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Applied Nonlinear Equations

Thursday 5:00-6:00 BA6183, Friday 3:00-5:00 BA6183

TA: Feodor Kogan

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 the questions of:

- existence of static, traveling wave, self-similar, topological and localized solutions;
- stability of the above solutions;
- pattern formation.

To deal with these questions, the course will develop some key mathematical techniques, like:

- the fixed point theorems,
- spectral analysis,
- bifurcation theory.

The following list gives an idea of the equations considered in this course:

- Allen-Cahn equation (material science),
- Ginzburg-Landau equation (condensed matter physics),
- Cahn-Hilliard (material science, biology),
- nonlinear Schrödinger equation (quantum and plasma physics, water waves, etc.)
- mean curvature flow (geometry, material sciences),
- Fisher-Kolmogorov-Petrovskii-Piskunov (combustion theory, biology),
- Keller-Segel equations (biology),
- Chern-Simons equations (particle and condensed matter physics).

Link to course description for more information.

Mathematical Concepts of Quantum Mechanics and Quantum Information

The goal of this course is to explain key concepts of Quantum Mechanics and to arrive quickly to the topics which are at the forefront of active research, the density functional theory and quantum information. The latter subjects have witnessed an explosion of research in the last decade and both involve deep and beautiful mathematics.

**Prerequisites for this course:**
I will cover all necessary definitions beyond
multivariable calculus and linear algebra, but without
familiarity with elementary ordinary and partial differential
equations, the course will be tough. Knowledge of elementary theory
of functions and operators would be helpful.
No physics background is required.

**Format:**
Lectures with occaisional class discussions.

**Syllabus:**

- Schrödinger equation
- Quantum observables
- Spectrum and evolution
- Spin and statistics
- Atoms and molecules
- Quasiclassical asymptotics
- Adibiatic theory and geometrical phases
- Feynman and Wiener path integrals
- Density functional theory
- Quantum statistics
- Quantum information
- Nano-devices

**Textbook**

S. Gustafson and I. M. Sigal, *Mathematical
Concepts of Quantum Mechanics*, Springer

For the material which is not in the book, I will post the lecture notes (link below)
or refer to on-line material.

Partial Differential Equations of Quantum Physics

In this course we will consider several key partial differential equations arising in quantum physics. Specifically, we will study the Hartree, Hartree-Fock, Gross-Pitaevskii (nonlinear Schrödinger), Ginzburg-Landau, Yang-Mills (-Higgs), Chern-Simons and Schrödinger and wave map equations, appearing in atomic, condensed matter and particle physics and playing an important role there.

We describe key properties of these equations, isolate their most important solutions and study existence and stability or instability of these solutions. The techniques will combine those of analysis, PDEs and geometry.

Partial Differential Equations I

In this course, we consider key partial differential equations of the first and second order and develop some basic techniques in showing existence, uniqueness and smoothness of their solutions. Specifically, we consider the transport, Laplace, heat and wave equations. These equations are important in mathematics and in applications, relevant to current research and the techniques needed to analyze these equations play the central role in PDEs and beyond.

**Syllabus:**

- The Laplace equation
- The heat equation
- First-order equations
- The wave equation

**References**

We follow

R. McOwen, *Partial Differential Equations*, 2nd edition, Prentice-Hall, 2003

Much of the material we cover could also be found in

L.C. Evans, *Partial Differential Equations*, AMS

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.

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 of non-evolution equations appear as static equations for the evolution ones. Given the time constraint, we have to be very selective about the equations we consider. The guiding principles are the importance of the equations in mathematics and in applications, relevance to the current research and the central role of the techniques needed to analyze these equations.

Introduction to Geometric Flows

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,

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

Introduction to Real Analysis

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:**

- 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.

Introduction to Quantum Field Theory

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:**

- Classical and Quantum Mechanics
- Classical fields and their quantization
- Quantization of Maxwell equations
- Quantum electrodynamics
- Non-linear sigma-model
- Gauge (Yang-Mill) 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. ISBN 0-471-14120-8

We
will also use S. Gustafson and I. M. Sigal, *Mathematical Concepts of Quantum
Mechanics*, 2nd edition, Springer, 2005

Mathematical Methods in Biology

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 2019