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Thursdays 3-6 pm (UC 177)
In this course we study some of the key nonlinear partial differential equations appearing in physics, geometry, material sciences, biology and engineering. We address questions of existence, long-time behaviour, formation of singularities, pattern formation, static, traveling wave, self-similar, topological and localized solutions and their stability. We also describe some of the central techniques in this area. The equations are selected according to their importance in applications, their current interest and providing illustration of key techniques used in this field. Specifically, we consider equations from the following list: Allen-Cahn equation (material science), Mean curvature flow (geometry), Ginzburg-Landau equation (condensed matter physics - superfluidity and superconductivity), Cahn-Hilliard (material science, biology), Fisher-Kolmogorov-Petrovskii-Piskunov (combustion theory, biology), Keller-Segel equations (biology), nonliear Schroedinger equation (Bose-Einstein condensation, nonlinear optics), Chern-Simmons equations (quantum Hall effect, elementary particle physics), and Free boundary problem (material sciences).
Wednesdays 2-4 pm, Fridays 4-5 pm (SS 2135)
The goal of this course is to explain key concepts of Real Analysis with the view at applications. The course is about the same level as MAT357, but while MAT357 deals mainly with theory, the present course aims at developing interesting applications.
Syllabus:
Texts
Kenneth R. Davidson and Allan P. Donsig,
Real Analysis and Applications, Springer, 2010.
Wednesdays 5-6 pm, Fridays 2-4 pm (BA 2179)
In this course we will study mean curvature, Ricci and harmonic map flows. We also plan to describe the curvature flow of networks of plane curves. We will give careful definitions of these flows, present existence results and results on formation of singularities (e.g. collapse to a point and neck-pinching) and soliton dynamics. We will also introduce main techniques, such as parabolic existence theory, maximum principles and monotonicity (entropy) formulae.
We will explain all needed notions from Differential Geometry and Partial Differential Equations, but knowledge of these subjects on an introductory level is required for this course
Prerequisites for this course:
Differential
Geometry of Curves and Surfaces; Elementary PDEs
References for the course:
K.Ecker,
Regularity theory for
mean curvature flow
,
Birkhaeuser, 2004; ISBN 08 176 32433
P.Topping, Lectures on the Ricci flow , London math society lecture notes series 325, Cambridge Univ Press, 2006; ISBN 0-521-68947-3. (The book can be downloaded from the webpage of Prof. Peter Topping)
Original papers
In this course we study some of the key partial differential equations appearing in Geometry, Topology, Physics, Material Sciences, Biology and Engineering. We will address questions of existence, long-time behaviour, formation of singularities, pattern formation, static, traveling wave, self-similar, topological and localized solutions and their stability. The equations are selected according to their importance in mathematics and its applications, their current interest and the way they illustrate key techniques used in this field.
We can digress to other fields of mathematics, such as differential geometry. The course will consist of the instructor lectures and presentations by students ranging from 20 to 50min.
The goal of this course is to explain key concepts of Quantum Field Theory and to arrive quickly to some topics which are at the forefront of active research. We will aim at physically relevant and mathematically interesting theories. 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: The course will concentrate on mathematical foundations of Quantum Field Theory. No serious knowledge of physics is necessary for this course. What is needed are the mathematical foundations of Quantum Mechanics, as e.g. in APM421HF Mathematical Concepts in Quantum Mechanics course. The latter include Functional Analysis, Partial Differential Equations and Probability, all on an elementary level. At the end of the course I plan to use some geometrical and topological techniques.
Syllabus:
Texts
K. Huang,
Quantum Field Theory: From Operators to Path Integrals, John Wiley, New York, 1998.
ISBN 0-471-14120-8
We will also use
S. Gustafson and I. M. Sigal,
Mathematical Concepts of Quantum Mechanics, 2nd edition, Springer, 2005
The goal of this course is to explain key concepts of Real Analysis with the view at applications. The course is about the same level as MAT357, but while MAT357 deals mainly with theory, the present course aims at developing interesting applications.
Syllabus:
Texts
Kenneth R. Davidson and Allan P. Donsig,
Real Analysis and Applications, Springer, 2010.
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.
Page last updated: January, 2012