Past lectures in the Landau lecture series

2011
 Prof.
Helmut Hofer
(School of Mathematics, Institute for Advanced Study, Princeton)

Details
June 2nd, 5th and 6th
 Lecture 1
From Celestial Mechanics to a Geometry Based on Area
Thursday, June 2nd
Department Colloquium

The mathematical problems arising from modern celestial mechanics, which
originated with Isaac Newton's Principia in 1687, have led to many
mathematical theories. Poincaré (18541912) discovered that a system of
several celestial bodies moving under Newton's gravitational law shows
chaotic dynamics. Earlier, Euler (170783) and Lagrange (17361813) found
instances of stable motion.
For example a spacecraft in the gravitational
fields of the sun, earth, and the moon provides an interesting system which
can experience stable as well as chaotic motion.
These seminal observations
have led to the theory of dynamical systems and to the field of symplectic
geometry, which is a geometry based on area rather than distance. It is
somewhat surprising
that these fields have developed separately, since both, in their modern
form, have their origin with Poincaré,
who had an highly integrated view point. Given the highly developed states
of both fields, and the background of some promising results, the time seems
ripe to bring them together around the core of Hamiltonian mechanics in a
field which perhaps should be called "Symplectic Dynamics". The talk will
conclude by giving some ideas what this field would be about.
 Lecture 2 & 3
Holomorphic Curves in Symplectic Geometry
and a Generalization of Fredholm Theory I & II
SundayMonday, June 5th & 6th
Seminar Talks

A meanwhile standard idea for producing geometric invariants
(f.e. Donaldson Theory, GromovWitten Theory, Symplectic Field Theory)
consists of counting solutions of nonlinear elliptic systems associated
to the geometric data. Although the basic idea is easy, the implementation
can be very difficult and involved, due to a usually large number of
technical
issues, which in more classical approaches to such type of problems
are more than "painful".
In symplectic geometry the partial differential equation in question is the
nonlinear CauchyRiemann equation.
If there weren't these inherent compactness and transversality
problems, the solution sets of the nonlinear CauchyRiemann operator would
be nice manifolds or orbifolds,
and the invariants could be achieved by integration of suitable differential
forms
over them. As it turns out, the arising difficulties can be overcome
by a drastic generalization of nonlinear Fredholm theory and new methods
for implementing it in concrete problems.

2009/10

Prof.
William B. Johnson (Texas A&M University, USA)

Details
 Five 20+ Year Old Problems in the Geometry of Banach Spaces
(Thursday, January 7th, 2010)
 Five More 20+ Year Old Problems in the Geometry of Banach Spaces
(Monday, January 11th, 2010)
 Dimension Reduction and Other Topics in Discrete Metric Geometry
(Wednesday, January 13th, 2010)
Invitation
Abstracts

2008/9

Prof. Paul Rabinowitz
(University of WisconsinMadison)
 It all began with Moser ...
Details
 Towards an Aubry Mather Theory for a class of PDE's
(Thursday, December 18th and 22nd, 2008)
 Variational gluing (Wednesday, December 24th, 2008)
Abstracts

2007/8
 Prof. Percy Deift
(New York University)
 Universality for mathematical and physical systems
Details
 Overview (Thursday, December 20th, 2007)
In the FIRST talk the speaker will recount some recent history of universality
ideas in physics starting with Wigner's model for the scattering of neutrons
off large nuclei and show how these ideas have led mathematicians to
investigate universal behavior for a variety of mathematical systems.
This is true not only for systems which have a physical origin, but also for
systems which arise in a purely mathematical context such as the Riemann
hypothesis, and a version of the card game solitaire called patience sorting.
 Analytical Tools (Monday, December 24th, 2007)
In the SECOND talk the speaker will describe some of the analytical tools that
are used in analyzing some of the mathematical systems discussed in the first
talk. The RiemannHilbert method plays a central role.
 Combinatorial Methods (Wednesday, December 26th, 2007)
In the THIRD talk the speaker will describe some of the combinatorial and
algebraic tools that are needed to analyze the above mathematical systems.
Here the RobinsonSchenstedKnuth algorithm and the theory of nonintersecting
paths play a key role.

2006/7
 Prof.
Benoit Perthame
(Ecole Normale Supérieure, Paris, France)

Details
 Transport equations in population biology (Dec. 7th, 2006)
 Kinetic formulations of conservation laws (Dec. 11th, 2006)
 Concentration phenomena in PDEs from biology (Dec. 13th, 2006)

2005/6
 Prof.
Jean Bourgain
(Institute For Advanced Study, Princeton, USA)
 The sumproduct phenomenon and applications
Details
 May 10th, 2006
 May 11th, 2006
 May 14th, 2006
Invitation

2004/5
 Prof.
Hans Föllmer
(Humboldt University, Berlin)
 Probablilistic aspects of financial risk
Details
 Stochastic analysis of financial options (March 3rd, 2005)
The price fluctuation of liquid financial assets is usually modeled as
a stochastic process
which satisfies some form of the "efficient markets hypothesis". Such
assumptions
can be made precise in terms of martingale measures. We discuss the
role of these martingale measures
in analyzing financial derivatives such as options, viewed as
nonlinear functionals of the underlying stochastic process. Uniqueness
of the martingale measure
provides the mathematical key to a perfect "hedge" of a financial
derivative by means of a dynamic trading strategy in the underlying
assets, and in particular to pricing formulas of BlackScholes type.
But for realistic models the martingale measure is no longer unique,
and intrinsic risks appear on the level of derivatives. We discuss
various mathematical approaches to the problem of pricing and hedging
in such a setting.
 Quantifying the risk: a robust view (March 6th, 2005)
In recent years, there has been an increasing focus on the problem of
quantifying the risk of a financial position, in particular from the
point of view of a supervising agency. We discuss some mathematical
developments in this area of financial risk management, in particular
representation results for convex risk measures which take model
uncertainty into account, and some robust optimization problems which
arise in this context.
 Dynamic risk measures (March 8th, 2005)
The problems of quantifying the risk of a stochastic payment stream and
of updating a risk assessment in the light of incoming information have
led to a theory of dynamic risk measures. We discuss some recent
developments, in particular the connections to the pricing problem for
American options and to the theory of backward stochastic differential
equations.
Invitation

2003/4
 Prof. Jeff Cheeger
(Courant Institute, N.Y.U.)
 The small scale structure of Riemannian Manifolds
Details
 Collapsed manifolds with bounded curvature (Jan. 15, 2004)
 4dimensional Einstein manifolds (Jan. 18, 2004)
 Einstein manifolds in general (Jan. 19, 2004)
Invitation

2002/3
 Prof.
Stefan Müller
(MaxPlanck Institute for Mathematics in the Sciences, Leipzig)

Details
 Geometric rigidity, curvature functionals and dimension reduction in
nonlinear elasticity (May 12, 2003)
 Mathematical problems in micromagnetics : a paradigm multiscale problems
(May 14, 2003)
 Convex integration, wild solutions of partial differential equations and
crystalline microstructure (May 15, 2003)
Invitation

2001/2
 Prof. Y. Eliashberg (Stanford University, I.A.S.)
Details
 Symplectic topology and Morse theory (April 21st, 2002)
 Symplectic topology and lowdimensional topology
(April 23rd, 2002)
 Symplectic topology and theory of several complex variables
(April 25th, 2002)
Invitation

2000/1
 Prof.
Dennis Sullivan
(City College (CUNY), New York ; SUNY at Stony Brook)
 Fluids, quanta and strings
Details
 Discrete models (Dec. 31, 2000)
 Algebraic quantization (Jan. 2, 2001)
 String topology (Jan. 4, 2001)

19992000
 Prof.
Elliott H. Lieb
(Princeton University)

Details
 The Second law of thermodynamics : a mathematical perspective
(March 12th, 2000)
 The Bose gas : a subtle manybody problem (March 13th, 2000)
 The quantum mechanical world view : a highly successful but still
incomplete theory (March 16th, 2000)

1998/9
 Prof.
Alain Connes (IHES)

Details
 Trace formulas and the zeros of the Riemann zeta function
(Jan. 3rd, 1999)
 Renormalization and Hopf algebras (Jan. 5th, 1999)
 NonCommutative geometry (Jan. 7th, 1999)
Invitation

1997/8
 Prof.
Peter Sarnak (Princeton University)
 Zeros of zeta functions and applications to arithmetic
(Jan. 15, 18 and 20th, 1998)

1996/7
 Prof.
Stefan Hildebrandt (Univ. Bonn)

Details
 Contact transformations, Huygens principle and the calculus of variations
(Nov. 28th, 1996)
 Nonlinear elliptic systems of partial differential equations and geometric
variational problems (Dec. 1st and 3rd, 1996)

1995/6
 Prof.
William P. Thurston (Cornell University)
 Threedimensional geometry and topology
Details
(Jan. 11th, 14th and 15th, 1996)
 On mathematics and its communication (Jan. 11, 1996)

1994/5
 Prof.
PierreLouis Lions (Ceremade, Univ. de Paris IX)
 Nonlinear partial differential equations and applications
Details
 On Euler and NavierStokes equations (Nov. 17, 1994)
 Mathematical models in image processing (Nov. 20, 1994)
 Compensated compactness and Hardy spaces (Nov. 22, 1994)

1993/4
 Prof.
Y.G. Sinai
(Princeton University ;
Landau Institute of Theoretical Physics,
Moscow)
 Nonlinear PDE and probability theory
Details
 Quasilinear onedimensional equations with random initial data and random
forces: Their origin and fractal properties (Nov. 25, 1993)
 Probabilistic problems for the onedimensional BurgersBrown equation
(Nov. 28, 1993)
 Burgers' equation with random forces and selfavoiding type random walks
(Nov. 30, 1993)

1992/3
 Prof.
Enrico Bombieri
(Institute for Advanced Study, Princeton)
 Prime numbers in arithmetic progressions
Details
 A survey (Nov. 5, 1992)
 Uniformity of distribution (Nov. 9, 1992)
 Beyond the Riemann hypothesis (Nov. 11, 1992)

1991/2
 Prof.
Jürgen Moser (19281999) (E.T.H. Zürich)

Details
 On Liouvilletype theorems for linear and nonlinear elliptic differential
equations (Oct. 29, 1991)
 The role of minimizers in geometry and in dynamical systems
(Oct. 31, 1991)
