The Landau Lecture series

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 (1854-1912) discovered that a system of several celestial bodies moving under Newton's gravitational law shows chaotic dynamics. Earlier, Euler (1707-83) and Lagrange (1736-1813) 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
Sunday-Monday, June 5th & 6th      Seminar Talks

A meanwhile standard idea for producing geometric invariants (f.e. Donaldson Theory, Gromov-Witten 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 Cauchy-Riemann equation. If there weren't these inherent compactness and transversality problems, the solution sets of the nonlinear Cauchy-Riemann 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.

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

Invitation      Abstracts

1. Towards an Aubry- Mather Theory for a class of PDE's
   (Thursday, December 18th and 22nd, 2008)
2. Variational gluing (Wednesday, December 24th, 2008)

Abstracts

1. 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.
2. 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 Riemann-Hilbert method plays a central role.
3. 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 Robinson-Schensted-Knuth algorithm and the theory of non-intersecting paths play a key role.

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

1. May 10th, 2006
2. May 11th, 2006
3. May 14th, 2006

Invitation

1. 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 non-linear 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 Black-Scholes 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.
2. 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.
3. 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

1. Collapsed manifolds with bounded curvature (Jan. 15, 2004)
2. 4-dimensional Einstein manifolds (Jan. 18, 2004)
3. Einstein manifolds in general (Jan. 19, 2004)

Invitation

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

Invitation

1. Symplectic topology and Morse theory (April 21st, 2002)
2. Symplectic topology and low-dimensional topology (April 23rd, 2002)
3. Symplectic topology and theory of several complex variables (April 25th, 2002)

Invitation

1. Discrete models (Dec. 31, 2000)
2. Algebraic quantization (Jan. 2, 2001)
3. String topology (Jan. 4, 2001)

1. The Second law of thermodynamics : a mathematical perspective (March 12th, 2000)
2. The Bose gas : a subtle many-body problem (March 13th, 2000)
3. The quantum mechanical world view : a highly successful but still incomplete theory (March 16th, 2000)

1. Trace formulas and the zeros of the Riemann zeta function (Jan. 3rd, 1999)
2. Renormalization and Hopf algebras (Jan. 5th, 1999)
3. Non-Commutative geometry (Jan. 7th, 1999)

Invitation

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

(Jan. 11th, 14th and 15th, 1996)

1. On mathematics and its communication (Jan. 11, 1996)

1. On Euler and Navier-Stokes equations (Nov. 17, 1994)
2. Mathematical models in image processing (Nov. 20, 1994)
3. Compensated compactness and Hardy spaces (Nov. 22, 1994)

1. Quasi-linear one-dimensional equations with random initial data and random forces: Their origin and fractal properties (Nov. 25, 1993)
2. Probabilistic problems for the one-dimensional Burgers-Brown equation (Nov. 28, 1993)
3. Burgers' equation with random forces and self-avoiding type random walks (Nov. 30, 1993)

1. A survey (Nov. 5, 1992)
2. Uniformity of distribution (Nov. 9, 1992)
3. Beyond the Riemann hypothesis (Nov. 11, 1992)

1. On Liouville-type theorems for linear and non-linear elliptic differential equations (Oct. 29, 1991)
2. The role of minimizers in geometry and in dynamical systems (Oct. 31, 1991)