Course Descriptions

MATHGA.1002001 Multivariable Analysis
3 Points, Mondays, 7:109:00PM, Gilles Francfort
Differentiation and integration for vectorvalued functions of one and several variables: curves, surfaces, manifolds, inverse and implicit function theorems, integration on manifolds, Stokes' theorem, applications.

MATHGA.1420001 Introduction To Math Analysis II
3 Points, Thursdays, 5:107:00PM, Gaoyong Zhang
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.

MATHGA.1420002 Introduction To Math Analysis II
3 Points, Fridays, 11:0012:50PM, Jumageldi Charyyev
Description TBA 
MATHGA.2012002 Advanced Topics In Numerical Analysis: High Performance Computing
3 Points, Mondays, 1:253:15PM, Benjamin Peherstorfer
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 handson 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.

MATHGA.2012003 Advanced Topics In Numerical Analysis: Randomized Numerical Linear Algebra
3 Points, Mondays, 11:0012:50PM, Michael O'Neil and Jonathan Weare
Prerequisites:
Graduatelevel 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. 
MATHGA.2012004 Advanced Topics In Numerical Analysis: Convex & Non Smooth Optimization
3 Points, Mondays, 5:107:00PM, Michael Overton
Description TBA 
MATHGA.2020001 Numerical Methods II
3 Points, Tuesdays, 5:107:00PM, Leslie Greengard
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 RungeKutta 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.

MATHGA.2048001 Scientific Computing In Finance
3 Points, Mondays, 5:107:00PM, Richard Lindsey and Mehdi Sonthonnax
Prerequisites:
Risk and Portfolio Management, Financial Securities and Markets, and Computing in Finance.Description:
This is a version of the course Scientific Computing (MATHGA 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 MATHGA 2043.001 and MATHGA 2048.001

MATHGA.2071001 Machine Learning & Computational Statistics (1st Half Of Semester)
1.5 Points, Mondays, 7:109:00PM, Ivailo Dimov
Prerequisites:
Multivariate calculus, linear algebra, and calculusbased probability. Students should also have working knowledge of basic statistics and machine learning (such as what is covered in Data Science & DataDriven Modeling).Course Description:
This halfsemester course (a natural sequel to the course “Data Science & DataDriven 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. Crossvalidation 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. Handson homework form an integral part of the course, where we analyze realworld datasets and model them in Python using the machine learning techniques discussed in the lectures. 
MATHGA.2110001 Linear Algebra I
3 Points, Tuesdays, 5:107:00PM, Michael Lindsey
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, CayleyHamilton 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/ WileyInterscience. 
MATHGA.2120001 Linear Algebra II
3 Points, Wednesdays, 5:107:00PM, Zhiyuan Zhang
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: PrenticeHall/ Pearson Education.Plus: Extensive instructor’s class notes.

MATHGA.2140001 Algebra II
3 Points, Tuesdays, 7:109:00PM, Fedor Bogomolov
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 PeterWeyl 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: PrenticeHall/ Pearson Education.Recommended Texts:
 Lang, S. (2005). Graduate Texts in Mathematics [Series, Bk. 211]. Algebra (3^{rd}ed.). New York, NY: SpringerVerlag.
 Serre, J.P. (1977). Graduate Texts in Mathematics [Series, Bk. 42]. Linear Representations of Finite Groups. New York, NY: SpringerVerlag.
 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: SpringerVerlag.
 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: SpringerVerlag.

MATHGA.2210001 Introduction To Number Theory I
3 Points, Wednesdays, 3:205:05PM, Yuri Tschinkel
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, zetafunction, prime number theorem, number fields, rings of integers, Dedekind zetafunction, introduction to analytic techniques: circle method, sieves. 
MATHGA.2320001 Topology II
3 Points, Mondays, 7:109:00PM, Sylvain Cappell
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.

MATHGA.2360001 Differential Geometry II
3 Points, Wednesdays, 1:253:15PM, Chao Li
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).

MATHGA.2420001 Advanced Topics: Seminar In AOS
3 Points, Fridays, 3:455:00PM, Edwin Gerber
Description TBA 
MATHGA.2420002 Advanced Topics In PDE: Mean Field Limits (February 3 Thru March 24th)
1.5 Points, Thursdays, 3:205:05PM, Sylvia Serfaty
Prerequisite:
Knowledge of PDE and probability theory.Description:
This course will be concerned with recent developments in the derivation of meanfield 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 meanfield 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.
.

MATHGA.2420003 Advanced Topics In Probability: Directed Polymers (2nd Half Of Semester)
1.5 Points, Tuesdays, 11:0012:50PM, Yuri Bakhtin
Description:
The term “directed polymers” refers to a class of Gibbs distributions on polymer chains with local selfinteraction 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.

MATHGA.2420004 Advanced Topics In Applied Mathematics: Working Group In Modeling And Simulation
3 Points, Thursdays, 12:302:00PM, Aleksandar Donev and Miranda HolmesCerfon and Leif Ristroph
Description TBA 
MATHGA.2420005 Advanced Topics In Geometry: Quantitative Geometry And Topology (1st Half Of Semester)
1.5 Points, Tuesdays, 11:0012:50PM, Robert Ji Wai Young
Prerequisites:
Some familiarity with manifolds and differential geometry.Description:
This course is an introduction to quantitative and asymptotic techniques in geometry and topology. Quantitativegeometry 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. 
MATHGA.2420006 Advaned Topics: Introduction To The Theory Of Elliptic Curves (1st Half Of Semester)
1.5 Points, Mondays, 11:0012:50PM, Alena Pirutka
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. 
MATHGA.2420007 Advanced Topics: Quantitative Unique Continuation (2nd Half Of Semester)
1.5 Points, Wednesdays, 1:253:15PM, Fanghua Lin
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..

MATHGA.2420008 Advanced Topics In Probability: Moment Generating Functions And Riemann Hypothesis (1st Half Of Semester)
1.5 Points, Wednesdays, 1:253:15PM, Charles Newman
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 RodgersTao (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), 595614). 
MATHGA.2420009 Advanced Topics In PDE: Hyperbolic And Dispersive PDEs: Asymptotic Methods (1st Half Of Semester)
1.5 Points, Mondays, 1:253:15PM, Jalal Shatah
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 (MATHGA.2500).
This course will cover basic material that may not be covered in a regular PDE sequence,
 Waves and rays. This will be a refresher class on ray tracing, dispersion, characteristics and bicharacteristics.
 Energy method. This will include, Noether’s theorem, energy estimates for symmetric hyperbolic equations.
 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.
 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.
 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.

MATHGA.2420010 Advanced Topics In Anaylsis: AlmostPeriodic Functions And Generalized Harmonic Anaylsis (2nd Half Of Semester)
1.5 Points, Thursdays, 3:205:05PM, Sinan Gunturk
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. 
MATHGA.2420011 Advanced Topics In Numerical Analysis: Bayesian Inverse Problems (1st Half Of Semester)
1.5 Points, Wednesdays, 5:107:00PM, Georg Stadler
Prerequisites:
Graduatelevel Linear Algebra, Basic Probability, Numerical Methods. Having taken the Inverse Problems course, which mainly focused on deterministic problems and PDEconstrained optimization in the fall 2021 is an advantage, but not necessary.Description:
This halfcourse 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 KarhunenLoeve 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. 
MATHGA.2440001 Real Variables II
3 Points, Tuesdays, 3:205:05PM, Raghu Varadhan
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. 
MATHGA.2460001 Complex Variables II
3 Points, Tuesdays, 5:107:00PM, Percy Deift
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 SchwarzChristoffel 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: McGrawHill. 
MATHGA.2470001 Ordinary Differential Equations
3 Points, Wednesdays, 11:0012:50PM, Fengbo Hang
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 SturmLiouville theory. Comparison theorem for second order equations. Asymptotic behavior of nonlinear systems. Perturbation theory and PoincaréBendixson 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. 
MATHGA.2500001 Partial Differential Equations
3 Points, Wednesdays, 3:205:05PM, Scott Armstrong
Prerequisites:
MATHGA 2490 (Introduction to Partial Differential Equations) and MATHGA 2430 (Real Variables), or equivalent background. Masters students should consult the course instructor before registering for this class.Description:
Undergraduate and MSlevel 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 selfadjointness, 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, steepestdescent pde's); viscosity solutions of firstorder equations.Main Texts:
 L.C. Evans, Partial Differential Equations, American Mathematical Society
 M. Renardy and R. Rogers, An Introduction to Partial Differential Equations, SpringerVerlag

MATHGA.2550001 Functional Analysis
3 Points, Mondays, 9:0010:50AM, Percy Deift

MATHGA.2620001 Advanced Topics In PDE: Elliptic Free Boundary Problems
3 Points, Tuesdays, 1:253:15PM, Fanghua Lin
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:
 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.ã€€
 Various research and survey papers by L.Caffarelli and others.

MATHGA.2704001 Applied Stochastic Analysis
3 Points, Wednesdays, 1:253:15PM, Miranda HolmesCerfon
Prerequisites:
Basic Probability (or equivalent masterslevel probability course), Linear Algebra (graduate course), and (beginning graduatelevel) 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

MATHGA.2708001 Algorithmic Trading & Quantitative Strategies
3 Points, Tuesdays, 7:109:00PM, Petter Kolm and Lee Maclin
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, backtesting 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. 
MATHGA.2710001 Mechanics
3 Points, Tuesdays, 11:0012:50PM, Robert Kohn
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: onedimensional 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 HamiltonJacobi 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.

MATHGA.2751001 Risk & Portfolio Management
3 Points, Wednesdays, 7:109:00PM, Gordon Ritter
Prerequisites:
Multivariate calculus, linear algebra, and calculusbased 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, delta1 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, extremevalue 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 calculusbased probability. 
MATHGA.2752001 Active Portfolio Management
3 Points, Wednesdays, 5:107:00PM, Jerome Benveniste
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 meanvariance 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. 
MATHGA.2753001 Advanced Risk Management
3 Points, Wednesdays, 7:109:00PM, Ken Abbott and Irena Khrebtova
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 valueatrisk 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. 
MATHGA.2755001 Project & Presentation
3 Points, Wednesdays, 5:107:00PM, Petter Kolm
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. 
MATHGA.2791001 Financial Securities And Markets
3 Points, Mondays, 7:109:00PM, Alireza Javaheri
Prerequisites:
Multivariate calculus, linear algebra, and calculusbased 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; riskneutral valuation; the lognormal hypothesis; binomial trees; the BlackScholes formula and applications; the BlackScholes partial differential equation; American options; one factor interest rate models; swaps, caps, floors, swaptions, and other interestbased 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 calculusbased probability. 
MATHGA.2793001 Dynamic Asset Pricing (2nd Half Of Semester)
1.5 Points, Mondays, 7:109:00PM, Bruno Dupire and Montacer Essid
Prerequisites:
Calculusbased 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) BlackScholesMerton 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 termstructure models such as Vasicek and HullWhite. It is important that students taking this course have good working knowledge of calculusbased 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. 
MATHGA.2798001 Interest Rate & Fx Models
3 Points, Thursdays, 5:107:00PM, Fabio Mercurio and Leif Andersen and Irena Khrebtova
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 fixedincome models most frequently used in the finance industry, and their applications to the pricing and hedging of interestbased derivatives. The second part covers the foreign exchange derivatives markets, with a focus on vanilla options and firstgeneration (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 widelyused derivative instruments. 
MATHGA.2799001 Modeling And Risk Management Of Bonds And Securitized Products (2nd Half Of Semester)
1.5 Points, Thursdays, 7:109:00PM, Rodney SunadaWong
Prerequisites:
Stochastic Calculus, and Financial Securities and Markets or equivalent knowledge of basic bond mathematics and bond risk measures (duration and convexity).Description:
This halfsemester course is designed for students interested in Fixed Income roles in frontoffice 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 MortgageBacked Securities. Students will build pricing models for mortgages, passthroughs, 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 Peertopeer / MarketPlace Lending.

MATHGA.2800001 Trading Energy Derivatives (1st Half Of Semester)
1.5 Points, Thursdays, 7:109:00PM, Ilia Bouchouev
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 higherorder options Greeks. Examples and case studies are based on actual market episodes using real market data. 
MATHGA.2801001 Advanced Topics In Equity Derivatives (2nd Half Of Semester)
1.5 Points, Wednesdays, 7:109:00PM, Alireza Javaheri
Prerequisites:
Financial Securities and Markets, Stochastic Calculus, and Computing in Finance or equivalent programming experience.Description:
This halfsemester 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 metamathematical topics such as the practical and regulatory aspects of setting up a hedge fund will also be covered. 
MATHGA.2802001 Market Microstructure (1st Half Of Semester)
1.5 Points, Wednesdays, 7:109:00PM, Merrell Hora
Prerequisites:
Financial Securities and Markets, Risk and Portfolio Management, and Computing in Finance or equivalent programming experience.Description:
This is a halfsemester course covering topics of interest to both buyside traders and sellside execution quants. The course will provide a detailed look at how the trading process actually occurs and how to optimally interact with a continuous limitorder 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 pretrade market impact estimation, posttrade slippage analysis, optimal execution strategies and dynamic noarbitrage models. We cover AlmgrenChriss model for optimal execution, Gatheral’s nodynamicarbitrage 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 limitorderbook simulator and analyze real NYSE TAQ data.

MATHGA.2840004 Advanced Topics In Applied Math: The Mechanics Of Active Matter
3 Points, Wednesdays, 9:0010:50AM, Michael Shelley
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 largescale 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 coarsegraining the microscopic dynamics to derive a macroscopic model, typically partial differential equations that reflect conservation laws for mass and momentum, plus evolution equations for socalled 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. 
MATHGA.2840005 Advanced Topics In Applied Math: Mathematical Tools For Data Science
3 Points, Thursdays, 2:003:40PM, Carlos FernandezGranda
Description TBA 
MATHGA.2840006 Advanced Topics In Applied Math: Mathematical Tools For Data Science Lab
3 Points, Mondays, 3:454:45PM, TBA
Description TBA 
MATHGA.2851001 Advanced Topics In Math Biology: Physiological Control Mechanisms
3 Points, Wednesdays, 11:0012:50PM, Charles Peskin
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 wholeorganism level and also from the cellular level. The wholeorganism 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 wholeorganism mechanism, since it involves circulating hormones, and of particular mathematical interest since it involves symmetry breaking and the control of an integervalued quantity). Celllevel topics are the control of cell volume (which has as a byproduct 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 
MATHGA.2856001 Adv Top In Math Physiology: Neuronal Networks
3 Points, Thursdays, 1:253:15PM, John Rinzel
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, decisionmaking, feature detection in sensory systems, cortical dynamics (gamma and other oscillations, updown 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. 
MATHGA.2901001 Essentials Of Probability
3 Points, Wednesdays, 7:109:00PM, Charles Newman
Prerequisites:
Calculus through partial derivatives and multiple integrals; no previous knowledge of probability is requiredDescription:
The onesemester 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

MATHGA.2902002 Stochastic Calculus Optional Problem Session
3 Points, Thursdays, 5:307:00PM, TBA
Description TBA 
MATHGA.2903001 Stochastic Calculus (2nd Half Of Semester)
1.5 Points, Tuesdays, 7:109:00PM, Alexey Kuptsov
Prerequisites:
MATHGA 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. FeynmanKac and CameronMartin Formulas. Applications as time permits.Optional Problem Session:
Thursdays, 5:307:00.Text:
Durrett, R. (1996). Probability and Stochastics Series [Series, Bk. 6]. Stochastic Calculus: A Practical Introduction. New York, NY: CRC Press. 
MATHGA.2912001 Probability Theory II
3 Points, Wednesdays, 9:0010:50AM, Yuri Bakhtin
Prerequisites:
Probability Limits Theorems 1Description:
Stochastic processes in continuous time. Brownian motion. Poisson process. Processes with independent increments. Stationary processes. Semimartingales. Markov processes and the associated semigroups. 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

MATHGA.2931001 Advanced Topics In Probability: Concentration Of Measure, Theory And Applications
3 Points, Thursdays, 11:0012:50PM, Eyal Lubetzky and Paul Bourgade
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 logconcave 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 sineGordon model. 
MATHGA.2962001 Mathematical Statistics
3 Points, Wednesdays, 6:008:30PM, Jonathan Weare
Description TBA 
MATHGA.3004001 Atmospheric Dynamics
3 Points, Tuesdays, 3:205:05PM, Olivier Pauluis
Description TBA 
MATHGA.3011001 Advanced Topics In Geophysical Fluid Dynamics
3 Points, Wednesdays, 11:0012:50PM, Shafer Smith and Oliver Buhler
Description TBA