Eulerian tours with application to reconstruction of  DNA sequences from
fragments. Euler characteristic formula. Map coloring problems and   4-color theorem. Trivalent  planar graphs with  application to fullerenes -  new forms  of carbon. Hamiltonian tours. Ramsey Theory and Erdos's probabilistic method. Matchings and Marriage Theorem.

please see http://www.math.uh.edu/~mike/4331-4332/

This course is a self-contained introduction to a very active and exciting area of applied mathematics which deals the representation of signals and im ages. It addresses fundamental and challenging questions like: how to effici ently and robustly store or transmit an image or a voice signal? how to remo ve unwanted noise and artifacts from data? how to identify features of inter ests in a signal? Students will learn to basic theory of Fourier series and wavelets which are omnipresent in a variety of emerging applications and tec hnologies including image and video compression, electronic surveillance, remote sensi ng and data transmission. Basic applications will also be covered in the cou rse.

Course outline:

Inner product spaces

• Inner product spaces.
• The spaces of square integrable functions and square summable series.
• Schwarz and triangle inequalities.
• Orthogonal projections and the least squares fit.

Fourier series and transform

• Computation of Fourier series.
• Convergence of Fourier series.
• The Fourier transform.
• Convolutions.
• Linear filters.
• The sampling theorem: Analog to digital and digital to analog conversi ons.
• From analog to digital filters.
• The Discrete Fourier transform (DFT), FFT, its use for the approximate computation of integral Fourier transforms.

Wavelets

• The Haar wavelet.
• Multiresolution analysis.
• The scaling relation.
• Properties of the scaling function.
• Decomposition and reconstruction.
• Wavelet design in the frequency domain.
• The Daubechies wavelet.

Grading:
  Grades will be based on homework assignments and on two exams (midter m and final).

Math 2331 (Linear Algebra), Math 3331 (Differential Equations)

Ability to do computer assignments in one of FORTRAN, C, Pascal, Matlab, Maple, Mathematica, and etc.

The first semester is not a prerequisite.

Topics covered include linear systems of equations, vector spaces, linear transformation, and matrices.

After a short review on Polynomials (Chapter 4) and Determinants (Chapter 5), the course will cover in depth Chapter 6 (Elementary Canonical Forms) and Chapter 7 (The Rational and Jordan Forms)

This course is an introduction to mathematical modeling of various financial instruments, such as options, futures, etc. The topics covered include: calls and puts, American and European options, expiry, strike price, drift and volatility, non-rigorous introduction to continuous-time stochastic processes including Wiener Process and Ito calculus, the Greeks, geometric Brownian motion, Black-Scholes theory, binomial model, martingales, filtration, and self financing strategy.

Introduction to groups, rings and fields. Additional notes will be made
available on http://www.math.uh.edu/~klaus/

Number theory is a subject that has interested people for thousand of years. This course is a one-semester long graduate course on number theory. Topics to be covered include divisibility and factorization, linear Diophantine equations, congruences, applications of congruences, solving linear congruences, primes of special forms, the Chinese Remainder Theorem, multiplicative orders, the Euler function, primitive roots, quadratic congruences, representation problems and continued fractions.

There are no specific prerequisites beyond basic algebra and some ability in reading and writing mathematical proofs. The method of presentation in this course is quite different. Rather than simply presenting the material, students first work to discover many of the important concepts and theorems themselves. After reading a brief introductory material on a particular subject, students work through electronic materials that contain additional background, exercises, and Research Questions. The research questions are typically more open ended and require students to respond with a conjecture and proof. We the present the theory of the material which the students have worked on, along with the proofs. The homework problems contain both computational problems and questions requiring proofs. It is hoped that students, through this course, not only learn the material, learn how to write the proofs, but also gain valuable insight into some of the realities of mathematical research by developing the subject matter on their own.

Simple and multiple linear regression, inference in multiple regression,
regression diagnostics, and influence measures, model selection, generalized linear models.

The course covers topics from the theory of groups, rings, fields, and
modules.

After a short review on Polynomials (Chapter 4) and Determinants (Chapter 5), the course will cover in depth Chapter 6 (Elementary Canonical Forms) and Chapter 7 (The Rational and Jordan Forms)

please see http://www.math.uh.edu/~mike/4331-4332/

G.B. Folland, Real Analysis: Modern Techniques and Their Applications (Pure and Applied Mathematics: A Wiley-Interscience Series of Texts, Monographs and Tracts).

Recommended reading: Lebesgue Integration on Euclidean Spaces, Frank Jones, Jones & Bartlett. Real Analysis, H.L. Royden, (3rd Edition), Prentice Hall. Real and Complex Analysis, W. Rudin, McGraw Hill. Measure Theory, D. L. Cohn, Birkhauser.

This semester we will be continuing to develope the basic principles of measure, integration, and real analysis. This body of knowledge is essential to many parts of mathematics (in particular to analysis and probability), and falls within the category of "What every graduate student has to know".

We will cover the following topics:

• Signed and complex measures. 
• The Radon-Nikodym theorem.
• The duality of $L^p$ spaces.
• Differentiation and integration of measures and functions on R$^n$.
• Basic connections with probability theory (distributions, density,
  independence).
• The Riesz representation theorem.
• Banach and Hilbert spaces. 
• Convolutions. 
• The Fourier transform.
• and suggested topics by students.
  
  After each chapter we will schedule a problem solving workshop, based on the homework assigned for that chapter.
  

Introduction to complex analysis, Juniro Noguchi, AMS(Translations of
mathematical monographs, Volume 168).

This course is a continuation of the course Math 6322. It covers Schwarz's lemma, Riemann mapping theorem, Casorati-Weterstrass theorem, Weierstrass' factorization theorem, little and big Picard theorems, and Riemann surfaces.

Notes for the course will be available on the class web-site.

A reference book for thematerial is
L.D. Berkowitz, Convexity and Optimization in Rn , WileyInterscience 2002

This course will cover theoretical topics in finite dimensionalconvexity and optimization theory including constrained optimization, Lagrangian and duality theories.

The first half of the course will treat the analysis ofunconstrained multidimensional optimization problems, including minimization of convex functions andquadratic forms. Some basic inequalities of analysis will be proved.

The second half will cover convex constrained optimization problems including principles for findingeigenvalues and eigenfunctions of symmetric matrices and the solution of problems with linear equality or inequality constraints. Some theory of Lagrange and KKT multipliers will be described and an introduction to duality theory will be given. Some geometric and otherapplications will be treated.

This is an introduction to the modern control theory of dynamic systems, focusing on typical and characteristic results. Linear and nonlinear continuous-time and discrete-time systems are dealt with for finite state space sets, in either a deterministic or a stochastic framework. Continuous-time stochastic control problems, encountered in modern control theory, and discrete-time Markovian decision problems, typical in operations research, are both treated. Simulation-based approximation techniques for dynamic programming are discussed.

The course is concerned with the development, analysis, and implementation of numerical methods for the solution of initial- and boundary-value problems for systems of ordinary differential equations:

1. Foundations of the theory of ODEs.

2. Numerical methods for non-stiff ODEs.

3. Numerical methods for stiff ODEs.

4. Numerical methods for Differential-Algebraic Equations.

5. Numerical methods for Boundary Value Problems.

Fundamental techniques in high performance scientific computation. Hardware architecture and floating point performance. Pointers and dynamic memory allocation. Data structures and storage techniques related to numerical algorithms. Parallel programming techniques. Code design. Applications to numerical algorithms for the solution of systems of equations, differential equations and optimization. Data visualization. This course also provides an introduction to computer programming issues and techniques related to large scale numerical computation.

The course is an introduction to mathematical statistics. Topics include random sample, data reduction, point estimation, hypothesis testing, interval estimation, and asymptotic statistical analysis.

Arbitrage Theory in Continuous Time, 2nd edition by Tomas Bjork, Oxford University Press 2004

The course is an introduction to continuous time models in finance. We first cover tools for pricing cintingency claims. They include stochastic calculus, Brownian motion, change of measures, and martingale representation theorem. We then apply these ideas in pricing financial derivatives whose underlying assets are equities, foreign exchanges, and fixed income securities. In addition, we will study single factor and multifactor HJM models, and models involves jump diffusions and mean reversion.

This is a topics course that serves as an introduction to graph C*-algebras.  We will begin by introducing C*-algebras and discussing their basic
properties.  Then we will move on to the main focus of the course: We will describe how one may build a C*-algebra from a directed graph, and show that the properties of these graph C*-algebras are reflected in the properties of the associated graph.  This will allow us to read off much of the structure of the C*-algebra from the graph, as well as take complicated operator algebra questions about the C*-algebra and translate them into (easier to answer) graph questions.  Although no prior knowledge of C*-algebras is required, some basic knowledge about operators on Hilbert space will be useful.

The study of rare random events began with Cramer's work on large deviations between empirical and theoretical means of independent random variables. Since then large deviations theory has been extended to a wide range of stochastic processes : Markov chains, diffusions, dynamic systems of interactive particles or cells. We will present the main concepts of large deviations theory and the major theoretical results, and show how they enable practical numerical applications in engineering and in other fields such as evolutionary genetics or simulated annealing.

We'll finish the book of John Lee and cover some other selected topics.

No required textbook.

Lecture notes for the first half of the class will be based on A First
Course in Stochastic Processes, (Karlin and Taylor).

This course will cover a wide range of topics in stochastic processes and applied probability. Main emphasis will be on applied topics in continuous-time stochastic processes and stochastic differential equations (SDEs). Computational projects with Matlab will be given. The following topics will be covered - continuous time Markov chains, Poisson process, Renewal process and the renewal equation, diffusion process, backward and forward equations, connection between partial differential equations and diffusions, adiabatic elimination of fast variables in SDEs, Elements of Queueing theory.

Boolean cubes and Euclidean balls in high dimensions, integration with respect to Gaussian and surface measures of the sphere in high dimensions. Laws of large numbers for independent random variables and random processes. Randomization techniques and metric embeddings. Semigroups and the logarithmic Sobolev inequality on Euclidean spaces and on graphs with suitable connectivity properties.

Recommended:

William Fulton, Joe Harris: Representation Theory: A First Course (Graduate Texts in Mathematics/Readings in Mathematics)

Additional material will be handed out or placed on reserve in the library.

Introduction to Lie groups and algebras we will study the classical semisimple Lie groups and their finite dimensional representations. We will describe the abstract theory behind these: Cartan subalgebras, root systems, highest weight representations, the Weyl character formula.

None

Title: Introduction to Smooth Manifolds
Author: John M. Lee
Publisher: Springer-Verlag

We will study smooth manifolds and structures associated with smooth manifolds. Topics include smooth manifolds, smooth maps, the tangent space, vector bundles, immersions, submersions, embeddings, submanifolds, tensors, differential forms, integration on manifolds, de Rham cohomology, flows, the Lie derivative, and foliations. In addition to this material, we will study aspects of Lie groups, Lie algebras, and Riemannian geometry.

The notes will be largely enough. Having said that, I will give to
the students the reprint of a review article on the splitting written by E. Dean, G.Guidoboni, H. Juarez and myself

Splitting methods have made possible the solution of complicated problems of Science and Engineering. Indeed, there are situations where the only metods available for the solution of a given problem are bases on operator splitting. The main goal of this course is to introduce the student to operator splitting methods. The following methods will be discussed: Douglas-Rachford, Peaceman-Rachford, Lie, Strang, Marchuk-Yanenko. These methods will be applied to the solution of problems from mathematical physics, mechanics and finance, such as Nonlinear Schrodinger and Navier-Stokes equations and parabolic variational inequalities from finance.