COURSE PURPOSE: Introduction to basic concepts of graph theory.

COURSE CONTENT: Two contributions of Euler - Eulerian tours and Euler characteristic formula. Hamiltonian tours. Chromatic number. Planar graphs including applications to fullerenes and carbon chemistry. Four-color conjecture and 5-color theorem. Decision problems and introduction to computational complexity emphasizing algorithmic solutions. Ramsey-like results including discussion of their applications to foundations of mathematics. Erdos' probabilistic methods with applications to Ramsey theory. Time permitting we will discuss eigenvalues and their significance in chemistry.

EVALUATION AND GRADING: The final grade will be the average of grades received on the first two tests, but active class participation and volunteering solutions of homework problems may be used in the calculation of the final grade to increase it by up to half a point.

BIBLIOGRAPHY: Many topics discussed in this course are included in most of discrete math textbooks.

Sequences and series of functions and uniform convergence. Functions of several variables and the inverse and implicit function theorems. Measure theory, in particular Lebesgue measure for the real line and Euclidean n-space.

An extensive set of additional notes as in 4331 will be provided.

Students should be referred to my web page from the department web pages: http://www.math.uh.edu/~mike/MATH4331

This is a continuation of the study of Differential Geometry from Math 4350. I plan to start with Chapter 4 of the text and continue with Chapter 5. An approximate course outline is as follows:

Chapter 4. The Intrinsic Geometry of Surfaces
4.2 Isometries; Conformal Maps
4.3 Gauss's Theorem Egregium and the Equations of Compatibility
4.4 Parallel Transport; Geodesics
4.5 The Gauss-Bonnet Theorem and its Applications
4.6 The Exponential Map. Geodesic Polar Coordinates
4.7 Further Properties of Geodesics. Convex Neighborhoods

Chapter 5. Global Differential Geometry
5.2 The Rigidity of the Sphere
5.3 Complete Surfaces. Theorem of Hopf-Rinow
5.4 First and Second Variations of the Arc Lenght; Bonnet's Theorem
5.5 Jacobi Fields and Conjugate Points
5.6 Covering Spaces; the Theorems of Hadamard
5.7 Global Theorems for Curves; the Fary-Milnor Theorem
5.8 Surfaces of Zero Gaussian Curvature
5.9 Jacobi's Theorems

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 conversions.

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 (midterm and final). You can improve the grade of your midterm exam with the final, but this implies that for those who choose to do so the final is comprehensible. The midterm gives you 80 pts and the final 110pts. Each homework assignment gives a different number of points (5 for each problem). All grades are summed and divided by the total number of pts you can collect in the course. This gives you your final grade. A quotient of 0.43 or more is D- , of 0.46 or more is D, of 0.54 or more is C, of 0.64 is B-, of 0.80 or more is A- , of 0.85 or more is A.

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 iterative methods for solving linear systems, approximation theory, numerical solutions of nonlinear equations, iterative methods for approximating eigenvalues, and elementary methods for ordinary differential equations with boundary conditions and partial differential equations.

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

This course is a continuation of MATH 4377 which was taught from the same text.

Topics to be covered include Determinants, Elementary Canonical Forms, and the Rational and Jordan Forms.

This course is an introduction to mathematical modeling of topics in investment science and finance. The prerequisites are Calculus III and probability; MATH 2433 and 3338. Students will be expected to use PCs and spreadsheet programs for financial problem-solving.

The course will treat topics in portfolio theory and the capital asset pricing model. Also an introductions to futures, swaps and asset dynamics, including calls and puts. Also American and European options, expiry, strike price, drift and volatility leading to the Black-Scholes pricing formula. The course will include some elementary use of results from optimization theory and stochastic processes.

Evolutionary Theory is a course designed for advanced undergraduate and graduate students who want to understand the mathematical and biological reasoning that underlies evolutionary theory. The course will cover the major theoretical approaches used to study the mechanics of evolution. These will include classical one- and two- locus models, diffusion theory, coalescent theory, quasispecies theory, quantitative genetics, and game theory. Each subject will be illustrated by focusing on those results that have the greatest impact on our thinking about the workings of evolution. These mathematical treatment of these major results will be will be complemented by illustrations, showing exactly how the mathematics relates to the biological insights that they yield. Thus, the students will learn something of the actual mathematical machinery of the theory while gaining a deeper understanding of the evolutionary process. Throughout, emphasis will be placed on the fundamental relationships between the different branches of theory, illustrating how all of these branches are united by a few basic, universal, principles.

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.

Software: The Geometer's Sketchpad, student edition

Topics:


Chapter 2:     Foundations:  Points, Lines, Segments, Angles, ...
                        Axiomatics and Proof
                        The Role of Examples and Models
                        Finite Geometries
                        Incidence Axioms
                        Distance and the Ruler Postulate
                        Angle Measure and the Protractor Postulate
                        Plane Separation

Chapter 3:     Foundations:        Triangles, Quadrilaterals, Circles
                        Congruence Relations
                        Taxicab Geometry
                        Other Congruence Criteria
                        Exterior Angle Theorems
                        An Introduction to Hyperbolic Geometry
                        An Introduction to Spherical Geometry
                        Quadrilaterals

Chapter 4:    Alternate Concepts for Parallelism
                        Euclidean Parallelism
                        Parallel Projection
                        Similar Triangles
                        Circle Theorems

Chapter 6      Alternate Concepts for Parallelism:  Non-Euclidean Geometry
                        Hyperbolic Geometry

Modules over Principal Ideal Domains with applications to finitely generated abelian groups and normal forms of matrices; Sylow theory, Universal algebraic constructions, like co-products, ultraproducts and ultrapowers of the real numbers.

This course is a continuation of MATH 4377 which was taught from the same text.

Topics to be covered include Determinants, Elementary Canonical Forms, and the Rational and Jordan Forms.

Sequences and series of functions and uniform convergence. Functions of several variables and the inverse and implicit function theorems. Measure theory, in particular Lebesgue measure for the real line and Euclidean n-space.

An extensive set of additional notes as in 4331 will be provided.

Students should be referred to my web page from the department web pages: http://www.math.uh.edu/~mike/MATH4331

Absolutely continuous functions and the extension of the fundamental theorem of calculus.

A short overview of cardinal and ordinal numbers/well ordering of sets. An introduction to Topology: Open, closed sets and related concepts; Continuity, Compactness, Connectedness, Countability, Separability. Arzela-Ascoli and Stone-Weierstrass theorems.

A short overview of Functional Analysis: Bounded operators on normed spaces, Banach spaces, Banach spaces and the Baire Category theorem, Lp-spaces.

Hilbert spaces and orthonormal bases. The L2-spaces of periodic functions; Fourier series.

Locally convex topological vector spaces and weak topologies: A brief introduction; The Banach-Alaoglu theorem.

Locally compact spaces: Their topology, Regular measures on LCS; The Riesz Representation theorem.

The Integral Fourier transform: Definition, main properties, inversion, Plancherel's theorem.

A) Required text:

Introduction to Dynamical Systems
Authors: Michael Brin and Garrett Stuck
Publisher: Cambridge University Press

B) Recommended

1) An Introduction to Ergodic Theory
Author: Peter Walters
Publisher: Springer

2) Introduction to the Modern Theory of Dynamical Systems
Authors: Anatole Katok and Boris Hasselblatt
Publisher: Cambridge University Press

This course is an introduction to modern dynamical systems. Topics covered include basic examples and constructions, symbolic dynamics, ergodic theory, hyperbolic dynamics, low dimensional and complex dynamics.

This course will start with the representation theory of finite groups. We will study groups acting on sets, induced representations, irreducible representations, the left regular representation, group algebras and character theory. We will then study topological groups, compact topological groups, Haar measures, amenable groups and finish with the Peter-Weyl theory.

It will be assumed that the student has a good foundation in linear algebra and is familiar with the basics of topology and measure theory.

There is no required text for the course. Two good references for the material are

1. L.D. Berkowitz, Convexity and Optimization in R^n, Wiley Interscience2002
2. John L Troutman, Variational Calculus with Elementary Convexity (2nd ed)Springer.

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

The first half of the course will cover theoretical issues in finite dimensional optimization, including existence results and necessary and sufficient optimality conditions. Also an introduction to the theoryof convex sets and functions, convex constrained optimization, conjugate functions and duality. This theory will be used to prove various well-known inequalities and to study linear eigenvalue problems.

The second half of the course will treat variational problems for 1-d integral functionals and their applications to two-point boundary value problems. This will include the derivation of the Euler-Lagrange equations and Hamiltonian formulations of the equations.

Also Sturm-liouville eigenvalue problems and some isoperimetric problems.

This course will be a mix of mathematics and practicalities in numerical optimization. We will look at the following topics: linear programming, (small and large scale) nonlinear programmming, and (depending on the students' interests) a short introduction to dynamic programming. This is the second semester of a two semester course but it will be self-contained and so the first semester is not a prerequisite.

We will focus on numerical linear algebra, including direct methods for the solution of linear systems, eigenvalue problems, iterative methods for the solution of large linear systems. We will also discuss numerical solutions of boundary value problems for ordinary differential equations and cover briefly of numerical solutions of three basic partial differential equations.

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.

This course is an introduction to mathematical statistics. It assumes a knowledge of probability at the level of Rosenthal's book "A first look at rigorous probability".

Topics covered include random samples, point and interval estimation, hypothesis testing, analysis of variance and regression, and if time permits asymptotic evaluations.

"Arbitrage Theory in Continuous Time" by Tomas Bjork, Oxford University Press, 2004, ISBN 0-19-927126-7.

Reference book: "Financial Calculus: An introduction to Derivative Pricing" by Martin Baxter and Andrew Rennie, Cambridge University Press, 1996, ISBN 0521552893

This is a continuation of the course enetitled "Discrete-Time Models in Finance." The course studies the roles played by continuous-time stochastic processes in pricing derivative securities. Topics incluse stochastic calculus, martingales, the Black-Scholes model and its variants, pricing market securities, interest rate models, arbitrage, and hedging.

This is the continuation of Math 6395 Complex Geometry and Analysis (I) offered by Dr. Ji last semeter. We'll first cover most part of the Chapter 0 and Chapter 1 of Griffith & Harris' book. We then will go through some part of Demailly's note: L² vanishing theorems for positive line bundles and adjunction theory; lecture notes of a CIME course on "Transcendental Methods of Algebraic Geometry" (Cetraro, Italy, July 1994), arXiv:alg-geom/9410022.

Note that for the materials which overlap with Dr. Ji's, we will briefly review them, so even you didn't take the first part, it is still possible to take the second part.

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, Large deviations theory, Wentzel-Freidlin theory for SDEs.

I encourage students to sign up even if you did not take the first semester, since the semesters will be fairly disjoint, so long as you know sme basics about Banach and Hilbert spaces.

I. Operator theory on Hilbert and Banach spaces
We discuss for example Fredholm theory, Sturm-Liouville systems, and spectral theory.

II. Algebras and spectral theory.
Banach algebras. Commutative Banach algebras and the Gelfand transform. The characterization of commutative $C^*$-algebras and the functional calculus for normal operators. The spectral theorem for normal operators. The Fourier Transform for locally compact groups.

III. Unbounded operators.

IV. Students requests.

This course describes the basic notions and constructions of differential geometry, and some of the more advanced results. It includes: manifolds, the inverse and implicit function theorems, submanifolds, partitions of unity; tangent bundles, vector fields, the Frobenius theorem, Lie derivatives, vector bundles; differential forms, tensors and tensor fields on manifolds; exterior algebra, orientation, integration on manifolds, Stokes' theorem; Lie groups. A few additional topics might be also covered, depending on the interest of the audience.

recommended only for the theoretical part of the Course:

D. Braess, Finite elements: Theory, fast solvers, and applications in solid mechanics. Cambridge University Press. 1997

S. Brenner and R. Scott. The mathematical theory of finite element methods. Springer-Verlag, 2002

The following text may be helpful:

R.GLOWINSKI, J.L. LIONS & J. HE, Exact and Approximate Controllability for Distributed Parameter Systems: A Numerical Approach, Cambridge University Press, Cambridge, 2008.

The main goal of this course is to introduce the student to a very active research area of Applied and Computational Mathematics with relevance to a large variety of problems from natural and engineering sciences. The discussion will include:

1)Motivation for control and controllability studies.
2)Mathematical aspects and optimality conditions via adjoint equation methods
3)Iterative solution of the control problems
4)Memory saving techniques for actual computations
5)An introduction to Riccati equation based control methods.
6)Control of diffusion systems, of wave models
7)Bilinear control of Schrödinger equation type models.