Text: Modern Techniques and Their Applications, 2nd Edition, Gerald Folland
Prerequisites: Undergraduate analysis at the level of Rudin's "Principles of Mathematical Analysis," chapters 1--9, including basic point set topology, metric space, continuity, convergence and uniform convergence of functions.
Description: Basic real variable function theory, measure and integration theory prerequisite to pure and applied analysis. Topics: Riemann and Lebesgue-Stieltjes integration; measure spaces, measurable functions and measure; Lebesgue measure and integration; convergence theorems for integrals; Lusin and Egorov theorems; product measures and Fubini-Tonelli theorem; Lp-spaces. Other topics and applications (such as Lebesgue's differentiation theorem, signed measures, absolute continuity and Radon-Nikodym theorem) as time permits.
Text: Function Theory of One Complex Variable: Third Edition (Graduate Studies in Mathematics) *by Robert E. Greene and Steven G. Krantz
Publisher: American Mathematical Society; 3rd edition (March 29, 2006)
ISBN-13: 978-0821839621
Prerequisites:
Description:
The beginning of the study of one complex variable is certainly one of
the loveliest mathematical subjects. It's the magnificent result of
several centuries of investigation into what happens when R is
replaced by C in "calculus". Among the consequences were the
creation of numerous areas of modern pure and applied mathematics, and
the clarification of many foundational issues in analysis and
geometry. Gauss, Cauchy, Weierstrass, Riemann and others found this
all intensely absorbing and wonderfully rewarding. The theorems and
techniques developed in modern complex analysis are of great use in
all parts of mathematics.
The course will be a rigorous introduction with examples and proofs
foreshadowing modern connections of complex analysis with differential
and algebraic geometry and partial differential equations. Acquaintance
with analytic arguments at the level of Rudin's Principles of
Modern Analysis is necessary. Some knowledge of algebra and
point-set topology is useful.
The course will include some appropriate review of relevant topics,
but this review will not be enough to educate the uninformed
student adequately. A previous "undergraduate" course in
complex analysis would also be useful though not necessary.
There are many excellent books about this subject. The official text
will be Function Theory of One Complex Variable, by Greene and
Krantz (American Math Society, 3rd edition, 2006). The
course will cover most of Chapters 1 through 5 of the text, parts of
Chapters 6 and 7, and possibly other topics. The titles of these
chapters follow.
1: Fundamental Concepts; 2: Complex Lines Integrals; 3: Applications
of the Cauchy Integral; 4: Meromorphic functions and Residues; 5: The
Zeros of a Holomorphic Function; 6: Holomorphic Functions as Geometric
Mappings.
Text: Reed-Simon I,II and E.B.Davies' book on Spectral theory
Prerequisites: Real Analysis, ODE, Linear Algebra
Description: The course will be focused on Spectral ans scattering theory techniques in
partial differential equations and mathematical physics.
It begins with review of basic notions and results from Functional
analysis, including the spectral theorem for unbounded operators.
Then, applications of compact operators in PDE and Spectral theory.
basic constructions of Hamiltonians in Quantum Mechanics. Large time
behavior: from Global existence to decay and scattering.
Examples and open problems.
Text: The course material will be mostly drawn from "Partial Differential
Equations" by Lawrence C. Evans, published by AMS, 2002; and "Partial
Differential Equations: Methods and Applications, Second Edition" by
Robert McOwen, Prentice Hall, 2002.
The former puts more emphasis on the theory, while the latter devotes some
spaceto working out applications of the theory in some interesting cases,
while leaving some full discussion of the theory to references.
You may obtain one or both of the texts. I will put these two and some
additional books on reserve in the math library:
* Jeffrey Rauch, Partial Differential Equations, Springer, 1997.
* G.B. Folland, Introduction to Partial Differential Equations,
Princeton University Press, 1976.
* F. John, Partial Differential Equations, 4th ed., Springer-Verlag,
1982.
Prerequisites: A strong background on advanced calculus involving multivariables (esp. Green's Theorem and Divergence Theorem). We will also use some basic facts of Lp function spaces and the usual integral inequalities (mostly completeness and Holder inequalities in L2 setting).
These topics are covered in the first semester graduate real variable course (640:501).
Description: This is the first half of a year-long introductory graduate course on PDE.
PDE is an enormously vast field. PDEs arise from very diverse fields: from
classical to modern physics, to more applied sciences such as material
sciences, mathematical biology, and signal processing, etc, and from the
more pure aspects of mathematics such as complex analysis and geometric
analysis.
This introductory course should be useful for students with a variety of
research interests: physics and mathematical physics, applied analysis,
numerical analysis, complex analysis, differential geometry, and, of
course, partial differential equations.
For an introductory course, it is more important to examine some important
examples to certain depth, to introduce the formulation, concepts, most
useful methods and techniques through such examples, than to concentrating
on presentation and proof of results in their most general form.
This is the way the course will be conducted.
The beginning weeks of the course aim to develop enough familarity and
experience to the basic phenomena, approaches, and methods in solving
initial/boundary value problems in the contexts of the classical prototype
linear PDEs of constant coefficients: the Laplace equation, the D'Alembert
wave equation, the heat equation and the Schroedinger equation.
Fourier series/eigenfunction expansions, Fourier transforms, energy
methods, and maximum principles will be introduced. More importantly,
appropriate methods are introduced for the purpose of establishing
qualitative, characteristic properties of solutions to each class of
equations. It is these properties that we will focus on later in extending
our beginning theories to more general situations, such as variable
coefficient equations and nonlinear equations.
Next we will discuss some notions and results that are relevant in
treating general PDEs: characteristics, non-characteristic Cauchy problems
and Cauchy-Kowalevskaya theorem, wellposedness.
Towards the end of the semester, we will begin some introductory
discussion on the extension of the energy methods to variable coefficient
wave/heat equations, and/or the Dirichlet principle in the calculus of
variations.
The purpose here is to motivate and introduce the notion of weak solutions and Sobolev spaces, which will be more fully developed in the second semester.
Subtitle: Hyperbolic PDEs and the Mathematical Foundations of Relativistic Physics
Text: None will be required, but the material is largely drawn from the following books by D. Christodoulou: The Action Principle and Partial Differential Equations (1999, Princeton University Press) And Mathematical Problems of General Relativity I (2008, European Mathematical Society)
Prerequisites: Common Sense, or permission of the instructor
Description: This will be a self-contained introductory course on hyperbolic partial differential equations and the geometry of space-time. We'll cover the basic theory of hyperbolic PDE's, emphasizing equations that arise in continuum physics, i.e. Maxwell's equations of electromagnetics, Euler's equations of fluid dynamics, and Einstein's equations of General Relativity, before focusing on the latter. Although welcomed and very much appreciated, no previous knowledge of physics or of partial differential equations is assumed. Some prior knowledge of differential geometry and functional analysis is helpful. The following is an outline of the course:
0. Hyperbolicity: Definitions of Hyperbolicity in the Theory of Maps, Geometry of Characteristics, Energy Estimates, Domain of Dependence Theorem, Local Well-posedness, Formation of Singularities.
1. The Geometry of Space-time: Causal structure, curvature and gravitation, the energy tensor and the matter equations of motion, the Einstein equations: variational formulation, derivation of the constraints and the evolution equations, maximal hypersurfaces and the Newtonian limit.
2. The Cauchy problem for Einstein Vacuum Equations: The Symbol and Characteristics of EVE. Local Existence in Wave Coordinates.
3. Formation of Singularities: The Penrose Singularity Theorem. Black Holes. Naked Singularities. Cosmic Censorship Conjectures.
4. Conservation Laws and Noether's Theorem: Lagrangian and Hamiltonian formulations. The Noether current in the theory of maps. Asymptotic flatness. The definition of global energy, momentum and angular momentum. Witten’s Proof of the Positive Energy Theorem.
5. Reduction under Symmetry: Homogeneous and Isotropic solutions. Spherically symmetric solutions. One- and Two-Killing Field Reductions of Einstein-Maxwell Equations. Ernst Potentials. Kerr and Newman solutions. Wave Maps.
Prerequisites: Students should have some background knowledge on Remannian geometry and real analysis.
Description: This course will be an introduction to the complex potential theory and
its applications in differential and algebraic geometry. The course will
begin with a slow-paced introduction to complex manifolds and the
constant scalar curvature equations on Riemann surfaces. Topics include
complex Monge-Ampere equations, the Calabi conjecture, pluripotential
theory and their relations to algebraic geometry and the Ricci flow.
Text: There will be no textbook for the course. Below are some nice references:
1. Marvin J. Greenberg, J. R. Harper, Algebraic Topology: A First course. Publisher: Westview Press .
2. A. Hatcher: algebraic topology, excellent collection of exercises. $30 in
paperback from Cambridge University Press, as well as online here
3. James W. Vick, Homology Theory: An Introduction to Algebraic Topology (Graduate Texts in Mathematics), Springer.
Prerequisites:
Description: This course will be an introduction to algebraic topology and basic manifold theory. The plan is to cover the following topics: fundamental group, Van Kampen's Theorem, covering spaces, simplicial and singular homology, cohomology, Brouwer's fixed-point theorem, and the Jordan-Brouwer separation theorem.
Text: An Invitation to Morse Theory - Liviu Nicolaescu,
Morse Theory - Milnor
Prerequisites: Algebraic topology as taught in math 540.
Description:
Morse theory is concerned with the relation between the topology of
differentiable manifolds and the critical points of real-valued functions defined
on those manifolds. The theory has applications to engineering, topology, differential geometry, PDE, geometric group theory, etc. There are even discrete versions of Morse theory that have applications to combinatorics.
The plan is to start with a brief discussion of differential manifolds, followed
by lectures from Nicolaescu's book with appropriate detours into Milnor. From the table of contents in Nicolaescu:
1.1 Local structure of Morse functions
1.2 Existence of Morse functions
2.1 Surgery, handle attachment, and cobordisms
2.2 The topology of sublevel sets
2.3 Morse inequalities
2.4 Morse-Smale Dynamics
2.5 Morse-Floer Homology
2.6 Morse-Bott functions
2.7 Min-Max theory
3.1 The Cohomology of Complex Grassmannians
3.2 Lefschetz Hyperplane Theorem
3.3 Symplectic manifolds and Hamiltonian flows
3.4 Morse theory of moment maps
3.5 S1-Equivariant localization
Text:Symmetry, Representations, and Invariants (Springer,
2009), by Roe Goodman and Nolan R. Wallach.
Prerequisites: Real analysis, linear algebra, abstract algebra, and elementary topology at
the beginning graduate or honors undergraduate level. No prior knowledge of
Lie algebras, Lie groups, or representation theory will be assumed.
Description: This course will be an introduction to Lie groups, Lie algebras, algebraic
groups, and finite-dimensional representation theory.
Course Outline:
The classical linear groups (complex and real forms)
Closed subgroups of GL(n, R) as real Lie groups
Linear algebraic groups and rational representations
Structure of complex classical groups and their Lie algebras: maximal torus,
roots, adjoint representation
Semisimple Lie algebras: structure and classification
Highest weight theory for representations of semisimple Lie algebras
Reductivity of classical groups
Additional topics from Lie groups and representation theory will be covered
depending on the interests of the class and the time available.
Text: Jacobson, "Basic Algebra", Volumes 1 and 2, second edition.
Students may be able to obtain used copies online (be sure it is the second edition) through addall.com or other websites. In the fall, photocopies will be available for purchase.
Prerequisites: Any standard course in abstract algebra for undergraduates.
Description: This is a standard graduate level course for beginners. We will consider a lot of examples.
Group Theory: Basic concepts, examples and theorems.
Groups acting on sets: orbits, cosets, stabilizers.
Basic Ring Theory: Fields, principal ideal domains (PIDs), matrix rings, division algebras, field of fractions.
Classification of finitely generated abelian groups, and modules over a PID. Application to linear algebra: rational and Jordan canonical form.
Bilinear Forms: Alternating and symmetric forms, invariants.
Text:Main text: J. Lepowsky and H. Li, Introduction to Vertex Operator
Algebras and Their Representations, Progress in Math., Vol. 227,
Birkhauser, Boston, 2003.
Supplementary text: I. Frenkel, J. Lepowsky and A. Meurman, Vertex
Operator Algebras and the Monster, Pure and Applied Mathematics,
Vol. 134, Academic Press, 1988.
Prerequisites: Only basic algebra. Also, some familiarity with Lie algebras would be helpful but is not necessary. Students potentially interested in this course are encouraged to consult me.
Description: This course will develop the axiomatic theory of vertex operator
algebras, including representation theory, from a contemporary point
of view. Important examples, including vertex operator algebras based
on lattices and on affine Lie algebras, and the vertex operator
algebra underlying ``monstrous moonshine,'' will be discussed. Using
this theory one can raise new questions and address a range of
problems relating to many areas of mathematics and to conformal field
theory in physics.
Please note: The Lie Groups/Quantum Mathematics Seminar, which will
meet Fridays at 11:45, will sometimes be related to the subjects of
the course. Students planning to take the course should also try to
arrange to attend the seminar, although the seminar will not be
required for the course.
Subtitle: Topics in algebraic geometry -- intro to toric geometry.
Text: No textbook required
Prerequisites: Some familiarity with varieties and schemes.
Description:
This is an introductory course in toric geometry. The pace will be dictated by the background of the participants who are expected to have some knowledge of algebraic, differential or arithmetic geometry. We will focus on the basics and examples.
The first part of the course will cover the following topics:
Convex rational polytopes, fans and rational polyhedral cones. Toric varieties and morphisms. Toric birational geometry; smooth toric varieties and divisors
Homogeneous coordinate ring construction. Singular toric varieties and resolution of singularities
The second part of the course will be tailored to the interests of the students in the class. Possible topics include:
Monomial ideals.
Cohomology rings of toric varieties and Betti numbers.
Text: David Eisenbud, Commutative Algebra with a view to Algebraic Geometry, Springer.
Prerequisites: Any graduate course in abstract algebra, or permission of the instructor
Description:
Commutative algebra is broadly concerned with solutions of structured sets of polynomial and analytic equations, and the study of pathways to methods and algorithms that facilitate the efficient processing in large scale
computations with such data.
This course will be an introduction to commutative algebra, with applications to algebraic geometry, combinatorics and computational algebra.
Topics:
1) (If needed by audience) Noetherian rings: Rings of polynomials,
Hilbert basis theorem, Dedekind domains, Finitely generated algebras
over fields, Noether normalization, Nullstellensatz.
2) The first part of the course will treat basic notions and results---chain conditions, prime ideals, flatness, Krull dimension,
Hilbert functions.
3) Required material from Homological Algebra--such as the derived functors of Hom and tensor products--will be given in class, not assumed.
4) The other half of the course will study in more detail rings of
polynomials and its geometry, and Groebner bases. It will open the
door to computational methods in algebra (a few will be studied).
Some other applications will deal with counting solutions of certain linear
diophantine equations.
This is an introductory course in Mathematical Logic aimed at graduate
students in mathematics rather than prospective logicians.
In Set Theory, we will discuss basic topics such as ordinals,
cardinals and the various equivalents of the Axiom of Choice; as well
as more advanced topics such as the club filter and stationary sets.
(The latter are needed to prove results such as the existence of
2ℵ1 nonisomorphic dense linear
orders of cardinality ℵ1.)
In Model Theory, we will begin with basic results such as the
Completeness and Compactness Theorems. Then we will cover some more
advanced topics focusing on the structure of the countable models of
a complete theory in a countable language.
M. Ram Murty and Jody Esmonde, Problems in Algebraic Number Theory, Springer-Verlag Graduate Texts in Mathematics, volume 190.
Other reference books:
Lang, Algebraic Number Theory, Springer GTM, volume 110
Ireland and Rosen, A Classical Introduction to Modern Number Theory, GTM,
volume 84.
Prerequisites: *Permission of instructor required for students not enrolled in the mathematics Ph.D. program.
Description:Topics: This will be an introductory graduate course, designed to cover the main prerequisites for our further graduate course offerings, such as
Iwaniec's usual graduate courses. I will start with algebraic topics, but present them with an analytic perspective. I will then turn to analytic
techniques proper, such as the proof of the prime number theorem. If time
permits, we will discuss the basics of modular and maass forms, and perhaps elliptic curves. My overall aim is to cover many topics and describe the role of the key ideas involved.
1. Elementary Number Theory
2. Euclidean Rings
3. Algebraic Numbers and Integers
4. Integral Bases
5. Dedekind Domains
6. The Ideal Class Group
7. Quadratic Reciprocity
8. The Structure of Units
9. Higher Reciprocity Laws
10. Zeta Functions
11. Prime Number Theorem
Subtitle: Analytic Theory of Automorphic L-functions
Text: In many cases I will distribute my personal notes on the subjects during the course. There is no one book which covers all the material, so I shall refer to specific publications when needed.
Prerequisites: knowledge of the spectral theory will be helpful
Description: This course is a continuation of the course given in the Fall 2009 on Spectral Theory of Automorphic Forms (see the description pasted below). Although the knowledge of the spectral theory will be helpful, a new student may still be able to follow and learn the new material by simply accepting basic theorems without studying their development. I will often recall these basic theorems during my course. The topics of the spring 2010 course include:
1. The theory of Hecke operators
2. Analytic properties of automorphic L-functions
3. Rankin-Selberg convolution L-functions
4. Symmetric power L-functions
5. Non-vanishing on the boundary of the critical line
6. Spectral power-moments of L-functions
7. Subconvexity bounds of L-functions on the critical line
8. Central values of L-functions
9. Applications to problems of equidistribution
These lectures will be on Tuesdays and Fridays 12:00-1:20pm in Hill 124.
Prerequisites: Topics the students should know, together with the courses in which they are taught at Rutgers, are: Introductory Linear Algebra (640:250); Multivariable Calculus (640:251); Elementary Differential Equations (640:244 or 640:252); Advanced Calculus for Engineering(Laplace transforms, sine and cosine series, introductory pde)(640:421).
Students who are not prepared for this course should consider taking 640:421.
Description: A first semester graduate course intended primarily for students in mechanical and aerospace engineering, biomedical engineering, and other engineering programs. Power series and the method of Frobenius for solving differential equations; nonlinear differential equations and phase plane methods; vector spaces of functions, Hilbert spaces, and
orthonormal bases; Fourier series and Sturm-Liouville theory; Fourier and Laplace transforms; separation of variables and other elementary solution methods for the linear differential equations of physics: the heat, wave, and Laplace equations.
Text: Gilbert Strang, "Linear Algebra and its Applications", 4th edition, ISBN #0030105676, Brooks/Cole Publishing, 2007
Prerequisites: Familiarity with matrices, vectors, and mathematical reasoning at the level of advanced undergraduate applied mathematics courses.
Description: Note: This course is intended for graduate students in science, engineering and statistics.
This is an introductory course on vector spaces, linear transformations, determinants, and canonical forms for matrices (Row Echelon form and Jordan canonical form). Matrix factorization methods (LU and QR factorizations, Singular Value Decomposition) will be
emphasized and applied to solve linear systems, find eigenvalues, and diagonalize quadratic forms. These methods will be developed in class and through homework assignments using MATLAB. Applications of linear algebra will include Least Squares Approximations, Discrete Fourier Transform, Differential Equations, Image Compression, and Data-base searching.
Grading: Written mid-term exam, homework, MATLAB projects, and a
written final exam.
Text: A. Quarteroni, R. Sacco, and F. Saleri, Numerical Mathematics, Springer (second edition)
OR K. Atkinson, An Introduction to Numerical Analysis, Wiley, 1989 (second edition)
Prerequisites: Advanced calculus, linear algebra, and familiarity with differential equations.
Description: This is the first part of a general survey of the basic topics in numerical analysis -- the study and analysis of numerical algorithms for approximating the solution of a variety of generic problems which occur in applications. In the fall semester, we will consider the approximation of functions by polynomials and piecewise polynomials, numerical integration, and the numerical solution of initial value problems for ordinary differential
equations, and see how all these problems are related.
In the spring semester (642:574), we will study the numerical solution of linear systems of equations, the approximation of matrix eigenvalues and eigenvectors, the numerical solution of nonlinear systems of equations, numerical techniques for unconstrained function minimization, finite difference and finite element methods for two-point boundary value problems, and finite difference methods for some model problems in partial differential equations.
Despite the many solution techniques presented in elementary calculus and differential equations courses, mathematical models used in applications often do not have the simple forms required for using these methods. Hence, a quantitative understanding of the models requires the use of numerical approximation schemes. This course provides the mathematical background for understanding how such schemes are derived and when they are likely to work.
To illustrate the theory, in addition to the usual pencil and paper problems, some short computer programs will be assigned. To minimize the effort involved, however, the use of Matlab will be encouraged. This program has many built in features which make programming easy, even for those with very little prior programming experience.
Text: There is no one text; we will make use of several books that will be on reserve in the library.
Prerequisites: The course assumes a level of mathematical maturity consistent with having had a good course in linear algebra (such as 640:350) and real analysis (such as 640:411)
at the undergraduate level. A prior introduction to combinatorics, and rudimentary probability are all occasionally helpful.
Description: This is the first part of a two-semester course surveying basic topics in combinatorics. Topics in the first semester will be some subset of:
Prerequisites: Basic courses in probability and graph theory or 642:582 or 642:581 or permission of instructor 642:581
Description: We are going to study basic random objects such as random graphs, random matrices, random walks etc and their applications. I am going to introduce essential techniques in these areas, including martingales, concentration, correlation, fourier analyis, convexity etc.
Some highlights: Chromatic number of random graphs, (Sharp-) Threshold phenomenon, Limiting distributions, Randomized algorithms, Pseudo-random graphs.
Text:Professor's notes will be made available online.
No textbook will be used, but the website contains
notes written by the instructor (to be updated) as well as additional material.
Prerequisites: Linear algebra, differential equations, and basic probability, though some
topics that require more advanced prerequisites will be also covered.
Please e-mail instructor if you have any questions.
Description:
The field of (Molecular) Systems Biology mostly concerns itself with individual cells, or small collections of cells, seen as networks involving DNA, RNA, proteins, metabolites, and small molecules. An example is the study
of signal transduction pathways in cells, and their disruption in cancer.
It is widely recognized by leading biologists that the typical "reductionist"
approach is not powerful enough to describe, analyze, and interpret the
complex behaviors of such networks. Quantitative (i.e, mathematical)
formalisms, concepts, tools, and models are required, and there is a major
role to be played by mathematicians in applying and adapting known theory to
model and understand specific systems.
This course will provide an introduction to mathematical techniques as well as to the relevant biology. No background in biology will be expected from
students. To make the course accessible to a wide audience, only minimal math prerequisites will be needed in order to follow most of the material.
There are a very large number of possible topics to choose from, and the
syllabus will evolve based on student's interest and input. Possible topics
include the dynamics of cell signaling networks, including memories, switches, adaptation, and oscillators, as well as biological phenomena of chemotaxis, pattern formation, and neural transmission. If there is interest, we can also discuss synthetic biology.
Text: Stephen E. Shreve, Stochastic Calculus for Finance II: Continuous-Time Models, Springer Verlag, 2004, ISBN 0-387-40101-8.
Supplemental Texts: Stephen E. Shreve, Stochastic Calculus for Finance I: The Binomial Asset Pricing Model, Springer Verlag, 2004; John C. Hull, Options, Futures, and other Derivatives, 6th Edition, Prentice Hall, 2006.
Prerequisites: Ordinary differential equations (01:640:244 or 01:640:252), multivariable calculus (01:640:251), linear algebra (01:640:250), and undegraduate probability theory with calculus (01:640:477 or 01:960:381). An undergraduate course on analysis (01:640:311-312 or 01:640:411-412) or engineering mathematics (01:640:421) or partial differential equations (01:640:423) is recommended but not required.
Please visit the prerequisites page for descriptions of Rutgers undergraduate course prerequisites. A solid understanding of undergraduate probability at the level of the textbook by Sheldon Ross, A First Course in Probability, is especially important. Given this background, the course should be accessible to Mathematical Finance master's degree students and graduate students in Computer Science, Economics, Finance, Engineering, Mathematics, Physics, Operations Research, and Statistics.
Description: This course is an introduction to the mathematical theory of derivative security (or option) pricing. Fundamental concepts are briefly introduced first using the discrete-time binomial model: financial markets, derivative securities, arbitrage, hedging and replicating portfolios, risk-neutral probabilities, risk-neutral pricing formula, and market completeness. Basic ideas of probability and stochastic processes are reviewed for finite probability spaces and discrete-time processes: conditional expectation, martingales, and Markov processes. After this introduction to finance using discrete-time models, the emphasis shifts to continuous-time models and the main part of the course. Topics covered include a summary of probability measure theory and conditional expectation, Brownian motion and quadratic variation, martingales, Ito integral, stochastic calculus, replicating portfolios and hedging, Black-Scholes-Merton formulae for a European-style call option price, change of measure and Girsanov's Theorem, risk-neutral pricing pricing theory, no-arbitrage and existence of risk-neutral measure, market completeness and uniqueness of risk-neutral measure, Markov property, Feyman-Kac theorem and the connection between stochastic calculus and partial differential equations, and local volatility and stochastic volatility models.
Description: In addition to equity, interest rates, FX, and commodity derivatives, credit derivatives play an increasingly important role in financial markets. The course will include a review of jump processes; the basic theory of single name credit derivative modeling; structual, reduced form or intensity models; credit default swaps; default correlation, multiname credit derivative modeling; top down versus bottom up models; basket credit derivatives; collaterized debt obligations; and tranche options. The goal of the course is to cover most of the material in "Credit Risk Modeling" by David Lando (Princeton University Press, 2004) or "Credit Derivatives Pricing Models" by Philipp Schonbucher (Wiley, 2004).
All course content – lecture notes, homework assignments and solutions, exam solutions, supplementary articles, and computer programs – are posted on Sakai and available to registered students.
Selected Topics in Mathematical Finance: Portfolio Theory
Luiza/ P. Limratanamongkol Miranyan
Text: A. Meucci, Risk and Asset Allocation, Springer, 2008
R. Grinold and R. Kahn, Active Portfolio Management: A Quantitative Approach for Producing Superior Returns and Controlling Risk, McGraw-Hill, second edition, 1999
Prerequisites: Math 16:642:622 and Stat 16:960:563 (or equivalent graduate course on regression analysis).
Co-requisites
Stat 16:960:565 (or equivalent graduate course on time series) recommended but not required.
Description: The course will introduce all aspects of risk management and portfolio management, from theoretical foundations to the most advanced developments, emphasizing the mathematical and statistical techniques involved. Main topics: Brief review of multivariate statistics (matrix-variate continuous and discrete distributions, location-dispersion ellipsoid, copula/marginal factorization). The quest for invariance (random walk, ARMA, GARCH, generalized processes in discrete and continuous time, foundations of statistical arbitrage). Brief review of multivariate estimation (non-parametric, non-normal maximum-likelihood, shrinkage, robust, Bayesian and generalized evaluation techniques). Dimension reduction and CAPM (generalized r-square, explicit and implicit factor models, CAPM, APT, principal component analysis). Pricing (exact and Greeks approximation, analytical and Monte Carlo. Investment objectives: (total return, P&L and prospect theory, benchmark allocation. Risk assessment (stochastic dominance, expected utility, value at risk, expected shortfall, coherent measures, spectral measures, marginal decomposition of risk). Classical allocation (sub-optimal two-step mean-variance approach, alternative trade-offs). Estimation risk (robust/SOCP optimization, shrinkage/Bayesian allocations, Black-Litterman and beyond). Implementation (liquidity, transaction costs, foundations of optimal execution. Applications discussed during the course are implemented in MATLAB (standard, statistics and optimization toolboxes required).
All course content – lecture notes, homework assignments and solutions, exam solutions, supplementary articles, and computer programs – are posted on Sakai and available to registered students.
