| Home | Abridged C.V. | Publications | Research Results | Information for Students | Current Courses |
September, 2006
The goal of this course is to explain key concepts of Quantum Mechanics and to arrive quickly to some topics which are at the forefront of active research, such as Bose-Einstein condensation, control of chemical reactions and quantum information. We will try to be as self-contained as possible and rigorous whenever the rigour is instructive. Whenever the rigorous treatment is prohibitively time-consuming we give an idea of the proof, if such exists, and/or explain the mathematics involved without providing all the details.
Prerequisites for this course: some familiarity with elementary ordinary and partial differential equations and elementary theory of functions and operators.
Syllabus:
References
S. Gustafson and I. M. Sigal, Mathematical Concepts of Quantum Mechanics,
Springer
A. S. Holevo, Probabilistic and Statistical Aspects of Quantum Theory, Amsterdam, The Netherlands: North Holland.
In this course we discuss two key groups of biological models which were intensively studied in the last few years. The first group deals with collective behaviour of interacting biological organisms such as cells and bacteria (e.g. chemotaxis). The goal here is to describe such phenomena as aggregation (congregation of cells or bacteria into tightly bound, rigid colonies) and developmental pattern formation.
The second group of models deals with mechanisms through which networks of interacting biomolecules (proteins or genes) carry out the essential functions in living cells. Among the questions which are addressed here is how the genetic and biochemical networks withstand considerable variations and random perturbations of biochemical parameters. The complexity and high inter-connectedness of these networks makes the question of the stability in their functioning of special importance.
Finally we will discuss mathematical models of the dynamics of HIV-1 and of cancer growth.
The models above are expressed in terms of Markov chains and stochastic ordinary differential equations. In addition, in the first case, reaction-diffusion equations (e.g. Keller-Segel equations) and stochastic particle dynamics are used. This mathematical background together with its biological interpretation will be developed in the course.
Prerequisites for this course: some familiarity with elementary ordinary and partial differential equations and elementary probability theory. No knowledge of biology is required.