Previous Courses
MAT1060HF
Partial Differential Equations I
In this course, we consider key partial differential equations, special classes of their solutions and their stability. Concentrating on the Laplace, heat, Schrödinger and wave equations, important in mathematics and in applications and relevant to the current research, we develop some basic techniques in showing existence, uniqueness and smoothness of their solutions.
Syllabus
- Reaction-diffusion (heat), Schrödinger and wave equations and their key solutions
- Sobolev spaces and second-order elliptic equations
- Linear heat, Schrödinger and wave equations
- Spectral theory and PDEs
- Gradient and Hamiltonian systems
Texts
- L.C. Evans, Partial Differential Equations, AMS, 2nd edition, 2010
- I.M. Sigal, Lectures on Applied Partial Differential Equations, 2019
MAT1508HF / APM446HF
Applied Nonlinear Equations
This course concentrates on partial differential equations appearing in physics, material sciences, biology, geometry, and engineering. It deals with evolution equations, mostly nonlinear, and addresses existence of static, traveling wave, self-similar, topological and localized solutions; stability of the above solutions; and pattern formation. To deal with these questions, the course develops key mathematical techniques, such as fixed point theorems, spectral analysis, and bifurcation theory. Equations considered include the Allen-Cahn, Ginzburg-Landau, Cahn-Hilliard, nonlinear Schrödinger, mean curvature flow, Fisher-Kolmogorov- Petrovskii-Piskunov, Keller-Segel, and Chern-Simons equations.
MAT1739
Partial Differential Equations of Quantum Physics
In this course we consider several key partial differential equations arising in quantum physics: the Hartree, Hartree-Fock, Gross-Pitaevskii (nonlinear Schrödinger), Ginzburg-Landau, Yang-Mills (-Higgs), Chern-Simons, and Schrödinger and wave map equations, which appear in atomic, condensed matter and particle physics. We describe key properties of these equations, isolate their most important solutions, and study existence and stability or instability of these solutions, combining techniques of analysis, PDEs and geometry.
Partial Differential Equations of Quantum Physics (ETH Hönggerberg, Zürich)
In this course we cover several fundamental equations of quantum physics: the Schrödinger equation, which lies at the foundation of Quantum Mechanics; the Gross-Pitaevskii, Landau-Lifshitz and Hartree and Hartree-Fock equations, playing an important role in condensed matter physics; the Ginzburg-Landau equations of superconductivity; and the Yang-Mills equations of particle physics.
MAT1061HS
Partial Differential Equations II
In this course we develop some basic techniques in solving partial differential equations and analyzing their solutions. The long-term goal is to understand principal evolution equations. We establish an existence theory for the selected class of equations, describe their key properties, isolate their most important solutions, and study stability or instability of these solutions. Some non-evolution equations appear as static equations for the evolution ones. Given the time constraint, we are selective about the equations considered, guided by their importance in mathematics and applications, relevance to current research, and the central role of the techniques needed to analyze them.
MAT1063HS
Introduction to Geometric Flows
In this course we study mean curvature, Ricci and harmonic map flows, and also plan to describe the curvature flow of networks of plane curves. We 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, and introduce main techniques such as parabolic existence theory, maximum principles and monotonicity (entropy) formulae. All needed notions from Differential Geometry and Partial Differential Equations are explained, but knowledge of these subjects at an introductory level is required.
Prerequisites: Differential Geometry of Curves and Surfaces; Elementary PDEs
Texts
- K. Ecker, Regularity theory for mean curvature flow, Birkhäuser, 2004
- P. Topping, Lectures on the Ricci flow, London Math Society Lecture Notes Series 325, Cambridge Univ Press, 2006
- Original papers
MAT337H1
Introduction to Real Analysis
The goal of this course is to explain key concepts of Real Analysis with a view to 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
- Vector and normed spaces
- Metric spaces
- Spaces of functions
- Contraction mapping principle
- Applications to ordinary differential equations
- Approximation by polynomials
- Fourier transform
- Wavelets
- Optimization
- Applications to probability theory
Texts
- Kenneth R. Davidson and Allan P. Donsig, Real Analysis and Applications, Springer, 2010
MAT1730HS
Introduction to Quantum Field Theory
The goal of this course is to explain key concepts of Quantum Field Theory and to arrive quickly at topics at the forefront of active research, aiming at physically relevant and mathematically interesting theories. We try to be self-contained and rigorous whenever the rigour is instructive; where a rigorous treatment would be prohibitively time-consuming we give an idea of the proof, if one exists, and/or explain the mathematics involved without all the details.
Prerequisites: The course concentrates on mathematical foundations of Quantum Field Theory. No serious knowledge of physics is necessary. What is needed are the mathematical foundations of Quantum Mechanics, as e.g. in APM421HF Mathematical Concepts in Quantum Mechanics. These include Functional Analysis, Partial Differential Equations and Probability, all on an elementary level. Toward the end of the course some geometrical and topological techniques are used.
Syllabus
- Classical and Quantum Mechanics
- Classical fields and their quantization
- Quantization of Maxwell equations
- Quantum electrodynamics
- Non-linear sigma-model
- Gauge (Yang-Mills) fields
- Path integral
- Symmetry breaking
- Quantum fields as stochastic processes and path integral
Texts
- K. Huang, Quantum Field Theory: From Operators to Path Integrals, John Wiley, New York, 1998
- S. Gustafson and I. M. Sigal, Mathematical Concepts of Quantum Mechanics, 2nd edition, Springer, 2005
MAT1880HS
Mathematical Methods in Biology
In this course we discuss two key groups of biological models intensively studied in recent years. The first deals with collective behaviour of interacting biological organisms such as cells and bacteria (e.g. chemotaxis), describing phenomena such as aggregation and developmental pattern formation. The second deals with mechanisms through which networks of interacting biomolecules (proteins or genes) carry out essential functions in living cells, including how genetic and biochemical networks withstand variations and random perturbations of biochemical parameters. We also discuss mathematical models of the dynamics of HIV-1 and of cancer growth. The models are expressed in terms of Markov chains and stochastic ordinary differential equations; reaction-diffusion equations (e.g. Keller-Segel equations) and stochastic particle dynamics are also used.
Prerequisites: Some familiarity with elementary ordinary and partial differential equations and elementary probability theory. No knowledge of biology is required.