An introduction to real analysis. The first semester MATH 4331 will focus mainly on 1-variable analysis and include topics from
(a) basic properties of the real number system,
(b) series, sequences and uniform convergence,
(c) properties of functions (analyticity, smoothness, nowhere differentiability, Weierstrass approximation theorem),
(d) classical function theory (Gamma function, infinite products),
(e) Euler-Maclaurin formula and asymptotics.

The second semester MATH 4332 will provide an introduction to metric spaces and compactness that develops from results proved in the first semester (for example, the compactness of a closed interval). Also included will be a short review of multivariable calculus and applications of metric space techniques (the contraction mapping lemma) to the inverse and implicit function theorems and the existence theorem for ordinary differential equations. If there is time (in either semester 1 or 2), there will also be some introductory lectures on Fourier series - in particular, the verification that under appropriate assumptions, the Fourier series does converge to the function.

Overall the aim of MATH 4331/2 is to provide an introduction to some of the powerful techniques, results and ideas of analysis as developed over the past three hundred years

This year-long course will introduce the theory of the geometry of curves and surfaces in three-dimensional space using calculus techniques, exhibiting the interplay between local and global quantities.

Topics include:

• Curves in the plane and in space
• Global properties of curves and surfaces in three dimensions
• The first fundamental form
• Curvature of surfaces
• Gaussian curvature and the Gaussian map
• Geodesics
• Minimal surfaces
• Gauss' Theorem Egregium (The "Pleasing Theorem")
• Gauss-Bonnet theorem.

We will develop and analyze numerical methods for approximating the solutions of common mathematical problems. This is an introductory course and will be a mix of mathematics and computing.

The emphasis this semester will be on solving nonlinear equations, interpolation, numerical integration, initial value problems of ordinary differential equations, and direct methods for solving linear systems of algebraic equations.

Note: This is the first semester of a two semester course. The emphasis the second semester will be in particular on iterative methods for solving linear systems, approximation theory, numerical solutions of nonlinear equations, and elementary methods for ordinary differential equations with boundary conditions and partial differential equations.

The course will cover the last two chapters of the Rosen book on Boolean Algebras and Modeling Computation. In particular the relationship between Boolean Algebras and Logic Gates will be discussed. We will also study Boolean Functions and the representation of finite free Boolean Algebras. Finite State Machines and Turing Machines serve as models of computations. We will study Kleene's Theorem for a concise description of finite state machines. . We will also discuss the halting problem for Turing machines.

Course Organization and Grading: Students will receive on a regular basis homework assignments. There will be a midterm and a final. Homework counts for 30%, midterm for 30% and final for 40% of the final grade.

During the first semester we will cover the basic theory of groups, rings and fields with strong emphasis on principal ideal domains. We will also discuss the most important algebraic constructions from a universal algebraic as well as from a categorical point of view.

The second semester will be mainly on modules over principal ideal domains, Sylow theory, free algebras and co-products and ultraproducts.

Course Organization and Grading: Students will receive on a regular basis homework assignments. There will be a midterm and a final. Homework counts for 30%, midterm for 30% and final for 40% of the final grade.

"Matrix Analysis", by Roger A. Horn and Charles R. Johnson, Cambridge University Press ISBN 0-521-38632-2
NOTE: This book is available in paperback.

An introduction to real analysis. The first semester MATH 6312 will focus mainly on 1-variable analysis and will include topics from
(a) basic properties of the real number system,
(b) series, sequences and uniform convergence,
(c) properties of functions (analyticity, smoothness, nowhere differentiability, Weierstrass approximation theorem),
(d) classical function theory (Gamma function, infinite products),
(e) Euler-Maclaurin formula and asymptotics.

The second semester MATH 6313 will provide an introduction to metric spaces and compactness that develops from results proved in the first semester (for example, the compactness of a closed interval). Also included will be a short review of multivariable calculus and applications of metric space techniques (the contraction mapping lemma) to the inverse and implicit function theorems and the existence theorem for ordinary differential equations. If there is time (in either semester 1 or 2), there will also be some introductory lectures on Fourier series - in particular, the verification that under appropriate assumptions, the Fourier series does converge to the function.

Overall the aim of MATH 6312/3 is to provide an introduction to some of the powerful techniques, results and ideas of analysis as developed over the past three hundred years

Donald Cohn, Measure Theory, Birkhauser

Recommended text: Gerald Folland, Real Analysis; Modern Techniques and their Applications, published by Wiley-Interscience.

Recommended Texts:

• Differential Equations, Dynamical Systems and Linear Algebra by M. Hirsch and S. Smale (available at Amazon or in the library)
• Ordinary Differential Equations by V. I. Arnold, M.I.T press, 1998 (paperback)
• Geometrical Methods in the Theory of Ordinary Differential Equations by V. I. Arnold, Springer Verlag, 2nd Edition 1988.
• Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields by J. Guckenheimer and P. Holmes (Applied Mathematical Sciences Vol 42) Springer Verlag.
• Mathematical Methods of Classical Mechanics by V. I. Arnold, Springer Verlag, 2nd Edition.

It is not neceesary to buy anything of the references, though Ordinary Differential Equations by V. I. Arnold and Differential Equations, Dynamical Systems and Linear Algebra by M. Hirsch and S. Smale would be most useful to own. The books are useful for reference and lecture notes will be based on these texts and a variety of sources.

This is the first semester of a two semester course. The topics for this first semester will include the theory of finite dimensional linear and nonlinear optimization and numerical methods. This course will be a mix of mathematics and practicalities.

This will include:

Theory and algorithms for unconstrained optimization:
(i) Newton and Quasi-Newton methods,
(ii) Linesearch and Trust Region methods.

Theory and algorithms for constrained optimization:
(i) Linear Programming (Interior Point methods, Simplex methods),
(ii) Quadratic Programming,
(iii) Equality and Inequality linear constraints,
(iv) Barrier and Augmented Lagrangian methods,
(v) Sequential Quadratic Programming.

J. Stoer and R. Bulirsch: Introduction to Numerical Analysis, 3rd ed., Springer-Verlag, New York, 2002.

Further reference: Numerical Mathematics, by A. Quarteroni, R. Sacco, and F. Saleri. Springer-Verlag, 2000

Kernel Methods in Computational Biology, B. Schölkopf, K. Tsuda, J.-P. Vert.

Reference book : selected chapters in Introduction to Support Vector Machines N. Cristianini and J. Shawe-Taylor Cambridge University Press, 2000,ISBN: 0 521 78019 5.

Automatic Learning of unknown functional relationships Y = f(X) between multidimensional inputs X and outputs Y, involves algorithms dedicated to the intensive analysis of large finite "training sets" of "examples" of inputs/outputs pairs (X,Y). This formalism applies to perceptive tasks involving automatic classification / identification of complex patterns such as shapes, sounds and speech, handwriting, texts, and has more recently been widely extended to classification / diagnosis in proteomics and genomics.

New machine learning algorithms have emerged in the last 10 years, as well as powerful mathematical concepts, focused on key learning features : generalisation, accuracy, speed, robustness. The mathematics involved rely on the use of positive definite kernels associated to Hilbert space representations of data, information and probability theory, quantification of complexity, functional approximation. The course will present major kernel based learning architectures : Support Vector Machines and selected applications to proteomics and genomics.

Source and channel coding from the very basics to currently active research in an intuitive, but mathematically sound manner. The material will be interspersed with simple programming projects and experiments.

Topics: Entropy, mutual information, source coding, information rate and ergodicity, arithmetic codes, channel capacity, data processing inequality, rate distortion theory, cryptography, pseudo-random number generation, vector-valued generalizations.

Stochastic differential equations arise when some randomness is allowed in the coefficients of a differential equation. They have many applications, including mathematical biology, theory of partial differential equations, differential geometry and mathematical finance.

This is an introduction to the theory and applications of stochastic differential equations. A knowledge of measure theory is strongly recommended but is not required. First we will review measure theory, probability spaces, random variables and stochastic processes. Brownian motion will be discussed in some detail. Then we will introduce the Ito integral and relevant aspects of martingale theory as a method to formulate and solve stochastic differential equations. Numerical schemes will be also discussed. Applications will include mathematical finance (arbitrage and option pricing).

Lecture notes will be provided.
Recommended book: Pedersen's "Analysis Now" or Conway's "A course in Functional Analysis".

Although we will be starting from scratch, if you wish to do some preliminary reading you could read the middle section in Royden's book on Real Analysis (on the Hahn Banach theorem, and so on). We will be starting with a little topology - so you might glance through any good basic book on topology to familiarize yourself with "compactness" , "locally compact", continuous functions between topological spaces, the basic theory of metric spaces. The reason I review some topology is to familiarize the students with the use of `nets' (=generalized sequences) in topology.

We will mostly avoid measure theory, so don't be too concerned if you lack that background. The tests and exam will be based on the notes given in class, and on the homework. After each chapter we will schedule a problem solving workshop, based on the homework assigned for that chapter.

Final grade is approximately based on a total score of 400 points consisting of homework (100 points), a semester test (100 points), and a final exam (200 points). The instructor may change this at his discretion.

The first semester will be a leisurely and general presentation, starting from scratch, of the basic facts in Linear Analysis, Banach spaces and Hilbert spaces. The second semester will be a more technical development of the theory of linear operators on Hilbert space; Algebras and spectral theory (focusing on the spectral theorem). Some Fourier series and transforms. Unbounded operators. We will also cover some topics which the students request.