IB method implicit so that

larger time steps can be taken.  A Fourier-Spectral version of the IB

method will also be discussed.

 

Course requirements include homework and a project, but no exam.

Students may collaborate on the homework and on the project.

Project presentations to the class are encouraged.

 

Text:

The Immersed Boundary Method.  Lecture notes freely available at:

http://www.math.nyu.edu/faculty/peskin/ib_lecture_notes/index.html

 

Abstract:  

Many natural questions of statistical physics deal with dynamics in random landscapes, in high dimension.

Their behavior in large time is very rich and has kept busy a few generations of physicists and mathematicians. 

I will cover quickly the state of knowledge today for the behavior in very long time scales and possibly aging for some of these classical questions, including dynamics of spin glasses, biased random walks on percolation clusters, for various models of random walks among random obstacles and the simple Bouchaud Trap Model. 

If time permits, I will then turn to the question of the dynamics of optimization for the natural algorithms of Machine Learning, which are again random dynamics in high-dimensional and random landscapes. There the natural time scales have to be much shorter. I will describe recent work to explain what is now understood in a few interesting learning questions.

I will naturally list a few open problems in these two related domains

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

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

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

Required text:

Probability and Random Processes, 3rd edition by G.Grimmett and D. Stirzaker, Oxford Press 2001 (Note: this is NOT the newer 4th edition).