Course Descriptions

Rigorous treatment of limits and continuity. Riemann integral. Taylor series. Absolute and uniform convergence. Elements of ordinary and partial differential equations. Functions of several variables and their derivatives. The implicit function theorem, optimization, and Lagrange multipliers. Theorems of Gauss, Stokes, and Green. Fourier series and integrals

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 lower complexity bound and optimal gradient method, the alternating direction method of multipliers, the nuclear norm and matrix completion, 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, BFGS and the stochastic gradient method, for nonsmooth, nonconvex optimization. Homework will be assigned, both mathematical and computational.

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.

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.

Theory of curves and surfaces. Riemannian geometry: manifolds, differential forms, and integration. Covariant derivatives and curvature. Differential geometry in the large. Curvature, geodesics, Jacobi fields, comparison theorems, and Gauss-Bonnet theorem.

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.

Course Description:

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.

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

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).

The course is targeted at Mathematics PhD students. 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.

What effects the large scale circulation of the atmosphere?   Like the antiquated heating system of a New York apartment, solar radiation unevenly warms the Earth, leading to gradients in energy in both altitude and latitude.  But unlike the simple convection of air in your drafty home, the effects of rotation, stratification, and moisture lead to exotic variations in weather and climate, giving us something to chat about over morning coffee ... and occasionally bringing modern life to a standstill.

The goals of this course are to describe and understand the processes that govern atmospheric fluid flow, from the Hadley cells of the tropical troposphere to the polar night jet of the extratropical stratosphere, and to prepare you for research in the climate sciences.   We will explore how stratification and rotation regulate the atmosphere's response to gradients in heat and moisture.  Much of our work will be to explain the zonal mean circulation of the atmosphere, but in order to accomplish this, we’ll need to learn a great deal about deviations from the zonal mean: eddies and waves.  It turns out that eddies and waves, planetary, synoptic (weather system size) and smaller in scale, are the primary drivers of the zonal mean circulation. There will also be a significant numerical modeling component to the course.  You will learn how to run atmospheric models on NYU's High Performance Computing facility, and then design and conduct experiments to test the theory developed in class for a final course project.  

Some experience with fluids (geophysical fluids in particular) is extremely helpful, but the course will be taught so that a student familiar with multidimensional calculus and differential equations can fully participate.  

This course explores the application of data-driven techniques in climate science, focusing on statistical methods, machine learning, and data assimilation. Students will learn how to analyze large climate datasets, extract meaningful patterns, and develop predictive models. Through hands-on exercises, we will cover topics such as observational data processing, feature selection, and neural networks for climate prediction. The course is designed for students with an interest in leveraging data science to address key challenges in climate research.