### Fractal geometry : mathematical methods, algorithms, applications

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Some examples of individual staff members activities now follow. Jonathan Blackledge has been working on research contracts with Nokia in Finland since January This work involves the use of stochastic self-affine models for speech recognition and synthesis and the use of fractal modulation for secure digital communications.

### Frontiers in Massive Data Analysis

It is related to another government funded project with Peninsula Technikon Cape Town South Africa supporting PhD student Mr R Key to investigate different stochastic models self-affine or otherwise for use in telecommunications systems. This collaboration has resulted in the establishment of a new postgraduate research programme on encryption techniques for E-commerce based in Russia. It has also resulted in the establishment of a project started in September and funded by the Russian Ministry of Education to investigate the use of non-linear dynamical systems for modelling financial time series with PhD student Ms I Lvova.

He has collaborated for many years with Hong Kong University and has attracted a postdoctoral fellow funded by the Chinese government Dr Zhang Zong to work on methods of oscillatory quadrature and its applications to computing inverse integral transforms. Dr A Khramtovsky. In the course I will show many applications of Functional Analysis in different areas of mathematics.

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Linear transformations on finite dimensional vector spaces are studied in a semi-abstract setting. The emphasis is on topics and techniques which can be applied to other areas, e. Topics will include spectral theory of self-adjoint mappings, calculus of matrix valued functions, matrix inequalities, convexity, duality theorem and normed linear spaces. The course is the first term of a two-term graduate algebra sequence. It covers the theory of groups, the theory of fields as well as Galois theory.

Highlights of the course will include: Sylow's theorems, the structure of finitely generated abelian groups, the fundamental theorem of Galois theory, and applications such as the solvability of polynomial equations by radicals and geometric constructions with a ruler and a compass.

This is a third course in algebra. It covers the classical results on the structure and representation theory of associative algebras culminating with modern developments such as the theory of quiver algebras and categorification. Highlights of the course include: Wedderburn-Artin theory, the structure of central simple algebras, the structure of finite dimensional algebras, semisimple algebras, the character theory of finite groups, theorems of Maschke, Frobenius, Burnside, quiver algebras and quiver representations, reflection functors, Gabriel's theorem, tensor categories, fusion categories.

### Account Options

This is a fourth course in algebra. It covers introductory topics in algebraic geometry, number theory, and representation theory, selected by the instructor. This course studies the mathematical analysis and practical implementation of discontinuous Galerkin methods for approximating elliptic, parabolic, and hyperbolic partial differential equations. The course will cover several topics in discretizations of partial differential equations and the solution of the resulting algebraic systems.

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Topics in discretizations will include mixed finite element methods, finite volume methods, mimetic finite difference methods, and local discontinuous Galerkin methods. Topics in solvers will include domain decomposition methods and multigrid methods. Applications to flow and transport in porous media, as well as coupled fluid and porous media flows will be discussed. The Advanced Scientific Computing sequence covers topics chosen at the leading edge of current computational science and engineering for which there is sufficient interest.

The course focuses on the fundamental mathematical aspects of numerical methods for stochastic differential equations, motivated by applications in physics, engineering, biology, economics. It provides a systematic framework for an understanding of the basic concepts and of the basic tools needed for the development and implementation of numerical methods for SDEs, with focus on time discretization methods for initial value problems of SDEs with Ito diffusions as their solutions.

The course material is self-contained. The topics to be covered include background material on probability, stochastic processes and statistics, introduction to stochastic calculus, stochastic differential equations and stochastic Taylor expansions. The numerical methods for time discretization of ODEs are briefly reviewed, then methods for time discretization for SDEs are introduced and analyzed. A first course in differential geometry.

Topics may include the geometry of curves and surfaces eg. Gauss map, fundamental forms, curvature , differentiable manifolds, Lie groups, tangent and tensor bundles, vector fields, and Riemannian structures.

## Fractal Geometry: Mathematical Methods, Algorithms, Applications

This course is a continuation of Differential Geometry 1. Applications may also be included. This course will cover Sobolev spaces, second order elliptic equations, weak solutions, linear evolution equations, semigroup theory, and Hamilton-Jacobi theory, and other topics in nonlinear PDE. PDE 1 Math is not a pre-requisite but a good background in analysis is necessary.

This is the first course in a two-term sequence designed to acquaint students with the fundamental ideas involved in the study of ordinary differential equations.

Basic existence and uniqueness of solutions as well as dependence on parameters will be presented. Students will also be introduced to geometric concepts such as stability of fixed points and invariance. This first term will provide an excellent introduction to ODE theory for students interested in applied mathematics. This course, which follows Math , presents a dynamical systems approach to the study of ordinary differential equations. Topics include geometric theory including proofs of invariant manifold theorems, flows on center manifolds and local bifurcation theory, the method of averaging, Melnikov's method, and an introduction to Smale horseshoes and chaos theory.

The prerequisite for this course is a one undergraduate semester course in complex variables with a grade of B or higher. The grade will be determined from assigned homework problems. This course covers methods that are useful for solving or approximating solutions to problems frequently arising in applied mathematics, including certain theory and techniques relating to the spectral theory of matrices, integral equations, differential operators and distributions, regular perturbation theory, and singular perturbation theory. This course is for all graduate students not under the direct supervision of a specific faculty member.

In addition to a student's formal course load, this study is for preparation for the preliminary, comprehensive and overview examinations.

This course will introduce students to the subject of calculus of variations and some of its modern applications. Topics to be covered include necessary and sufficient conditions for weak and strong extrema, Hamiltonian vs Lagrangian formulations, principle of least action, conservation laws and direct methods of calculus of variations. Extensions to the functionals involving higher-order derivatives, variable regions and multiple integrals will be considered. The course will emphasize applications of these ideas to numerical analysis, mechanics and control theory.

Prerequisite s : single-variable and multivariable calculus, some exposure to ordinary and partial differential equations. All other concepts, such as function spaces and the necessary background for the applications, will be introduced in the course. Beginning graduate students and advanced undergraduates are welcome.

The course objective is to introduce students to formulating, debugging and solving finite element simulations of practical applications, with a focus on the equations of fluid flow. FEniCS is less tightly integrated, consisting of a collection of functions for specifying the mathematical formulation as well as functions for interfacing with other packages for mesh generation, post processing, and numerical solution. This course will focus on using Python.

Python is a widely-used language with applications far removed from finite element modelling and can be the subject of multiple-semester courses. Although previous experience with Python would be valuable, it is not necessary. The basics of the language plus those features necessary for this course will be presented during the lectures. Previous experience with finite element methods will be valuable, but is not required because the theory will be summarized during the lectures. Various boundary conditions and finite elements will be presented, as well as the effect of these choices on solution methods.

This course is an introduction to the theoretical and computational aspects of the finite element method for the solution of boundary value problems for partial differential equations. Emphasis will be on linear elliptic, self-adjoint, second-order problems, and some material will cover time dependent problems as well as nonlinear problems.

Topics include: Sobolev spaces, variational formulation of boundary value problems, natural and essential boundary conditions, Lax-Milgram lemma, approximation theory, error estimates, element construction, continuous, discontinuous, and mixed finite element methods, and solution methods for the resulting finite element systems. Prerequisite s : Good undergraduate background in linear algebra and advanced calculus.

Familiarity with partial differential equations will be useful.

This course provides an introduction to the mathematical subjects required for the mathematical finance program, and assumes that the student has an undergraduate degree with some technical component e. Students are expected to have knowledge of Multivariable Calculus and Linear Algebra, and any sections on these topics will be presented as review.

## Research - Mälardalen University Sweden

No financial background is required, but many of the examples and llustrations of the mathematics will be drawn from economics and finance. The course with its pre-sequel MATH present fundamental principles and standard approaches used in mathematical finance. This course will investigate the mathematical modeling, theory and computational methods in modern finance. The main topics will be i basic portfolio theory and optimization, ii the concept of risk versus return and the degree of efficiency of markets, iii discrete models in options.

This course describes a number of topics related to mathematical biology. This year we will cover several areas of interest including pattern formation in reaction-diffusion and advection models with applications to immunology, chemotaxis, etc; evolutionary dynamics such as the evolution of cooperation, some game theory, and replicator dynamics; and some cell physiology modeling such as the cell cycle and simple circadian models. The prerequisites are some simple differential equations, a bit of Fourier transforms, and some knowledge of software to numerically solve the various equations.

The main goal of the course is to understand the structure and classification of complex semisimple Lie algebras as well as their basic representation theory and the relationship with Lie groups. Highlights will include, the theorems of Engel, Cartan and Weyl, root systems, the Harish-Chandra isomorphism and various formulas for characters and weight multiplicities. This course covers special topics in pure mathematics.