Academic year 2021-2022 : Mathematical Physics, Analysis and Geometry.

The program of this year is devoted to the part of mathematical physics which is linked to analysis and geometry. Most theories in physics are formulated in terms of partial differential equations. Examples are electrodynamics (Maxwell equations), quantum mechanics (Schrödinger equation), general relativity (Einstein equations) and fluid mechanics (Navier Stokes equations). Existence, uniqueness and long term behavior of the solutions of these equations are studied by sophisticated tools from functional analysis. Examples are Sobolev spaces, Spectral Analysis, the theory of semigroupes etc. The structure of the equations is in many cases very much linked to geometry. The most striking example are the Einstein equations, where the solution itself is a lorentzian manifold, but geometry is also very important in the study of many other of these equations. For example quantum systems are in certain high energy limits well descriped by classical systems. This observation lead to the development of microlocal and semiclassical analysis which itself is very much based on symplectic geometry. All of the above mentioned equations are very active fields of research and the aim of the program is to introduce the students to the most modern methods in the study of these equations. The courses of the first semester provide the necessary background on analysis and geometry as well as an introduction to partial differential equations. The two advanced courses are devoted to quantum chaos and the equations of fluid mechanics.



1 Basic courses


1.1 Partial differential equations (Eric Dumas and Christophe Lacave)

This fundamental course is a general introduction to the wide field of partial differential equations, with a special emphasis on evolution problems and semigroup theory. The aim is to present a few standard techniques which make it possible to solve the Cauchy problem for a large class of linear or semilinear evolution equations. Such problems occur in a variety of situations, including fluid mechanics, nonlinear optics, quantum physics, population dynamics, and general relativity. In particular, this course will give the necessary material to tackle the equations of fluid mechanics, which will be studied in greater depth in a advanced course during the second semester. No previous knowledge of the subject is required, but the students are supposed to be familiar with Lebesgue's integration theory, calculus in Lp spaces, and elementary functional analysis. Basic tools such as fundamental solutions of classical linear PDEs will be recalled in the first lectures, and essential notions in Sobolev spaces and distribution theory will also be reviewed. Moreover, all necessary notions in operator theory and spectral theory will be studied in another fundamental course, to be taught in parallel.

Preliminary program :

- Representation formulas. Fundamental solution of a few linear PDEs, such as the heat equation, the wave equation, and the Schrodinger equation.

-Sobolev spaces. Elementary distribution theory, Sobolev spaces. Embeddings and traces.

- Second order elliptic equations. Weak solutions of linear elliptic equations of second order, existence and uniqueness, interior and boundary regularity.

- Linear hyperbolic systems. Plane waves, introduction to geometrical optics, propagation of singularities.

- Linear and semilinear evolution equations. Strongly continuous and analytic semigroups, Cauchy problem and regularity properties. Mild and classical solutions of abstract semilinear evolution equation in Banach spaces.

- Nonlinear parabolic equations. The nonlinear heat equation : comparison theorems, global existence, asymptotic behavior of solutions, blow-up criteria. Introduction to reaction-diffusion systems, travelling waves.

- Nonlinear dispersive equations. Nonlinear Schrodinger equation, Strichartz estimates, scattering theory, blowup phenomena.

-Nonlinear hyperbolic equations. First order equations and systems, method of characteristics, entropy criterion for weak solutions, Burgers and Vlasov-Poisson equations. Nonlinear wave equations, Strichartz estimates, scattering and blow-up.

References :
H. Brezis, Functional Analysis, Sobolev Spaces and Partial Di erential Equations, Springer, 2010.
K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer, 1999.
L. C. Evans, Partial di erential equations, Graduate Studies in Mathematics 19, AMS, 1998.
D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, 2001.
W. Strauss, Partial Di erential Equations : An Introduction, Wiley, 2008.
M. Taylor, Partial Differential Equations I-III, second edition, Springer, 2011.


1.2 Spectral and semiclassical analysis (Alain Joye)
The aim of this basis course consists in presenting some of the fundamental tools from spectral analysis with a view to their applications to problems ranging from the analysis of nonlinear evolution equations to semiclassical analysis.

Preliminary program
-Linear operators on Banach and Hilbert spaces. Resolvent, spectrum. Adjoint operators, selfadjoint operators. Spectral theorem.
-One parameter semigroups. Stone's and Trotter's theorem, ergodic theorem. Characterization of contraction semigroups, perturbation formulas.
-Semiclassical analysis. Pseudodifferential operators, Weyl quantization, Egorov's theorem, Garding's inequality, and semiclassical functional calculus.

References :
E. B. Davies, Linear Operators and their Spectra, Cambridge University Press (2007).
T. Kato, Perturbation Theory of Linear Operators, CIM, Springer (1981).
A. Martinez, An Introduction to Semiclassical and Microlocal Analysis, Springer (2002)
M. Reed, B. Simon, Methods of Modern Mathematical Physics, Academic Press (1972).
M. Zworski, Semiclassical analysis, Graduate Studies in Mathematics, 139, AMS (2012)

1.3 Riemannian and symplectic geometry (Emmanuel Russ and Dietrich Häfner)
This course is an introduction to Riemannian and symplectic geometry. After having introduced the basic objects of the theory like riemannian and symplectic manifolds, curvature etc., we describe how these tools can be used to study partial differential equations.

Preliminary program
-Manifolds. Vector bundles, tensor bundles, tensor fields. Riemannian manifolds. Riemannian metrics, model spaces in Riemannian geometry.
-Riemannian connections. Exponential map, geodesics. Length, distance, minimizing curves, completedness.
-Curvature. Ricci and scalar curvature. Manifolds of positive curvature, of negative curvature.
-Symplectic manifolds. Definition, examples. Lagrangian submanifolds. Hamiltonian vector fields.
-Link to semiclassical analysis and partial differential equations. Propagation of singularities. Semiclassical approximation of solutions of linear evolution equations at high frequencies.

References :
M. do Carmo, Riemannian geometry, Mathematics : Theory and Applications, Birkhauser, 1992.
S. Gallot, D. Hulin, J. Lafontaine, Riemannian geometry, Universitext, Springer, 2004.
J. M. Lee, Riemannian manifolds, an introduction to curvature, Graduate Texts in Mathematics, 176, Springer, 1997.
M. Taylor, Partial differential equations 1-3, Applied mathematical sciences, Springer 2011.


2 Advanced courses

2.1 Mathematical fluid mechanics (Thierry Gallay)
The aim of this advanced course is to give a self-contained introduction to the mathematical theory of incompressible fluids, starting from the derivation
of the fundamental equations of motion and ending with an overview of current research in this very active field. No previous knowledge of fluid mechanics is required, all the prerequisites being covered by the PDE course in the first semester. The emphasis is put on the qualitative behavior of the solutions of the evolution problem, especially in domains without boundaries where the vorticity equation can be used to gain some insight into the dynamics. Flows with localized vorticity, such as isolated vortices in 2D or vortex laments in 3D will receive a special attention. In contrast, important questions regarding fluid-solid interactions, free surfaces, or fully developed turbulence will not be addressed, due to limited time.

Preliminary program :
-Derivation of the fundamental equations. Continuity equation, momentum and energy balance, Euler equations for perfect fluids. Internal friction, rate-of-strain tensor, Navier-Stokes equations for viscous fluids. Boundary conditions.
-Perfect fluids. Classical solutions of Euler equations in a smooth domain. Vorticity formulation, Yudovich's theory for weak solutions in R2. Flow past an obstacle, the d'Alembert paradox. Shear flows and stability criteria.
-Viscous fluids without boundaries. The Cauchy problem for the Navier-Stokes equations in R2 and R3. Critical function spaces, Fujita-Kato theory, and the Millenium Problem.
-Boundary layers and inviscid limit. The Navier-Stokes equations in a smooth two-dimensional domain. Prandtl's theory of boundary layers. Inviscid limit,
the Kato criterion. The drag force around an obstacle, Stokes' formula. The plane and cylindrical Poiseuille flows.
-Vortices and filaments. Isolated vortices in inviscid planar fluids, the point vortex system. Oseen vortices and long-time asymptotics of two-dimensional viscous fluids. Vortex laments, binormal flow, existing results and conjectures.

References :
F. Boyer and P. Fabrie, Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models, Applied Mathematical Sciences 183,
Springer, 2013.
J.-Y. Chemin, Perfect Incompressible Fluids, Oxford Lecture Series in Mathematics and Its Applications 14, Clarendon Press, 1998.
P. Constantin and C. Foias, Navier-Stokes Equations, Chicago Lectures in Mathematics, 1989.
Y. Giga and A. Novotny (Ed.), Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, Springer, 2018.
A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow, Cambrige texts in applied mathematics 27, Cambridge 2001.
C. Marchioro and M. Pulvirenti, Mathematical Theory of Incompressible Nonviscous Fluids, Applied mathematical sciences 96, Springer, 1994.
P.-G. Lemarie Rieusset, The Navier-Stokes Problem in the 21st Century, Chapman and Hall/CRC, 2016.

2.2 Classical and quantum chaos (Frédéric Faure)
Chaotic classical dynamics is the property of some deterministic dynamical systems to posses sensitivity to initial conditions. An important example in geometry is the geodesic flow on negatively curved Riemannian manifolds. This is the Hamiltonian flow of a free particle on the manifold. The purpose of dynamical system theory is to predict long time behavior of particles. Chaos implies here that a typical trajectory seems unpredictable and behaves as random. It is then preferable to consider evolution of probability distributions under the flow. Then it appears that statistical properties of a typical trajectory can be computed and the evolution of probability distributions is predictable : it converges towards a uniform measure called equilibrium (this is the mixing property) and the transient evolution is governed by an effective Schrödinger equation (namely a wave equation in "quantum chaos"), whose discrete spectrum is called Ruelle-Pollicott resonances. We will describe this recent functional approach in dynamical systems theory that has given many interesting results in this field and is still very active. Quantum chaos is the manifestation of chaotic dynamics in quantum dynamics, i.e. evolution of functions (or waves) under the Schrödinger equation when the corresponding classical mechanics is chaotic. The corresponding
important example is the wave equation on negatively curved Riemannian manifolds. The purpose of quantum chaos theory is to predict long time behavior of waves, equivalently properties of stationary waves, i.e. eigenfunctions and eigenvalues of the Schrodinger operator. The diculty is interference phenomena that are complicated due to chaos. Some known results is the Theorem of "quantum ergodicity" that shows the equidistribution of most of eigenfunctions. Some important conjectures are the random matrix conjecture and the unique quantum ergodicity conjecture : they claim that (for a generic system) all eigenfunctions equidistribute and behave at high frequencies as eigenfunctions of a universal model of random matrices. Quantum chaos theory is important in physics of waves (acoustic, electromagnetism, quantum mechanics etc) but also in mathematics, in group theory, in number theory where a very conjectural manifestation would be statistics of random matrices in the distribution of prime numbers and in the zeros of the Riemann zeta function.

Preliminary program :
-Classical mechanics, deterministic chaos. Example and properties of chaotic (i.e. hyperbolic, Anosov dynamics) dynamical systems : linear hyperbolic automorphism on the torus (Arnold's cat map), dispersive billiards, geodesic flow on negatively curved manifold, Lorenz flow.
-Classical evolution of probability distributions using micro-local analysis. Ruelle-Pollicott discrete spectrum. Atiyah-Bott trace formula. Selberg and Ruelle zeta functions. Analogy with the distribution of primes.
-Quantum mechanics, quantum chaos. Geometric quantization. Propagation of singularities. Some properties of quantum chaos that can be deduced from short or finite time analysis versus frequency : Weyl law, theorem of quantum ergodicity. Trace formula of Duistermaat-Guillemin. Random waves and random matrix conjectures.

References :
V.I. Arnold. Les methodes mathematiques de la mecanique classique. Ed. Mir. Moscou, 1976.
M. Brin and G. Stuck. Introduction to Dynamical Systems. Cambridge University Press, 2002.
E.B. Davies. Linear operators and their spectra. Cambridge University Press, 2007.
K.J. Engel and R. Nagel. One-parameter semigroups for linear evolution equations, volume 194. Springer, 1999.
F. Faure. Introduction au chaos quantique. In journees X-UPS, Editions de l'ecole polytechnique 2014.
F. Faure. From classical chaos to quantum chaosspectrum. In link, Lectures notes for the school 22-26 April 2019 at CIRM. 2018.
F. Faure. Spectrum, traces and zeta functions in hyperbolic dynamics. In link, school 23-27 April 2018 at the University Cheikh Anta Diop in Dakar, Senegal. 2018.
A. Katok and B. Hasselblatt. Introduction to the Modern Theory of Dynamical Systems. Cambridge University Press, 1995.
Lee. Riemannian Manifolds :An Introduction to Curvature. Springer, 1997.
D McDu and D Salamon. Introduction to symplectic topology, 2nd edition. clarendon press, Oxford, 1998.
M. Nakahara. Geometry, topology and physics. Institute of Physics Publishing, 2003.
A. Cannas Da Salva. Lectures on Symplectic Geometry. Springer, 2001.
M. Dillard-Bleick Y. Choquet-Bruhat, C. Dewitt-Morette. Analysis, manifolds and physics. North-Holland, 1982.
M. Zworski. Semiclassical Analysis. Graduate Studies in Mathematics Series. Amer Mathematical Society, 2012.