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Course Descriptions: AY 2016-17
Course Schedule
Algebra and Number
Geometry and Topology
Numerical Analysis
Applied Mathematics
Probability and Statistics
All course descriptions are subject to change
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3 points per term. Fall and spring terms.
Tuesday, 5:10-7:00 R. Kleeman Monday, 5:10-7:00, D. Cai (spring).
Fall Term
Prerequisites: Undergraduate linear algebra or permission of the instructor.
Linear spaces, subspaces. Linear dependence, linear independence; span, basis, dimension, isomorphism. Quotient spaces. Linear functionals, Dual spaces. Linear mappings, null space, range, fundamental theorem of linear algebra. Underdetermined systems of linear equations. Composition, inverse, transpose of linear maps, algebra of linear maps. Similarity transformations. Matrices, matrix Multiplication, Matrix Inverse, Matrix Representation of Linear Maps, Determinant, Laplace expansion, Cramer's rule. Eigenvalue problem, eigenvalues and eigenvectors, characteristic polynomial, Cayley-Hamilton theorem. Diagonalization.
Text: Friedberg, S.H., Insel, A.J., & Spence, L.E. (2003). Linear Algebra (4th ed.). Upper Saddle River, NJ: Prentice-Hall/ Pearson Education.
Recommended Text: Lipschutz, S., & Lipson, M. (2012). Schaum's Outlines [Series]. Schaum's Outline of Linear Algebra (5th ed.). New York, NY: McGraw-Hill.
Note: Extensive lecture notes keyed to these texts will be issued by the instructor.
Spring Term
Prerequisites: Linear Algebra I or permission of the instructor.
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 decompositions. Complexification and diagonalization over R. Matrix norms, series and the matrix exponential map, applications to ODE. Bilinear and quadratic forms and their normal forms. The classical matrix groups: unitary, orthogonal, symplectic. Implicit function theorem, smooth surfaces in R^n and their tangent spaces. Intro to matrix Lie algebras and Lie groups.
Text: Friedberg, S.H., Insel, A.J., & Spence, L.E. (2003). Linear Algebra (4th ed.). Upper Saddle River, NJ: Prentice-Hall/ Pearson Education.
Plus: Extensive instructor's class notes.
3 points. Spring term.
Tuesday, 5:10-7:00, S. Marques.
Prerequisites: Undergraduate Linear Algebra or permission of the instructor.
Linear spaces, subspaces. Linear dependence, linear independence; span, basis, dimension, isomorphism. Quotient spaces. Linear functionals, dual spaces. Linear mappings, null space, range, fundamental theorem of linear algebra. Underdetermined systems of linear equations. Composition, inverse, transpose of linear maps, algebra of linear maps. Similarity transformations. Matrices, matrix Multiplication, Matrix Inverse, Matrix Representation of Linear Maps determinant, Laplace expansion, Cramer's rule. Eigenvalue problem, eigenvalues and eigenvectors, characteristic polynomial, Cayley-Hamilton theorem. Diagonalization.
Text: Lax, P.D. (2007). Pure and Applied Mathematics: A Wiley Series of Texts, Monographs and Tracts [Series, Bk. 78]. Linear Algebra and Its Applications (2nd ed.). Hoboken, NJ: John Wiley & Sons/ Wiley-Interscience.
MATH-GA 2111.001 LINEAR ALGEBRA (one-term format)
3 points. Fall term.
Thursday, 9:00-10:50, E. Tabak.
Prerequisites: Undergraduate linear algebra.
Linear algebra is two things in one: a general methodology for solving linear systems, and a beautiful abstract structure underlying much of mathematics and the sciences. This course will try to strike a balance between both. We will follow the book of our own Peter Lax, which does a superb job in describing the mathematical structure of linear algebra, and complement it with applications and computing. The most advanced topics include spectral theory, convexity, duality, and various matrix decompositions.
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 (2nd ed.). Hoboken, NJ: John Wiley & Sons/ Wiley-Interscience.
Recommended Text: Strang, G. (2005). Linear Algebra and Its Applications (4th ed.). Stamford, CT: Cengage Learning.
3 points per term. Fall and spring terms.
Thursday, 7:10-9:00, Y. Tschinkel (fall); Monday, 7:10-9:00, R. Young (spring).
Prerequisites: Elements of linear algebra and the theory of rings and fields.
Fall Term
Basic concepts of groups, rings and fields. Symmetry groups, linear groups, Sylow theorems; quotient rings, polynomial rings, ideals, unique factorization, Nullstellensatz; field extensions, finite fields.
Recommended Texts: Artin, M. (2010). Featured Titles for Abstract Algebra [Series]. Algebra (2nd ed.). Upper Saddle River, NJ: Pearson.
Chambert-Loir, A. (2004). Undergraduate Texts in Mathematics [Series]. A Field Guide to Algebra (2005 ed.). New York, NY: Springer-Verlag.
Serre, J-P. (1996). Graduate Texts in Mathematics [Series, Vol. 7]. A Course in Arithmetic (Corr. 3rd printing 1996 ed.). New York, NY: Springer-Verlag.
Spring Term
Representations of finite groups. Characters, orthogonality of characters of irreducible representations, a ring of representations. Induced representations, Artin's theorem, Brauer's theorem. Representations of compact groups and the Peter-Weyl theorem. Lie groups, examples of Lie groups, representations and characters of Lie group. Lie algebras associated with Lie groups. Applications of the group representations in algebra and physics. Elements of algebraic geometry.
Text: Fulton, W., Harris, J. (2008). Graduate Texts in Mathematics/ Readings in Mathematics [Series, Bk.129]. Representation Theory: A First Course (Corrected ed.). New York, NY: Springer-Verlag.
Recommended Texts: Lang, S. (2005). Graduate Texts in Mathematics [Series, Bk. 211]. Algebra (3rd ed.). New York, NY: Springer-Verlag.
Serre, J.P. (1977).Graduate Texts in Mathematics [Series, Bk. 42]. Linear Representations of Finite Groups. New York, NY: Springer-Verlag.
Reid, M. (1989). London Mathematical Society Student Texts [Series]. Undergraduate Algebraic Geometry. New York, NY: Cambridge University Press.
James, G., & Liebeck, M. (1993). Cambridge Mathematical Textbooks [Series]. Representations and Characters of Groups. New York, NY: Cambridge University Press.
Artin, M. (2010). Algebra (2nd ed.). Upper Saddle River, NJ: Prentice-Hall/ Pearson Education.
Sagan, B.E. (1991). Wadsworth Series in Computer Information Systems [Series]. The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions. Pacific Grove, CA: Wadsworth & Brooks/Cole.
Brocker, T., & Dieck, T. (2003). Graduate Texts in Mathematics [Series, Bk. 98]. Representations of Compact Lie Groups. New York, NY: Springer-Verlag.
3 points. Spring term.
Wednesday, 7:10-9:00, A. Pirutka.
Prerequisites: Undergraduate elementary number theory, abstract algebra, including groups, rings and ideals, fields, and Galois theory (e.g. undergraduate Algebra I and II).
This course is a graduate level introduction to algebraic number theory, in which we will cover fundamentals of the subject. Topics include: the theory of the valuation (p-adic numbers, completion, local fields, henselian fields, ramification theory, Galois theory of valuations) and Riemann-Roch theory.
For additional information, see the course website.
Text: Neukirch, J. (1999).Grundlehren der mathematischen Wissenschaften [Series, Book 322]. Algebraic Number Theory. New York, NY: Springer-Verlag.
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3 points per term. Fall and spring terms.
Thursday, 5:10-7:00, S. Cappell (fall); Tuesday, 7:10-9:00, S. Cappell, (spring).
Fall Term
Prerequisites: Any knowledge of groups, rings, vector spaces and multivariable calculus is helpful. Undergraduate students planning to take this course must have V63.0343 Algebra I or permission of the Department.
After introducing metric and general topological spaces, the emphasis will be on the algebraic topology of manifolds and cell complexes. Elements of algebraic topology to be covered include fundamental groups and covering spaces, homotopy and the degree of maps and its applications. Some differential topology will be introduced including transversality and intersection theory. Some examples will be taken from knot theory.
Recommended Texts: Hatcher, A. (2002). Algebraic Topology. New York, NY: Cambridge University Press.
Munkres, J. (2000). Topology (2nd ed.). Upper Saddle River, NJ: Prentice-Hall/ Pearson Education.
Guillemin, V., and Pollack, A.(1974). Differential Topology. Englewood Cliffs, NJ: Prentice-Hall.
Milnor, J.W. (1997). Princeton Landmarks in Mathematics [Series]. Topology from a Differentiable Viewpoint (Rev. ed.). Princeton, NJ: Princeton University Press.
Spring Term
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 Poincar duality. Products and ring structures. Vector bundles, tangent bundles, DeRham cohomology and differential forms.
3 points per term. Fall and spring terms.
Monday, 1:25-3:15, A. Pirutka (fall); Wednesday, 1:25-3:15, J. Cheeger (spring).
Fall Term
Differentiable manifolds, tangent bundle, embedding theorems, vector fields and differential forms. Introduction to Riemannian metrics, connections and geodesics.
Text: Lee, J.M. (2009). Graduate Studies in Mathematics [Series, Vol. 107]. Manifolds and Differential Geometry. Providence, RI: American Mathematical Society.
Spring Term
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.
MATH-GA2400.001 ADVANCED TOPICS IN GEOMETRY (Introduction to Hyperbolicity in Geometry, Dynamics and Group Theory)
3 points. Fall term.
TBA, M. Gromov.
This course will cover 3 topics:
1. Smale-Anosov dynamical systems: This will include basic stability and ergodicity theorems for Anosov and more genral Smale systems, including geodesic flows on manifolds with negative curvatures. The course will also prove basic results on Markov partitions and Markov codings.
2. Spaces with negative curvatures: The course will start with the classical material on Riemannian manifolds, then will prove basic properties of singular spaces of Busemann and Alexandrov with negative curvatures and explain the main constructions of these spaces.
3. Hyperbolic groups: The course will define these groups, prove their basic properties and discuss various classes of examples.
Texts: TBA.
3 points. Spring term.
Tuesday, 9:00-10:50, J. Cheeger.
Prerequisites: TBA.
Texts: TBA.
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3 points. Spring term.
Monday, 7:10-9:00, Y. Chen.
Differentiation and integration for vector-valued functions of one and several variables: curves, surfaces, manifolds, inverse and implicit function theorems, integration on manifolds, Stokes' theorem, applications.
3 points per term. Fall and spring terms.
Monday, 5:10-7:00, S. Serfaty (fall); Thursday, 5:10-7:00, TBA (spring).
Fall Term
Elements of topology on the real line. Rigorous treatment of limits, continuity, differentiation, and the Riemann integral. Taylor series. Introduction to metric spaces. Pointwise and uniform convergence for sequences and series of functions. Applications.
Recommended Text: Introduction to Real Analysis, William F. Trench.
Spring Term
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.
Recitation/ Problem Session: 7:15-8:30 (following the course in both terms).
MATH-GA 2430.001 REAL VARIABLES (one-term format)
3 points per term. Fall term.
Monday, Wednesday, 9:35-10:50, T. Austin.
Note: Master's students need permission of course instructor before registering for this course.
Prerequisites: A familiarity with rigorous mathematics, proof writing, and the epsilon-delta approach to analysis, preferably at the level of MATH-GA 1410, 1420 Introduction to Mathematical Analysis I, II.
Measure theory and integration. Lebesgue measure on the line and abstract measure spaces. Absolute continuity, Lebesgue differentiation, and the Radon-Nikodym theorem. Product measures, the Fubini theorem, etc. L^p spaces, Hilbert spaces, and the Riesz representation theorem. Fourier series.
Text: Main Text: Folland's "Real Analysis: Modern Techniques and Their Applications"
Secondary Text: Bass' "Real Analysis for Graduate Students"
3 points per term. Fall and spring terms.
Tuesday, 7:10-9:00, F. Hang (fall); Wednesday, 7:00-9:00 (spring), E. Hameiri.
Fall Term
Prerequisites (Complex Variables I): Advanced calculus (or equivalent).
Complex numbers; analytic functions, Cauchy-Riemann equations; linear fractional transformations; construction and geometry of the elementary functions; Green's theorem, Cauchy's theorem; Jordan curve theorem, Cauchy's formula; Taylor's theorem, Laurent expansion; analytic continuation; isolated singularities, Liouville's theorem; Abel's convergence theorem and the Poisson integral formula.
Text: Brown, J., & Churchill, R. (2008). Complex Variables and Applications (8th ed.). New York, NY: McGraw-Hill.
Spring Term
Prerequisites (Complex Variables II): Complex Variables I (or equivalent).
The fundamental theorem of algebra, the argument principle; calculus of residues, Fourier transform; the Gamma and Zeta functions, product expansions; Schwarz principle of reflection and Schwarz-Christoffel transformation; elliptic functions, Riemann surfaces; conformal mapping and univalent functions; maximum principle and Schwarz's lemma; the Riemann mapping theorem.
Text: Ahlfors, L. (1979). International Series in Pure and Applied Mathematics [Series, Bk. 7]. Complex Analysis (3rd ed.). New York, NY: McGraw-Hill.
MATH-GA 2451.001 COMPLEX VARIABLES (one-term format)
3 points. Fall term.
Tuesday, Thursday, 1:25-2:40, A. Cerfon.
Note: Master's students need permission of course instructor before registering for this course.
Complex numbers, the complex plane. Power series, differentiability of convergent power series. Cauchy-Riemann equations, harmonic functions. conformal mapping, linear fractional transformation. Integration, Cauchy integral theorem, Cauchy integral formula. Morera's theorem. Taylor series, residue calculus. Maximum modulus theorem. Poisson formula. Liouville theorem. Rouche's theorem. Weierstrass and Mittag-Leffler representation theorems. Singularities of analytic functions, poles, branch points, essential singularities, branch points. Analytic continuation, monodromy theorem, Schwarz reflection principle. Compactness of families of uniformly bounded analytic functions. Integral representations of special functions. Distribution of function values of entire functions.
Text: Ahlfors, L. (1979). International Series in Pure and Applied Mathematics [Series, Bk. 7]. Complex Analysis (3rd ed.). New York, NY: McGraw-Hill.
3 points. Spring term.
Tuesday, 9:00-10:50, TBA.
Prerequisites: Undergraduate background in analysis, linear algebra and complex variable..
Existence and uniqueness of initial value problems. Linear equations. Complex analytic equations. Boundary value problems and Sturm-Liouville theory. Comparison theorem for second order equations. Asymptotic behavior of nonlinear systems. Perturbation theory and Poincar-Bendixson theorems.
Recommended Text: Teschl, G. (2012). Graduate Studies in Mathematics [Series, Vol. 140]. Ordinary Differential Equations and Dynamical Systems. Providence, RI: AMS Chelsea Publishing/ American Mathematical Society.
3 points per term. Fall and spring terms.
Monday, 11:00-12:50, J. Shatah (fall); Tuesday, 9:00-10:50, P. Germain (spring)
Note: Master's students should consult course instructor before registering for PDE II in the spring.
Fall Term
Prerequisites: Knowledge of undergraduate level linear algebra and ODE; also some exposure to complex variables (can be taken concurrently).
A basic introduction to PDEs, designed for a broad range of students whose goals may range from theory to applications. This course emphasizes examples, representation formulas, and properties that can be understood using relatively elementary tools. We will take a broad viewpoint, including how the equations we consider emerge from applications, and how they can be solved numerically. Topics will include: the heat equation; the wave equation; Laplace's equation; conservation laws; and Hamilton-Jacobi equations. Methods introduced through these topics will include: fundamental solutions and Green's functions; energy principles; maximum principles; separation of variables; Duhamel's principle; the method of characteristics; numerical schemes involving finite differences or Galerkin approximation; and many more. For more information (including a tentative semester plan) see the syllabus here.
Recommended Texts: Guenther, R.B., & Lee, J.W. (1996). Partial Differential Equations of Mathematical Physics and Integral Equations. Mineola, NY: Dover Publications.
Evans, L.C. (2010). Graduate Studies in Mathematics [Series, Bk. 19]. Partial Differential Equations (2nd ed.). Providence, RI: American Mathematical Society.
Spring Term
Prerequisites: MATH-GA 2490.001 PDE I and MATH-GA 2430.001 Real Variables, or theTH-GA 2430.001 equivalent.
This course is a continuation of MATH-GA 2490 and is designed for students who are interested in analysis and PDEs. The course gives an introduction to Sobolev spaces, Holder spaces, and the theory of distributions; elliptic equations, harmonic functions, boundary value problems, regularity of solutions, Hilbert space methods, eigenvalue problems; general hyperbolic systems, initial value problems, energy methods; linear evaluation equations, Fourier methods for solving initial value problems, parabolic equations, dispersive equations; nonlinear elliptic problems, variational methods; Navier-Stokes and Euler equations.
Recommended Texts: Garabedian, P.R. (1998). Partial Differential Equations (2nd Rev. ed.). Providence, RI: AMS Chelsea Publishing/ American Mathematical Society.
Evans, L.C. (2010). Graduate Studies in Mathematics [Series, Bk. 19]. Partial Differential Equations (2nd ed.). Providence, RI: American Mathematical Society.
John, F. (1995). Applied Mathematical Sciences [Series, Vol. 1]. Partial Differential Equations (4th ed.). New York, NY: Springer-Verlag.
3 points. Fall term.
Thursday, 9:00-10:50, F. Lin.
Prerequisites: Linear algebra, real variables or the equivalent, and some complex function-theory would be helpful.
The course will concentrate on concrete aspects of the subject and on the spaces most commonly used in practice such as Lp (1? p ? ?), C, C?, and their duals. Working knowledge of Lebesgue measure and integral is expected. Special attention to Hilbert space (L2, Hardy spaces, Sobolev spaces, etc.), to the general spectral theorem there, and to its application to ordinary and partial differential equations. Fourier series and integrals in that setting. Compact operators and Fredholm determinants with an application or two. Introduction to measure/volume in infinite-dimensional spaces (Brownian motion). Some indications about non-linear analysis in an infinite-dimensional setting. General theme: How does ordinary linear algebra and calculus extend to d=? dimensions?
Recommended Texts: Lax, P.D. (2002). Pure and Applied Mathematics: A Wiley Series of Texts, Monographs and Tracts [Series, Bk. 55]. Functional Analysis (1st ed.). New York, NY: John Wiley & Sons/ Wiley-Interscience.
Reed, M., & Simon, B. (1972). Methods of Modern Mathematical Physics [Series, Vol. 1]. Functional Analysis (1st ed.). New York, NY: Academic Press.
3 points. Spring term.
Monday, 9:00-10:50, F. Hang.
Prerequisites: Real analysis; basic knowledge of complex variables and functional analysis.
Classical Fourier Analysis: Fourier series on the circle, Fourier transform on the Euclidean space, Introduction to Fourier transform on LCA groups. Stationary phase. Topics in real variable methods: maximal functions, Hilbert and Riesz transforms and singular integral operators. Time permitting: Introduction to Littlewood-Paley theory, time-frequency analysis, and wavelet theory.
Recommended Text: Muscalu, C., & Schlag, W. (2013). Cambridge Studies in Advanced Mathematics [Series, Bk. 137]. Classical and Multilinear Harmonic Analysis (Vol.1). New York, NY: Cambridge University Press. (Online version available to NYU users through Cambridge University Press.)
MATH-GA 2610.001 ADVANCED TOPICS IN PDE (Stochastic homogenization of elliptic equations)
3 points. Fall term.
Wednesday, 9:00-10:50, S. Armstrong.
The purpose of the course is to give an overview of some recent results in elliptic homogenization theory which focus on the random setting and are concerned with quantitative results. We will begin with an introduction to qualitative stochastic homogenization results, based on variational methods. Next we will discuss quantitative results based on the same variational methods which have a convex duality flavor, give explicit error estimates for the Dirichlet and Neumann problems, discuss variational methods for "non-variational" equations, develop a higher regularity theory and, time permitting, prove optimal error estimates and scaling limits for correctors. The course has few requisites apart from a basic knowledge of Sobolev spaces.
MATH-GA 2610.002 ADVANCED TOPICS IN PDE (Topic TBA)
3 points. Fall term.
Wednesday, 1:25-3:15, P. Germain.
Prerequisites: TBA.
Texts: TBA.
MATH-GA 2620.001 ADVANCED TOPICS IN PDE (Topic TBA)
3 points. Spring term.
Tuesday, 9:00-10:50, J. Shatah.
Prerequisites: TBA.
Texts: TBA.
MATH-GA 2650.001 ADVANCED TOPICS IN ANALYSIS (Dynamical Systems)
3 points. Fall term.
Monday, 5:10-7:00, L. Young.
Prerequisites: Real analysis at the graduate level.
This is the first semester of a year course on dynamical systems. It is an introductory sequence, requiring no prior knowledge of the subject. In the fall semester, I will cover mostly ergodic theory, a probabilistic approach to dynamical systems. Topics include ergodicity, mixing properties, entropy; ergodic theory of continuous and differentiable maps, Lyapunov exponents etc. In the spring semester, the focus will be on differentiable dynamical systems. Topics include invariant manifolds, hyperbolicity, and various models of chaotic systems.
Recommended Text (for Fall semester): Walters, P. (2000). Graduate Texts in Mathematics [Series, Bk. 79]. An Introduction to Ergodic Theory. New York, NY: Springer-Verlag.
MATH-GA 2650.002 ADVANCED TOPICS IN ANALYSIS (Signal Representations in Harmonic Analysis)
3 points. Fall term.
Monday, 1:25-3:15, S. Gunturk.
Prerequisites: Real analysis (with measure theory), basic knowledge of functional and harmonic analysis.
This course will cover a selection of classical and modern topics in applied harmonic analysis related to representation and processing of signals, including sampling theory, frames, time-frequency analysis, and wavelets.
Texts: None required. Relevant texts which are electronically available to the NYU community will be used as needed.
MATH-GA 2660.001 ADVANCED TOPICS IN ANALYSIS (Dynamical Systems)
3 points. Spring term.
Wednesday, 1:25-3:15, L. Young.
Prerequisites: Real analysis at the graduate level.
This is the first semester of a year course on dynamical systems. It is an introductory sequence, requiring no prior knowledge of the subject. In the fall semester, I will cover mostly ergodic theory, a probabilistic approach to dynamical systems. Topics include ergodicity, mixing properties, entropy; ergodic theory of continuous and differentiable maps, Lyapunov exponents etc. In the spring semester, the focus will be on differentiable dynamical systems. Topics include invariant manifolds, hyperbolicity, and various models of chaotic systems.
Texts: TBA.
3 points. Spring term.
Tuesday, Thursday, 1:25-3:15, F. Lin.
Prerequisites: TBA.
Texts: TBA.
3 points. Spring term.
Monday, 1:25-3:15, R. >Kohn.
Prerequisites: TBA.
Texts: TBA.
(Show Detailed Descriptions for All Courses)
3 points per term. Fall and spring terms.
Thursday, 5:10-7:00, G. Stadler (fall); Tuesday, 5:10-7:00, L. Greengard (spring).
Fall Term
Prerequisites:A good background in linear algebra, and some experience with writing computer programs (in MATLAB, Python or another language). MATLAB will be used as the main language for the course. Alternatively, you can also use Python for the homework assignments. You are encouraged but not required to learn and use a compiled language.
This course is part of a two-course series meant to introduce graduate students in mathematics to the fundamentals of numerical mathematics (but any Ph.D. student seriously interested in applied mathematics should take it). It will be a demanding course covering a broad range of topics. There will be extensive homework assignments involving a mix of theory and computational experiments, and an in-class final. Topics covered in the class include floating-point arithmetic, solving large linear systems, eigenvalue problems, interpolation and quadrature (approximation theory), nonlinear systems of equations, linear and nonlinear least squares, nonlinear optimization, and Fourier transforms. This course will not cover differential equations, which form the core of the second part of this series, Numerical Methods II.
Recommended Texts (Springer books are available online from the NYU network):
Deuflhard, P. & Hohmann, A. (2003). Numerical Analysis in Modern Scientific Computing. Texts in Applied Mathematiks [Series, Bk. 43]. New York, NY: Springer-Verlag.
Further Reading (available on reserve at the Courant Library):
Bau III, D., & Trefethen, L.N. (1997). Numerical Linear Algebra. Philadelphia, PA: Society for Industrial & Applied Mathematics.
Quarteroni, A., Sacco, R., & Saleri, F. (2006). Numerical Mathematics (2nd ed.). Texts in Applied Mathematics [Series, Bk. 37]. New York, NY: Springer-Verlag.
If you want to brush up your MATLAB:
Gander, W., Gander, M.J., & Kwok, F. (2014). Scientific Computing – An Introduction Using Maple and MATLAB. Texts in Computation Science and Engineering [Series, Vol. 11]. New York, NY: Springer-Verlag.
Moler, C. (2004). Numerical Computing with Matlab. SIAM. Available online.
Cross-listing: CSCI-GA 2420.001.
Spring Term
Prerequisites: Numerical linear algebra, elements of ODE and PDE.
This course will cover fundamental methods that are essential for numerical solution of differential equations. It is intended for students familiar with ODE and PDE and interested in numerical computing; computer programming assignments form an essential part of the course. The course will introduce students to numerical methods for (1) nonlinear equations, Newton's method; (2) ordinary differential equations, Runge-Kutta and multistep methods, convergence and stability; (3) finite difference and finite element methods; (4) fast solvers, multigrid method; and (5) parabolic and hyperbolic partial differential equations.
Text: LeVeque, R. (2007). Classics in Applied Mathematics [Series]. Finite Difference Methods for Ordinary and Partial Differential Equations. Philadelphia, PA: Society for Industrial and Applied Mathematics.
Cross-listing: CSCI-GA 2421.001.
MATH-GA 2011.001 ADVANCED TOPICS IN NUMERICAL ANALYSIS (Computational Fluid Dynamics)
3 points. Fall term.
Thursday, 9:00-10:50, A. Donev.
Prerequisites: TBA.This course will cover advanced numerical techniques for solving PDEs, with a particular focus on fluid dynamics. This includes advection-diffusion-reaction equations, compressible and incompressible Navier-Stokes equations, and fluid-structure coupling. Basic familiarity with temporal integrators for ODEs (multistep, Runge-Kutta), methods for solving PDEs (finite difference, finite volume, finite elements for parabolic and elliptic problems), iterative solvers for linear systems, and the Navier-Stokes equations will be assumed. Topics covered will include: higher-order spatio-temporal discretizations for advection-diffusion equations, artificial dissipation and dispersion, compressible flow (conservation laws, limiters, shock-capturing methods, boundary layers, turbulence), incompressible flow (projection methods, Stokes solvers, spectral methods), fluid-structure coupling (boundary-integral formulations, immersed boundary methods) geo-physical dynamics (shallow water, wave equations, turbulent flows).
Texts: TBA.
Cross-listing: CSCI-GA 2945.001
MATH-GA 2011.002 ADVANCED TOPICS IN NUMERICAL ANALYSIS (Numerical Optimization)
3 points. Fall term.
Tuesday, 5:10-7:00, M. Wright.
Prerequisites: TBA.
Texts: TBA.
Cross-listing: CSCI-GA 2945.002
MATH-GA 2011.003 ADVANCED TOPICS IN NUMERICAL ANALYSIS (ODE/Ordinary Differential Equations)
3 points. Fall term. Meets from 9/6 - 10/13.
Tuesday, Thursday, 1:25-3:15, N. Trefethen.
Prerequisites: TBA.
The course is intended for Courant Institute graduate students who are either mathematics students on the applied half of the spectrum or computer science students with an interest in numerical computation. ODEs, the starting point of analysis and dynamics, are one of the two foundation stones of applied mathematics (the other is linear algebra). This course will deepen your appreciation of them fundamentally.
This course will be based on a new textbook that I am 2/3 of the way through writing, Exploring ODEs, with coauthors Toby Driscoll and Asgeir Birkisson. The book aims to be interesting and informative for every applied or numerical mathematician. Its unusual missions are to be numerically-enabled, but not a numerical book and mathematically mature, but not technically advanced. To get an idea, including the table of contents, take a look at the talk I gave about the book project posted at https://people.maths.ox.ac.uk/trefethen/talks.html.
The course will be assessed by homework assignments. These will be computational (working in Chebfun), but will be about behavior of ODEs, not about algorithms or numerical analysis.
Cross-listing: CSCI-GA 2945.003
MATH-GA 2011.004 ADVANCED TOPICS IN NUMERICAL ANALYSIS (The Finite Element Method)
3 points. Fall term.
Tuesday, 9:00-10:50, O. Widlund.
This course will provide an introduction to the finite element method and its theoretical foundation.
Self-adjoint elliptic problem and calculus of variation. Sobolev spaces, Poincare's and Friedrichs' inequalities. Triangulations of bounded domains in two and three dimensions. Lagrange and Hermite finite elements, which are conforming in H^1. The biharmonic problem and H^2 conforming elements.
Error bounds for basic finite element methods: Cea's lemma and the result of Aubin and Nitsche. Nonconforming finite elements: the lemmas due to Strang. Isoparametric elements and spectral elements. Compressible and almost incompressible elasticity. Mixed methods and the inf-sup condition due to Babuska and Brezzi. Incompressible Stokes. Raviart-Thomas elements and other elements conforming in H(div). Nedelec elements and other conforming elements in H(curl). Solving the large linear and non-linear systems of algebraic equations resulting from finite element approximations. An introduction to domain decomposition and multigrid algorithms.
Cross-listing: CSCI-GA 2945.004
MATH-GA 2012.001 ADVANCED TOPICS IN NUMERICAL ANALYSIS (High-Performance Computing)
3 points. Spring term.
Thursday, 5:10-7:00, G. Stadler.
Prerequisites: TBA.
Texts: TBA.
MATH-GA 2012.002 ADVANCED TOPICS IN NUMERICAL ANALYSIS (Monte Carlo)
3 points. Spring term.
Monday, 5:10-7:00, J. Goodman.
Prerequisites: TBA.
Texts: TBA.
3 points. Fall term.
Thursday, 7:10-9:00, E. Fishler & L. Maclin
This course will introduce students to the software development process, including applications in financial asset trading, research, hedging, portfolio management, and risk management. Students will use the Java programming language to develop object-oriented software, and will focus on the most broadly important elements of programming - superior design, effective problem solving, and the proper use of data structures and algorithms. Students will work with market and historical data to run simulations and test strategies. The course is designed to give students a feel for the practical considerations of software development and deployment. Several key technologies and recent innovations in financial computing will be presented and discussed.
3 points. Fall term.
Thursday, 5:10-7:00, A. Rangan.
Prerequisites: Undergraduate multivariate calculus and linear algebra. Programming experience strongly recommended but not required.
This course is intended to provide a practical introduction to problem solving. Topics covered include: the notion of well-conditioned and poorly conditioned problems, with examples drawn from linear algebra; the concepts of forward and backward stability of an algorithm, with examples drawn from floating point arithmetic and linear-algebra; basic techniques for the numerical solution of linear and nonlinear equations, and for numerical optimization, with examples taken from linear algebra and linear programming; principles of numerical interpolation, differentiation and integration, with examples such as splines and quadrature schemes; an introduction to numerical methods for solving ordinary differential equations, with examples such as multistep, Runge Kutta and collocation methods, along with a basic introduction of concepts such as convergence and linear stability; An introduction to basic matrix factorizations, such as the SVD; techniques for computing matrix factorizations, with examples such as the QR method for finding eigenvectors; Basic principles of the discrete/fast Fourier transform, with applications to signal processing, data compression and the solution of differential equations.
This is not a programming course but programming in homework projects with MATLAB/Octave and/or C is an important part of the course work. As many of the class handouts are in the form of MATLAB/Octave scripts, students are strongly encouraged to obtain access to and familiarize themselves with these programming environments.
Recommended Texts: Bau III, D., & Trefethen, L.N. (1997). Numerical Linear Algebra. Philadelphia, PA: Society for Industrial & Applied Mathematics.
Quarteroni, A.M., & Saleri, F. (2006). Texts in Computational Science & Engineering [Series, Bk. 2]. Scientific Computing with MATLAB and Octave (2nd ed.). New York, NY: Springer-Verlag.
Otto, S.R., & Denier, J.P. (2005). An Introduction to Programming and Numerical Methods in MATLAB. London: Springer-Verlag London.
Cross-listing: CSCI-GA 2112.001.
3 points. Fall term.
Tuesday, 7:10-9:00, TBA.
Prerequisites: Scientific Computing or Numerical Methods II, Continuous Time Finance, or permission of instructor.
The classical curriculum of mathematical finance programs generally covers the link between linear parabolic partial differential equations (PDEs) and stochastic differential equations (SDEs), resulting from Feynmam-Kac's formula. However, the challenges faced by today's practitioners mostly involve nonlinear PDEs. The aim of this course is to provide the students with the mathematical tools and computational methods required to tackle these issues, and illustrate the methods with practical case studies such as American option pricing, uncertain volatility, uncertain mortality, different rates for borrowing and lending, calibration of models to market smiles, credit valuation adjustment (CVA), portfolio optimization, transaction costs, illiquid markets, super-replication under delta and gamma constraints, etc.
We will strive to make this course reasonably comprehensive, and to find the right balance between ideas, mathematical theory, and numerical implementations. We will spend some time on the theory: optimal stopping, stochastic control, backward stochastic differential equations (BSDEs), McKean SDEs, branching diffusions. But the main focus will deliberately be on ideas and numerical examples, which we believe help a lot in understanding the tools and building intuition.
Recommended text: Guyon, J. and Henry-Labordère, P.: Nonlinear Option Pricing, Chapman & Hall/CRC Financial Mathematics Series, 2014.
3 points. Fall term.
Thursday, 7:10-9:00, G. Ritter.
Prerequisites: Derivative Securities, Risk & Portfolio Management with Econometrics, and Computing in Finance (or equivalent programming experience).
A rigorous background in Bayesian statistics geared towards applications in finance, including decision theory and the Bayesian approach to modeling, inference, point estimation, and forecasting, sufficient statistics, exponential families and conjugate priors, and the posterior predictive density. A detailed treatment of multivariate regression including Bayesian regression, variable selection techniques, multilevel/hierarchical regression models, and generalized linear models (GLMs). Inference for classical time-series models, state estimation and parameter learning in Hidden Markov Models (HMMs) including the Kalman filter, the Baum-Welch algorithm and more generally, Bayesian networks and belief propagation. Solution techniques including Markov Chain Monte Carlo methods, Gibbs Sampling, the EM algorithm, and variational mean field. Real world examples drawn from finance to include stochastic volatility models, portfolio optimization with transaction costs, risk models, and multivariate forecasting.
3 points. Spring term.
Wednesday, 5:10-7:00, TBA.
Prerequisites: Risk and Portfolio Management with Econometrics, Derivative Securities, and Computing in Finance
This is a version of the course Scientific Computing (MATH-GA 2043.001) designed for applications in quantitative finance. It covers software and algorithmic tools necessary to practical numerical calculation for modern quantitative finance. Specific material includes IEEE arithmetic, sources of error in scientific computing, numerical linear algebra (emphasizing PCA/SVD and conditioning), interpolation and curve building with application to bootstrapping, optimization methods, Monte Carlo methods, and the solution of differential equations
Please Note: Students may not receive credit for both MATH-GA 2043.001 and MATH-GA 2048.001
(Show Detailed Descriptions for All Courses)
3 points. Fall term.
Monday, 1:25-3:15, D. Giannakis.
Prerequisites: Elementary linear algebra and differential equations.
This is a first-year course for all incoming PhD and Master students interested in pursuing research in applied mathematics. It provides a concise and self-contained introduction to advanced mathematical methods, especially in the asymptotic analysis of differential equations. Topics include scaling, perturbation methods, multi-scale asymptotics, transform methods, geometric wave theory, and calculus of variations
Recommended Texts: Barenblatt, G.I. (1996). Cambridge Texts in Applied Mathematics [Series, Bk. 14]. Scaling, Self-similarity, and Intermediate Asymptotics: Dimensional Analysis and Intermediate Asymptotics. New York, NY: Cambridge University Press.
Hinch, E.J. (1991). Camridge Texts in Applied Mathematics [Series, Bk. 6]. Perturbation Methods. New York, NY: Cambridge University Press.
Bender, C.M., & Orszag, S.A. (1999). Advanced Mathematical Methods for Scientists and Engineers [Series, Vol. 1]. Asymptotic Methods and Perturbation Theory. New York, NY: Springer-Verlag.
Whitham, G.B. (1999). Pure and Applied Mathematics: A Wiley Series of Texts, Monographs and Tracts [Series Bk. 42]. Linear and Nonlinear Waves (Reprint ed.). New York, NY: John Wiley & Sons/ Wiley-Interscience.
Gelfand, I.M., & Fomin, S.V. (2000). Calculus of Variations. Mineola, NY: Dover Publications.
3 points. Fall term.
Wednesday, 1:25-3:15, O. Buhler.
Prerequisites: Introductory complex variable and partial differential equations.
The course will expose students to basic fluid dynamics from a mathematical and physical perspectives, covering both compressible and incompressible flows. Topics: conservation of mass, momentum, and Energy. Eulerian and Lagrangian formulations. Basic theory of inviscid incompressible and compressible fluids, including the formation of shock waves. Kinematics and dynamics of vorticity and circulation. Special solutions to the Euler equations: potential flows, rotational flows, irrotationall flows and conformal mapping methods. The Navier-Stokes equations, boundary conditions, boundary layer theory. The Stokes Equations.
Text: Childress, S. Courant Lecture Notes in Mathematics [Series, Bk. 19]. An Introduction to Theoretical Fluid Mechanics. Providence, RI: American Mathematical Society/ Courant Institute of Mathematical Sciences.
Recommended Text: Acheson, D.J. (1990). Oxford Applied Mathematics & Computing Science Series [Series]. Elementary Fluid Dynamics. New York, NY: Oxford University Press.
3 points. Spring term.
Monday, 1:25-3:15, M. Holmes-Cerfon.
Prerequisites: Basic knowledge (e.g. undergraduate) of: probability, linear algebra, ODEs, PDEs, and analysis.
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.
Recommended Texts: Arnold, L. (1974). Stochastic differential equations: Theory and applications. New York: John Wiley & Sons.
Oksendal B. (2010). Universitext [Series]. Stochastic Differential Equations: An Introduction with Applications (6th Ed.). New York, NY: Springer-Verlag Berlin Heidelberg.
Koralov, L., & Sinai, Y.G. (2012). Universitext[Series]. Theory of Probability and Random Processes (2nd Ed.). New York, NY: Springer-Verlag Berlin Heidelberg.
Karatzas, I., & Shreve, S.E. (1991). Graduate Texts in Mathematics [Series, Vol. 113]. Brownian Motion and Stochastic Calculus (2nd Ed.). New York, NY: Springer Science+Business Media, Inc.
Kloeden, P., & Platen, E. (1992). Applications of Mathematics: Stochastic Modelling and Applied Probability [Series, Bk. 23]. Numerical Solution of Stochastic Differential Equations (Corrected 3rd Printing). New York, NY: Springer-Verlag Berlin Heidelberg New York.
Rogers, L.C.G. & Willams, D. (2000). Cambridge Mathematical Library [Series, Bks. 1-2]. Diffusions, Markov Processes, and Martingales: Foundations (Vol. 1, 2nd Ed.); and Diffusions, Markov Processes, and Martingales: Ito Calculus (Vol. 2, 2nd Ed.). New York, NY: Cambridge University Press.
Grimmett, G.R., & Stirzaker, D.R. (2001). Probability and Random Processes (3rd ed.). New York, NY: Oxford University Press.
Gardiner, C.W. (2009). Springer Series in Synergetics [Series, Bk. 13]. Stochastic Methods: A Handbook for the Natural and Social Sciences (4th Ed.). New York, NY: Springer-Verlag Berlin Heidelberg New York.
Risken, H., & Frank, T. (1996). Springer Series in Synergetics [Series, Bk. 18]. The Fokker-Planck Equation: Methods of Solution and Applications (1996 2nd Ed.). New York, NY: Springer-Verlag Berlin Heidelberg New York.
3 points. Fall term.
Monday, 7:10-9:00, R. Reider.
Prerequisites: Derivative Securities, Scientific Computing, Computing for Finance, and Stochastic Calculus.
The term "statistical arbitrage" covers any trading strategy that uses statistical tools and time series analysis to identify approximate arbitrage opportunities while evaluating the risks inherent in the trades (considering the transaction costs and other practical aspects). This course starts with a review of Time Series models and addresses econometric aspects of financial markets such as volatility and correlation models. We will review several stochastic volatility models and their estimation and calibration techniques as well as their applications in volatility based trading strategies. We will then focus on statistical arbitrage trading strategies based on cointegration, and review pairs trading strategies. We will present several key concepts of market microstructure, including models of market impact, which will be discussed in the context of developing strategies for optimal execution. We will also present practical constraints in trading strategies and further practical issues in simulation techniques. Finally, we will review several algorithmic trading strategies frequently used by practitioners.
3 points. Spring term.
Tuesday, 7:10-9:00, TBA.
Prerequisites: Computing in Finance, and Risk Portfolio Management with Econometrics, or equivalent.
In this course we develop a quantitative investment and trading framework. In the first part of the course, we study the mechanics of trading in the financial markets, some typical trading strategies, and how to work with and model high frequency data. Then we turn to transaction costs and market impact models, portfolio construction and robust optimization, and optimal betting and execution strategies. In the last part of the course, we focus on simulation techniques, back-testing strategies, and performance measurement. We use advanced econometric tools and model risk mitigation techniques throughout the course. Handouts and/or references will be provided on each topic.
3 points. Spring term.
Wednesday, 1:25-3:15, R. Kohn.
This course provides a basic mathematical
introduction to classical Newtonian mechanics, the
mechanics of rigid bodies, and quantum mechanics.
Prior knowledge in physics is not required. Key topics
include: variational formulation of classical
mechanics, Lagrangians, conserved quantities,
constrained motion; Hamilton's equations, phase space
evolution and Poincaré sections, Lyapunov exponents,
Liouville's theorem ;lLinear stability of fixed
points, integrable systems, homoclinic tangle,
Poincaré -Birkhoff theorem; Moments of inertia, Euler
angles, motion of rigid bodies, spin-orbit coupling;
representation of quantum states, correspondence
principle, observables and their spectrum, Schrödinger
equation, Ehrenfest's equations, uncertainty
principle, spin angular momentum, quantum description
of the hydrogen atom.
Recommended texts: Ciarlet, P.G. (1988). Studies in Mathematics & Its Applications: Mathematical Elasticity [Series, Vol. 1]. Three-dimensional Elasticity. New York, NY: Elsevier Science/ North-Holland.
Buhler, O. (2006). Courant Lecture Notes in Mathematics [Series, Bk. 13]. A Brief Introduction to Classical, Statistical, and Quantum Mechanics. Providence, RI: American Mathematical Society/ Courant Institute of Mathematical Sciences.
3 points. Fall and spring terms.
Tuesday, 7:10-9:00, P. Kolm (fall); Wednesday, 7:10-9:00, M. Avellaneda (spring).
Fall Term
Prerequisites: Univariate statistics, multivariate calculus, linear algebra, and basic computing (e.g. familiarity with MATLAB or co-registration in Computing in Finance).
A comprehensive introduction to the theory and practice of portfolio management, the central component of which is risk management. Econometric techniques are surveyed and applied to these disciplines. Topics covered include: factor and principal-component models, CAPM, dynamic asset pricing models, Black-Litterman, forecasting techniques and pitfalls, volatility modeling, regime-switching models, and many facets of risk management, both theory and practice.
Spring Term
Risk Management is arguably one of the most important tools for managing a trading book and quantifying the effects of leverage and diversification (or lack thereof).
This course is an introduction to risk-management techniques for portfolios of (i) equities and delta-1 securities and futures (ii) equity derivatives (iii) fixed income securities and derivatives, including credit derivatives, and (iv) mortgage-backed 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 historical back-testing of portfolios. We also review current risk-models and practices used by large financial institutions and clearinghouses.
If time permits, the course will also cover models for managing the liquidity risk of portfolios of financial instruments.
3 points. Spring term.
Monday, 5:10-7:00, TBA.
Prerequisites: Risk & Portfolio Management with Econometrics, Computing in Finance.
The first part of the course will cover the theoretical aspects of portfolio construction and optimization. The focus will be on advanced techniques in portfolio construction, addressing the extensions to traditional mean-variance optimization including robust optimization, dynamical programming and Bayesian choice. The second part of the course will focus on the econometric issues associated with portfolio optimization. Issues such as estimation of returns, covariance structure, predictability, and the necessary econometric techniques to succeed in portfolio management will be covered. Readings will be drawn from the literature and extensive class notes.
3 points. Spring term.
Monday, 7:10-9:00, TBA.
Prerequisites: Derivative Securities, Computing in Finance or equivalent programming.
The importance of financial risk management has been increasingly recognized over the last several years. This course gives a broad overview of the field, from the perspective of both a risk management department and of a trading desk manager, with an emphasis on the role of financial mathematics and modeling in quantifying risk. The course will discuss how key players such as regulators, risk managers, and senior managers interact with trading. Specific techniques for measuring and managing the risk of trading and investment positions will be discussed for positions in equities, credit, interest rates, foreign exchange, commodities, vanilla options, and exotic options. Students will be trained in developing risk sensitivity reports and using them to explain income, design static and dynamic hedges, and measure value-at-risk and stress tests. Students will create Monte Carlo simulations to determine hedge effectiveness. Extensive use will be made of examples drawn from real trading experience, with a particular emphasis on lessons to be learned from trading disasters.
Text: Allen, S.L. (2003). Wiley Finance [Series, Bk. 119]. Financial Risk Management: A Practitioner's Guide to Managing Market and Credit Risk. Hoboken, NJ: John Wiley & Sons.
3 points. Fall and spring Terms.
Monday, 5:10-7:00 (fall); Wednesday 5:10-7:00 (spring), P. Kolm.
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.
3 points. Fall term.
Wednesday, 7:10-9:00, TBA.
Prerequisites: Risk Management, Derivative Securities (or equivalent familiarity with market and credit risk models).
The course is divided into two parts. The first addresses the institutional structure surrounding capital markets regulation. It will cover Basel (1, MRA, 2, 2.5, 3), Dodd-Frank, CCAR and model review. The second part covers the actual models used for the calculation of regulatory capital. These models include the Gaussian copula used for market risk, specific risk models, the Incremental Risk Calculation (single factor Vasicek), the Internal Models Method for credit, and the Comprehensive Risk Measure.
3 points. Fall and spring terms.
Wednesday, 7:10-9:00, M. Avellaneda (fall); Monday, 7:10-9:00, TBA (spring).
An introduction to arbitrage-based pricing of derivative securities. 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.
3 points. Fall and spring terms.
Monday, 7:10-9:00, A. Javaheri (fall); Wednesday, 7:10-9:00, TBA (spring).
Prerequisites: Derivative Securities and Stochastic Calculus, or equivalent.
Fall Term
A second course in arbitrage-based pricing of derivative securities. The Black-Scholes model and its generalizations: equivalent martingale measures; the martingale representation theorem; the market price of risk; applications including change of numeraire and the analysis of quantos. Interest rate models: the Heath-Jarrow-Morton approach and its relation to short-rate models; applications including mortgage- backed securities. The volatility smile/skew and approaches to accounting for it: underlyings with jumps, local volatility models, and stochastic volatility models.
Spring Term
A second course in arbitrage-based pricing of derivative securities. Concerning equity and FX models: we'll discuss numerous approaches that are used in practice in these markets, such as the local volatility model, Heston, SABR, and stochastic local volatility. The discussion will include calibration and hedging issues and the pricing of the most common structured products. Concerning interest rate models: we'll start with a thorough discussion of one-factor short-rate models (Vasicek, CIR, Hull-White) then proceed to more advanced topics such as two-factor Hull-White, forward rate models (HJM) and the LIBOR market model. Throughout, the pricing of specific payoffs will be considered and practical examples and insights will be provided. We'll conclude with a discussion of inflation models.
3 points. Spring term.
Thursday, 7:10-9:00, TBA.
Prerequisites: Basic bond mathematics and bond risk measures (duration and convexity); Derivative Securities, Stochastic Calculus.
The first part of the course will cover the fundamentals of Securitized Products, emphasizing Residential Mortgages and Mortgage-Backed Securities (MBS). We will build pricing models that generate cash flows taking into account interest rates and prepayments. The first part of the course will also review subprime mortgages, CDO's, Commercial Mortgage Backed Securities (CMBS), Auto Asset Backed Securities (ABS), Credit Card ABS, and CLO's, and will discuss drivers of the financial crisis and model risk.
The second part of the course will focus on energy commodities and derivatives, from their basic fundamentals and valuation, to practical issues in managing structured energy portfolios. We develop a risk neutral valuation framework starting from basic GBM and extend this to more sophisticated multifactor models. These approaches are then used for the valuation of common, yet challenging, structures. Particular emphasis is placed on the potential pitfalls of modeling methods and the practical aspects of implementation in production trading platforms. We survey market mechanics and valuation of inventory options and delivery risk in the emissions markets.
Recommended Texts: Hayre, L. (2007). Wiley Finance [Series, Bk. 83]. Salomon Smith Barney Guide to Mortgage-backed and Asset-backed Securities. New York, NY: John Wiley & Sons.
Swindle, G. (2014). Valuation and Risk Management in Energy Markets. New York, NY: Cambridge University Press
Eydeland, A., & Wolyniec, K. (2002). Wiley Finance [Series, Bk. 97]. Energy and Power Risk Management: New Developments in Modeling, Pricing, and Hedging. Hoboken, NJ: John Wiley & Sons.
Harris, C. (2006). Wiley Finance [Series, Bk. 328]. Electricity Markets: Pricing, Structures and Economics (2nd ed.). Hoboken, NJ: John Wiley & Sons.
3 points. Fall term.
Wednesday, 7:10-9:00, TBA.
When a corporation borrows money there is a risk that it will not fulfill its obligation to repay the lenders in the future; this is credit risk. This course develops mathematical tools and models that are useful in analyzing, valuing and managing credit risk, both in its original form as found embedded in bonds and loans, and in derived forms as it exists in derivatives like asset swaps, credit default swaps (CDS), CDS indices and options, and tranched portfolio products like synthetic CDOs. We will discuss alternative notions of credit spread, and their dynamics and relationship to fundamental quantities like probability of default and loss given default. The consideration of portfolio products will require the introduction of notions of credit correlation including, but not limited to, the Gaussian default time copula.
Required Text: O'Kane, D. (2008). Wiley Finance [Series, Bk. 545]. Modeling Single-name and Multi-name Credit Derivatives. John Wiley & Sons, Hoboken, NJ.
3 points. Spring term.
Thursday, 5:10-7:00, TBA.
Prerequisites: Derivative Securities, Stochastic Calculus, and Computing in Finance (or equivalent familiarity with financial models, stochastic methods, and computing skills).
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.
3 points. Fall term.
Thursday, 1:15-3:15, A. Majda.
The course should be interesting for graduate students, and postdocs in pure and applied mathematics, physics, engineering, and climate, atmosphere, ocean science interested in turbulent dynamical systems as well as other complex systems.
Turbulent dynamical systems are ubiquitous complex systems in geoscience and engineering and are characterized by a large dimensional phase space and a large dimension of strong instabilities which transfer energy throughout the system. They also occur in neural and material sciences. Key mathematical issues are their basic mathematical structural properties and qualitative features, their statistical prediction and uncertainty quantification (UQ), their data assimilation, and coping with the inevitable model errors that arise in approximating such complex systems. These model errors arise through both the curse of small ensemble size for large systems and the lack of physical understanding. This is a research expository course on the applied mathematics of turbulent dynamical systems through the paradigm of modern applied mathematics involving the blending of rigorous mathematical theory, qualitative and quantitative modelling, and novel numerical procedures driven by the goal of understanding physical phenomena which are of central importance. The contents include the general mathematical framework and theory, instructive qualitative models, and concrete models from climate atmosphere ocean science. New statistical energy principles for general turbulent dynamical systems are discussed with applications, linear statistical response theory combined with information theory to cope with model errors, reduced low order models, and recent mathematical strategies for UQ in turbulent dynamical systems. Also recent mathematical strategies for online data assimilation of turbulent dynamical systems as well as rigorous results are briefly surveyed. Accessible open problems are often mentioned. This research expository book is the first of its kind to discuss these important issues from a modern applied mathematics perspective.
Texts: Chapter 1, 2, 4, 6, 7, 8 from Nonlinear Dynamics
and Statistical Theories for Basic Geophysical Flow, by A. J.
Majda and X. Wang, Cambridge Press 2006;
Introduction to Turbulent Dynamical Systems for Complex Systems, Frontiers in Applied Dynamical Systems, by A. J. Majda, 85 pages, Springer 2016.
3 points. Fall term.
Wednesday, 11:00-12:50, A. Bandeira.
A mostly self-contained research-oriented and fast-paced course designed for PhD students with an interest in doing research in theoretical aspects of algorithms that aim to extract information from data. These often lie in overlaps of (Applied) Mathematics with: Computer Science, Electrical Engineering, Statistics, and/or Operations Research. Each lecture will feature either one or two Mathematical Open Problem(s) with relevance in Data Science. The main mathematical tools used will be Probability and Linear Algebra, and a basic familiarity with these subjects is required. There will also be some (although knowledge of these tools is not assumed) Graph Theory, Representation Theory, Applied Harmonic Analysis, among others. The topics treated will include Dimension reduction, Manifold learning, Sparse recovery, Random Matrices, Approximation Algorithms, Community detection in graphs, and several others.
MATH-GA 2840.001 ADVANCED TOPICS IN APPLIED MATHEMATICS (Data Analysis Opt Transp)
3 points. Spring term.
Monday, 1:25-3:15, E. Tabak..
Prerequisite: TBA.
Texts: TBA.
MATH-GA 2851.001 ADVANCED TOPICS IN MATH BIOLOGY (Mathematical Neural Science)
3 points. Fall term.
Wednesday, 1:25-3:15, D. McLaughlin & D. Cai.
This course covers the mathematical representations of neuronal systems, and the mathematical methods used to analyze these representations. Many of these methods are the same as those used to analyze experimental data in neural science.
The course begins with the fundamental physiological properties of neurons, including neuronal and synaptic dynamics. The Hodgkin-Huxley equations for single neuronal dynamics, with synaptic couplings, will be briefly summarized -- as well as their reductions to simpler representations such as integrate and fire neurons and "exponential" integrate and fire neurons. The course then turns to its focus – representations of systems of excitatory and inhibitory neurons. It will cover, in detail, population dynamics of such neuronal networks, including coarse-grained representations (mean-field firing rate models, models which incorporate fluctuations, and models which incorporate multi-neuron effects), and functional and structural connectivity. The course strives to bring students from a background in either the physical/mathematical sciences or neural science quickly to research topics in theoretical neural science.
Topics to be covered will be taken from the following longer list:
- Basic Neurophysiology
- Anatomy of a Neuron
- Synapses and synaptic coupling
- Hodgkin-Huxley Equations
- Integrate and Fire Representations, including "exponential integrate and fire"
- Neuronal Coding – rate vs spike coding
- Signal Detection Theory
- Discrimination
- Nieman-Pearsen Lemma
- Spike train analysis
- Renewal processes
- Wiener-Khinchin Theorm
- Power spectrum
- Coarse-grained representations for populations of neurons
- Mean-field firing rate models
- Mean-field representations with fluctuations
- Other methods to model fluctuations
- Kinetic Theory
- Multi-neuron effects
- Fokker-Planck equation for a single neuron
- Kramer-Moyal Expansion
- Pawula Theorem
- Compressed Sensing
- System analysis
- Weiner Expansion
- Bussgang Theorem
- Receptive Fields
- Functional vs anatomical connectivity
- Granger causality
- Early visual system (Retina → LGN → V1)
- Receptive Fields
- Simple and complex cells
Texts: TBA.
MATH-GA 2852.001 ADVANCED TOPICS IN MATH BIOLOGY (The Art of Modeling in Cell and Developmental Biology)
3 points. Spring term.
Wednesday, 1:25-3:15, D. Tranchina.
Prerequisite: TBA.
Texts: TBA.
MATH-GA 2855.001 ADVANCED TOPICS IN MATH PHYSIOLOGY (Mathematical Aspects of Neurophysiology)
3 points. Fall term.
Tuesday, 1:25-3:15, J. Rinzel.
Prerequisite: TBA.
This course will focus on neurophysiology and biophysics at the cellular level; the mechanistic and mathematical descriptions of neuronal dynamics and input/output properties. Topics will include: ionic channels (current-voltage relations, gating kinetics, different types), Hodgkin-Huxley equations and reductions (the action potential, repetitive firing, bursting, propagation), dendrites (branching cable theory, passive and active membrane, spines), synapses (transmitter release, depression/facilitation, plasticity). Both analytical (perturbation and bifurcation methods) and numerical techniques will be described and used, serving as an applied introduction to these methodologies. Students will undertake computing projects related to the course material.
Texts: TBA.
3 points. Fall term.
Tuesday, 9:00-10:50, E. Gerber.
This course serves as an introduction to the fundamentals of geophysical fluid dynamics. No prior knowledge of fluid dynamics will be assumed, but the course will move quickly into the subtopic of rapidly rotating, stratified flows. Topics to be covered include (but are not limited to): the advective derivative, momentum conservation and continuity, the rotating Navier-Stokes equations and non-dimensional parameters, equations of state and thermodynamics of Newtonian fluids, atmospheric and oceanic basic states, the fundamental balances (thermal wind, geostrophic and hydrostatic), the rotating shallow water model, vorticity and potential vorticity, inertia-gravity waves, geostrophic adjustment, the quasi-geostrophic approximation and other small-Rossby number limits, Rossby waves, baroclinic and barotropic instabilities, Rayleigh and Charney-Stern theorems, geostrophic turbulence. Students will be assigned bi-weekly homework assignments and some computer exercises, and will be expected to complete a final project. This course will be supplemented with out-of-class instruction.
Recommended Texts: Vallis, G.K. (2006). Atmospheric and Oceanic Fluid Dynamics: Fundamentals and Large-scale Circulation. New York, NY: Cambrdige University Press.
Salmon, R. (1998). Lectures on Geophysical Fluid Dynamics. New York, NY: Oxford University Press.
Pedlosky, J. (1992). Geophysical Fluid Dynamics (2nd ed.). New York, NY: Springer-Verlag.
MATH-GA 3003.001 ADVANCED TOPICS IN ATMOSPHERE-OCEAN SCIENCE (Climate Change)
3 points. Spring term.
Tuesday, 1:25-3:15, S. Smith.
Prerequisites: TBA.
Texts: TBA.
MATH-GA 3010.001 ADVANCED TOPICS IN ATMOSPHERE-OCEAN SCIENCE (Geophysical Fluid Dynamics Laboratory)
3 points. Fall term.
Wednesday, 9:00-10:50, D. Holland.
Prerequisites: Undergraduate calculus, linear algebra, introduction to physics.
Description: This course introduces geophysical fluid dynamics from an experimental laboratory point of view. Laboratory instrumentation centers on a turntable equipped with velocity field (PIV, particle imaging velocimetry) and density field (LIF, laser induced florescence) measurement equipment. The course intertwines the laboratory observations with geophysical fluid dynamics theory, while numerical models are used as a platform to reconcile the observations with theoretical understanding.
Grading: Graded as a seminar course.
Recommended Texts: Atmosphere, Ocean and Climate Dynamics: an Introductory Text. Plumb & Marshall
Introduction to Geophysical Fluid Dynamics. Benoit Cushman-Roisin
Laboratory Experiments from the Following:
Details: http://paoc.mit.edu/labguide/
- Solid-body Rotation: The parabolic shape taken up by the free surface of water in solid body rotation is studied. Similarly, we study why rotating planets take on the shape of an oblate spheroid.
- Dye Stirring: Interleaving patterns are created by stirring colored dyes into a rotating fluid. Examples from atmospheric and oceanic flows are studied.
- Balanced Vortex: The balance of forces in the momentum equation in a rotating frame of reference is investigated by studying the flow of water down a drain-hole in a rotating system. The ideas are applied to intense atmospheric vortices such as hurricanes.
- Fronts: We create fronts in the laboratory by bringing together two bodies of water of differing densities in a rotating system. We study atmospheric fronts using meteorological data.
- Ekman Layers: Ageostrophic flow in a bottom Ekman layer is investigated in high and low pressure surfaces created in the laboratory. The same phenomenon is studied in analyzed surface fields of atmospheric highs and lows.
- General Circulation: The Hadley circulation and middle-latitude weather systems are studied in a rotating annulus experiment, using an ice bucket to represent the pole, and in atmospheric data.
- Convection: The evolution of convective boundary layers and convective plumes are studied as an analogue of dry atmospheric and oceanic convection.
- Flow over a Barrier: Investigates Rossby waves generated by flow over a barrier on a beta plane.
- Taylor Columns: The rigidity imparted to a fluid by rotation, as encapsulated in the Taylor-Proudman theorem, is investigated by studying flow over a submerged obstacle.
- Density Currents: The role of density differences in driving fluid motion is studied using an experiment first carried out by Marsigli in 1695.
- Ekman Pumping/Suction: cyclonic and anticylonic circulations are set up in the laboratory through the use of fans blowing air over the surface of the water to study the role of Ekman layers in inducing vertical motion.
- Ocean Gyres: western intensification of the wind-driven ocean circulation is studied by setting up a gyre in the presence of topographic beta.
- Thermohaline Circulation: The thermohaline circulation of the ocean is studied in a rendition of the classic Stommel-Arons experiment.
- Source/Sink Flow: flow from source to sink on a topographic beta plane is studied as an analogue of mid-depth/abyssal circulation in the ocean.
- Rossby Waves: Rossby waves are studied by investigating the 'westward' drift of a vortex induced by a melting ice cube in a rotating fluid of variable depth. Rossby waves in the ocean, as seen from satellite altimetry, are also presented.
MATH-GA 3010.002 ADVANCED TOPICS IN ATMOSPHERE-OCEAN SCIENCE (Climate Change)
3 points. Fall term.
Tuesday, 1:25-3:15, O. Pauluis.
Prerequisites: This course is aimed at graduate students who want to learn more about the scientific basis for climate change. It will require to read a substantial number of complex scientific papers, to evaluate them critically, and to discuss them constructively among colleagues..
Long before being recognized as a fundamental environmental challenge for to our society, climate change has been an topic of very active scientific research, dating back to the work of Fourier in the 19th Century. Climate change is fundamentally a multidisciplinary endeavors, touching on complex issues such a radiative transfer, mathematical modeling, remote sensing and biochemestry. In this course, we will focus on various scientific papers published overall the decades that have lead to the emergence of a scientific consensus that human activities are directly affecting the Earth's climate and will continue to do so through the course of the 21st Century. More specifically, we will address the following questions:
- How do greenhouse gases impact the Earth's energy budgets?
- Where does carbon come and go in the Earth system?
- How do we observe and attribute ongoing climate change?
- How do we assess future climate changes?
To do so, we will analyze and discuss a dozen of important scientific papers published over the last 60 years.
The students will be expected to critically analyze a dozen scientific papers, and actively participate to their discussion during class. Each student will lead several of these discussions. The final grade will be based on a project paper reporting on recent research articles in climate science.
Texts: TBA.
(Show Detailed Descriptions for All Courses)
3 points. Fall and spring terms.
Wednesday, 5:10-7:00, Y. Bakhtin (fall); Wednesday, 7:10-9:00, E. Vanden Eijnden (spring).
Prerequisites: Calculus through partial derivatives and multiple integrals; no previous knowledge of probability is required.
Fall Term
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.
Recommended Text: Probability Essentials, by J.Jacod and P.Protter. Springer, 2004.
Spring Term
The course introduces the basic concepts and methods of probability that are now widely used in scientific research. A feature of this course is that such concepts, usually presented in advanced mathematical settings, are here described in a more elementary framework requiring no knowledge of measure theory; nevertheless, analytical thinking and mathematical rigor will still be required at the level of a graduate course (undergraduate courses on relevant topics include MATH-UA 233, 234 and 235). Topics include: probability spaces, random variables, distributions, expectations and variances, law of large numbers, central limit theorem, Markov chains, random walk, diffusion processes including Brownian motion, and martingales.
Texts: Grimmett, G.R., & Stirzaker, D.R. (2001). Probability and Random Processes (3rd ed.). New York, NY: Oxford University Press.
Grinstead, C.M., & Snell, J.L. (1997). Introduction to Probability (2nd Rev. ed.). Providence, RI: American Mathematical Society.
3 points. Fall and spring terms.
Monday, 7:10-9:00, P. Bourgade(fall); Thursday, 7:10-9:00, TBA (spring).
Prerequisites: MATH-GA 2901 Basic Probability or equivalent.
Review of basic probability and useful tools. Bernoulli trials and random walk. Law of large numbers and central limit theorem. Conditional expectation and martingales. Brownian motion and its simplest properties. Diffusion in general: forward and backward Kolmogorov equations, stochastic differential equations and the Ito calculus. Feynman-Kac and Cameron-Martin Formulas. Applications as time permits.
Optional Problem Session: Wednesday, 5:30-7:00 (fall); TBA (spring).
Text: Durrett, R. (1996). Probability and Stochastics Series [Series, Bk. 6]. Stochastic Calculus: A Practical Introduction. New York, NY: CRC Press.
3 points per term. Fall and spring terms.
Wednesday, 11:00-12:50, E. Lubetzky (fall); Wednesday, 9:00-10:50, H. McKean (spring).
Prerequisites: A first course in probability, familiarity with Lebesgue integral, or MATH-GA 2430 Real Variables as mandatory co-requisite.
Fall Term
First semester in an annual sequence of Probability Theory, aimed primarily for Ph.D. students. Topics include laws of large numbers, weak convergence, central limit theorems, conditional expectation, martingales and Markov chains.
Text: Varadhan, S.R.S. (2001). Courant Lecture Series in Mathematics [Series, Bk. 7].
Spring Term
Independent increment processes, including Poisson processes and Brownian motion. Markov chains (continuous time). Stochastic differential equations and diffusions, Markov processes, semi-groups, generators and connection with partial differential equations.
Recommended Text: Varadhan, S.R.S. (2007). Courant Lecture Series in Mathematics [Series, Bk. 16]. Stochastic Processes. Providence, RI: American Mathematical Society/ Courant Institute of Mathematics.
MATH-GA 2931.001 ADVANCED TOPICS IN PROBABILITY (Extremes via second moments for logarithmic correlated fields: Gaussian and beyond)
3 points. Fall term. Meets from 9/8 - 10/27.
Tuesday, Thursday, 3:20-5:05, O. Zeitouni.
Prerequisites: A graduate course in probability. Prerequisites on Gaussian processes will be introduced as needed.
We will discuss the extremes of logarithmically correlated fields, starting with the classical case of branching random walks and moving to the general Gaussian case and beyond.
Texts: Notes will be made available, there is no textbook.
MATH-GA 2931.002 ADVANCED TOPICS IN PROBABILITY (Coarsening and Related Interacting Particle Systems)
3 points. Fall term. Meets from 9/7 - 10/21.
Wednesday, Friday, 1:25-3:15, C. Newman.
Prerequisites: Recommended: Probability Theory at the level of Probability Limit Theorems I and II
An introduction to some of the stochastic processes where the states, evolving in time, are assignments of, say, +1 or -1 to points in d-dimensional lattices or other graphs, which are motivated by nonequilibrium statistical physics at zero (or low) temperature. A typical transition/update rule is to agree with a strict majority of nearest neighbors or else, in the case of a tie, to toss a fair coin. We will discuss both known results and many open problems.
One old and one recent reference:
Nanda-Newman-Stein: Dynamics of Ising spin systems at zero temperature, A. M. S. Transl. (2) 198 (2000) 183-194.
Damron-Kogan-Newman-Sidoravicius: Coarsening with a frozen vertex, Elec. Commun. Prob. 21 (2016) 1-4.
MATH-GA 2931.003 ADVANCED TOPICS IN PROBABILITY (Entropy and Ergodic Theory)
3 points. Fall term.
Thursday, 1:25-3:15, T. Austin.
Prerequisites: TBA.
This will be an introduction to entropy and its many roles in different branches of mathematics, especially information theory, probability, ergodic theory and statistical mechanics. The aim is to give a quick overview of many topics, emphasizing a few basic combinatorial problems that they have in common and which are responsible for the ubiquity of entropy.
The course divides roughly into five parts:
- Shannon Entropy and Information Theory;
- Large Deviations and Measure Concentration;
- Statistical Mechanics and Thermodynamics;
- Ergodic Theory;
- The Thermodynamic Formalism.
Parts 1, 2 and 3 will introduce entropy in settings that can be described in terms of independent random variables. Parts 4 and 5 will generalize much of that story to stochastic processes having an invariance under the passage of time or spatial translation. The natural setting for those generalizations is provided by ergodic theory. In order to cover a wide range of subjects, I will have to sacrifice a lot of generality, and sometimes omit technical details. To make up for this, students will be expected to support the lectures with reading from a variety of sources.
Texts: TBA.
MATH-GA 2932.001 ADVANCED TOPICS IN PROBABILITY (Markov Chain Analysis)
3 points. Spring term.
Monday, 9:00-10:50, E. Lubetzky.
Prerequisites: TBA.
Texts: TBA.
3 points. Spring term.
TBA, H. McKean.
Prerequisites: TBA.
Texts: TBA.
Revised September 2016


text: The The Battle For South Am-- Chapter One: One For All wiki last edited by pikahyper on 07/31/13 05:57PM View full history Rai and the Future Force prepare for the Malev attack on North Am. Will the heroes be able to defeat this robotic menace?! And will Rai and Leeja's relationship finally be brought to light?

subset: commoncrawl


text:
Prison directory
Christian County
City Jail
Hopkinsville Jail is a City Jail used by the jurisdictions of Hopkinsville to confine inmates for short periods while awaiting trial or processing. As inmates are staying for only a short period, it has fewer amenities than a jail or prison. Inmates do, however, have access to bathroom facilities and are provided with meals during their stay. Hopkinsville Jail may also have a common area in which inmates can socialize. Use the inmate search feature to find an inmate here.
Connect with an Inmate
Facility Details
112 West First Street
Hopkinsville , KY 42240
Hopkinsville , KY 42240


text:
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Enjoy your favourite Abahna fragrance in the rest of your home, as well as your bathroom, with our elegant reed diffusers. This handy 200ml refill means you can continue to enjoy your favourite Abahna fragrance for even longer. Simply remove the reeds and gold ring from your diffuser bottle and carefully pour in the refill.
Only the highest quality ingredients are sourced from around the world and mixed in small batches to ensure intensity of fragrance.
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White Grapefruit & May Chang - reed diffuser set 200ml
White Grapefruit & May Chang scented candle 70g


text:
Sunday, 23 October, 2016
A message board where you can request and brainstorm words, ... Category: Entertainment\Writers Resources
Chicago, IL private dealer specializing in German Expressionist prints and ... Category: Visual Arts\Resources
A journal of paintings, photography, illustration, comics, multimedia, fiction, and ... Category: Visual Arts\Resources
A non-profit organization for working artists providing a forum for ... Category: Visual Arts\Resources
A clandestine group of underground performers staging site-specific, unsanctioned, and ... Category: Visual Arts\Public Art
Glasgow-based artist shows drawings, mixed media works, and sculpture. Abstract, ... Category: Visual Arts\Personal
Mixed-media artist and teacher presents works exploring the ecological and ... Category: Visual Arts\Personal
Artist and teacher shows surrealist-style artwork. Category: Visual Arts\Personal
New York artist presents galleries full of colorful abstract and ... Category: Visual Arts\Personal
Information on the Dutch painter associated with the Cobra movement. ... Category: Visual Arts\Personal
Digital paintings by this Venetian artist. Category: Visual Arts\Personal
Dutch surrealist presents an online gallery. Includes exhibitions listing. Category: Visual Arts\Personal
Artist with images and writings. Category: Visual Arts\Performance Art
Football players, cowboys and the American family all viewed with ... Category: Visual Arts\Painting
Presents landscape, nude, portrait, and figurative oil paintings. Category: Visual Arts\Painting
Four-Sided Abstract Art designed to turn 360 degrees on the ... Category: Visual Arts\Painting
Abstract sculpture, and painting. Also includes animation, and design. Category: Visual Arts\Multiple Media Artists
Online documentation of mail art project "Mail Art Show Reeperbahn" ... Category: Visual Arts\Mail Art and Artistamps
Showcase of paintings and computer collages, executed in a dadaistic ... Category: Visual Arts\Multiple Media Artists
Paintings and illustrations working with found objects on surfaces of ... Category: Visual Arts\Multiple Media Artists
Virtual gallery showcasing the work of various artists with avant-garde/modernist ... Category: Visual Arts\Galleries
Visual art ranging from surreal to graffiti in a variety ... Category: Visual Arts\Galleries
Cyber Art Gallery Eindhoven - One of the oldest galleries ... Category: Visual Arts\Galleries
Focuses on late 19th century, and 20th century European and ... Category: Visual Arts\Galleries
Fine art gallery representing contemporary British, European, and international artists, ... Category: Visual Arts\Galleries
Exhibits and sells the works of upcoming artists along with ... Category: Visual Arts\Galleries
Stone sculptures from Zimbabwe and bark paintings from Papua New ... Category: Visual Arts\Galleries
Gwydian's RealmComputer graphics by ranging from the absurd to the ... Category: Visual Arts\Computer Graphics
An interactive art exhibit of collage works and multimedia projects. Category: Visual Arts\Computer Graphics
Includes drawings, photograms, boxes, digital art, and text. Category: Visual Arts\Computer Graphics
An interactive art exhibit of collage works and multimedia projects. Category: Visual Arts\Collage
ASCII's from Bulletin Board days. 1994-2004. Category: Visual Arts\ASCII Art
From Northern Nevada, 30-minute show with news, skits, local events, ... Category: Television\Cable Television
Covering the artists involved in photomontage in the Berlin Dada ... Category: Photography\Reference
A library of his work. Category: Photography\Photographers
Submit your poetry site to this webring. Category: Online Writing\Web Rings
Small personal collection including a bio. Category: Online Writing\Poetry
Personal collection including rants. Category: Online Writing\Poetry
An illustrated hypertext fiction set in an alternative cosmology. Category: Online Writing\Fiction
A literary review incorporating poetry, fiction, memoirs, reviews and satire. Category: Online Writing\E-zines


text:
50 Best Anti-Aging Beauty Finds
+ Look-Great Secrets From Beauty-Biz Insiders
Susan Sarandon gives her facialist, Tracie Martyn, full credit for keeping her skin as smooth as it is. (We should all look this good at 58)
Tracie Martyn Enzyme Exfoliant
A brightening mask/scrub that tragets dull skin and dark spots
"There are amazing ingredients in this very effective mask - every exfoliant from salicylic acid to pineapple enzymes to malic, tartaric and lactic acids! This is the equivalent of a doctor's microdermabrasion. I'd even leave it on longer than the suggested maximum 15 minutes." - Dr. Grace Pak, M.D.
Tracie Martyn Enyzme Exfoliant
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Click here to know more about Tracie Martyn Enyzme Exfoliant
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Keywords: anti-aging, dull skin, dark spot, Enyzme Exfoliant, Susan Sarandon


text:
realmagick.com The shrine of knowledge.
Spirits Symbols
A selection of articles related to spirits symbols.
Original articles from our library related to the Spirits Symbols. See Table of Contents for further available material (downloadable resources) on Spirits Symbols.
- About Reiki Symbols
- "Instead of seeking many other medicines to cure their illness, I wish people would always honor their own body's healing ability and take good care of themselves." Meiji Emperor A symbol is a communication element intended to simply represent or...
Body Mysteries >> Reiki
- The Four Symbols of Usui Reiki Ryoho
- " It is not God's will merely that we should be happy, but that we should make ourselves happy." Immanuel Kant According to many traditional teachers the Reiki symbols are sacred and therefore secret and should only be seen by insiders, meaning...
Body Mysteries >> Reiki
- Discovery and Development of Reiki
- "The more we care for the happiness of others, the greater our own sense of wellbeing becomes." The Dalai Lama The traditional story of the discovery of Reiki is as historically correct as the biblical report of the life of Jesus - it is molded and...
Body Mysteries >> Reiki
- The Sacred Wheel: The Sabbats
- The Wheel of the Year honors the never ending cycle of life, death, and rebirth. It expresses the belief which Pagan religions hold that time is circular, not linear. The Sabbats are derived from the rich traditions of seasonal festivals as celebrated by...
Paganism & Wicca >> Holidays
- The Living Tradition of Thelema
- Do what thou wilt shall be the whole of the Law. What Crowley began, others must continue and develop or Thelema will become but a memory in the history of the Western Mystery Tradition. Yes, he was a Prophet and the Ipsissimus that one could say invented...
Mystic Sciences >> Magick
- What are Ghosts and Spirits?
- I was compelled to write this after seeing an atrocious TV special on ghosts last night. It was as corny as they come, with the 'scary'-voiced narrator reading while the scene was overacted by people wearing too much pale make-up in parody of spirits. The...
Parapsychology >> Spirits
- Orr, Emma Restall: Joint Chief of the British Druid Order
- Bobcat, Joint Chief of the British Druid Order, Priestess, author, poet and singer Emma Restall Orr has books to her credit including keystone works 'Principles of Druidry', 'Spirits of the Sacred Grove' & 'Ritual: A Guide To Life, Love And Inspiration', as...
Real Interviews >> Authors
Spirits Symbols is described in multiple online sources, as addition to our editors' articles, see section below for printable documents, Spirits Symbols books and related discussion.
Suggested News Resources
- The Last Lifestyle Magazine
- "When Kinfolk first started out, I was excited about it because I thought it would elucidate a lot on food culture but in a more social way," says Jenny Hong, a spirits buyer in New York City.
- Ranger Station Episode 28: Island Of Illusion, Part One
- While Zack tries to build up his confidence with an impromptu dance contest, Rita — watching, as always, from the moon — decides that now is the time to strike, destroying Angel Grove and breaking the Power Rangers' spirits at the same time.
- Rarely staged 1606 opera 'Eumelio' plays March 19-20
- Another challenge was interpreting the archaic notation, which includes some symbols no longer in use as well as others whose meaning has evolved. And along the way, at least one of the time signatures in a song was changed.
- The Spring Equinox and Death
- Ostara is a celebration of life. That is pretty clear. All the familiar symbols of Ostara – eggs, hares, chicks and flowers – represent fertility and life's rebirth in spring following the dark of winter.
- Mural painted as way to rebuild church after tragedy
- "We dealt with the sort of obvious symbols of spring and new life," Tucker said of the flowers, butterflies and budding tree painted where a car once crashed through the New Beginnings church wall in Elmira.
Great care has been taken to prepare the information on this page. Elements of the content come from factual and lexical knowledge databases, realmagick.com library and third-party sources. We appreciate your suggestions and comments on further improvements of the site.
Spirits Symbols Topics
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text:
Man accused of trying to get police to shootApril 25,2013
The Associated Press
WILLIAMSVILLE — Vermont state police responding to a report of a suicidal person have arrested a 71-year-old man accused of attempting to draw fire from them after shooting bullets from his handgun.
Welton Townsend of Williamsville has been charged with reckless endangerment and disorderly conduct.
Police responded to Townsend's home at about 6:10 p.m. Wednesday. They said a man later identified as Townsend left the home and threatened troopers with a "shootout."
Police surrounded the home and tried to convince him to speak to them. The man began to fire rounds from a handgun into the air. Police did not fire back and the man stopped firing shots.MORE IN This Just InMONTPELIER Neither of the two top Republican seeking office in Vermont this fall agrees with... Full StoryMONTPELIER Charges have been upgraded against two suspects in a deadly fire in Northfield last... Full StorySTOWE Ten months after two sons of the founder of Burton Snowboards caused an accidental fire... Full Story
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text:
Names of crash victims released
YOUNGSTOWN- The two men killed in a fiery crash Monday night in Berlin Township are Salem-area residents, according to the Mahoning County Coroner's Office.
The office identified the men as driver Jarod Todd Cameron, 23, of Beloit and passenger Ryan E. Zwingler, 29, of Green Township. They were identified through dental records after being trapped in their burning truck. Toxicology results are pending.
Both men worked for Bonnie Plants Inc., and were transporting plants, flowers and Styrofoam when the crash happened at about 11 p.m. on state Route 534 between Leffingwell Road and Western Reserve Road.
According to the highway patrol, Cameron's straight truck was southbound and allegedly went off the right side of the road, overcorrected and crossed center line, hitting a semi head on. Both vehicles were engulfed in flames. The driver of the semi, Mark Hoskins of Alliance, was taken to St. Elizabeth Health Center with non life-threatening injuries.
Zwingler's full obituary appears on 3A. Cameron's full obituary appeared in Thursday's edition.
Kevin Howell can be reached at firstname.lastname@example.com


text:
I am a 26 year old female and over the last week I have been getting a burning feeling in the left hand side of my chest around and under the breast area.
I feel lightheaded, sick, clammy hands and I sweat at night even if it's cold. My heart rate isn't fast but I do feel it pounding more then usual.
It happens mainly at night and sometimes when I am sitting down doing nothing.
It often radiates round to the side of my chest and under my arm pit and just a few hours ago my upper back area also started burning along with pins and needles.
I wouldn't say it's painful just uncomforterble and worrying!
My breathing isn't any different (I have asthma anyway) and it sometimes burns a bit more when I cough.
Please help!!!!
thank you
Know the five types of psoriasis and how to spot flares.
Newer diabetes treatments can suppress appetite and aid weight loss.
Try these tips to get your salivary glands back into action.
Constipation is a common side effect of opioid and narcotic pain medicines.
Is it sensitive skin or something else?


text:
Eastern Gateway Quality Rewarded
Officials and staff at Eastern Gateway Community College in Steubenville had to have suspected they would do well in a statewide comparison of enrollment among institutions of higher learning. After all, EGCC offers two attributes more and more young people are seeking – solid career training and reasonable prices.
But when the numbers were reported last week, they may have surprised even those at EGCC: The college's 2013 enrollment was 15 percent higher than in 2012. That was the highest increase of any of the 61 Ohio institutions of higher learning that were included in the report. In fact, even though it competed against colleges and universities throughout the state, EGCC's increase was the only one in double digits.
It just makes sense. Again, like many community colleges, EGCC has excellent career training programs. And for those planning to continue their educations, it provides an excellent two-year start.
All this is accomplished at very reasonable prices.
EGCC staff, officials and students should be proud. Those at other colleges and universities should take note of the secret to EGCC's success.


text:
Ok, the other day our old router broke, so we got a new one. I just tried to play Diddy Kong Racing online, and I entered the WEP key for the router into the DS, and went to test the connection. That is what was wrong. I see it testing, and the signal had 3 bars, then it drops to zero, and then the DS says there is an error. I think the router is not DS compatible, but I need to know.....
My Wii Games: Wii Sports, Wii Play, LoZTP, Red Steel, Rayman Raving Rabbids.
Wanted Games: Super Paper Mario, Excite Truck, Need for Speed Carbon, Medal of Honor Vanguard, Spiderman 3, Pirates 3
Friend Codes for DS!
Mario Kart: 193368313009
Diddy Kong Racing 090290007271
i have a problem too maybe if u help me ill help u. do u know how to find ur WEP code on a windows vista computer? [mines a xp but my moms is the vista and hers is wireless connection mine is wired]
I'm a Rising star, In this forum.
~Join the Zelda Fan Club (ZFC)~
Spoiler Alert!Originally Posted by Song in my head