Topic: A Discrete Time Optimisation Problem for the Lognormal Distribution and April 28

its Relation to Passport Options

Presenter: Professor F. Delbaen, Rutgers University

Place: Rutgers University Operation Research Department

Abstract: Passport options are financial instruments whose price is calculated through an optimisation problem in continuous time. We relate the problem to a discrete time optimisation for the lognormal distribution as well as to an $\Cal H^1$-like equality of martingale norms. This is joint work with Marc Yor.

Topic: The Fibrifold Notation and Classification for Three-Dimensional April 29

Space Groups

Presenter: Daniel Huson, Princeton University

Abstract: Two-dimensional symmetry groups are perhaps best described and classified using John Conway's orbifold notation (which is a simplification of the group signature introduced by A.M. Macbeath), based on specifying the features of the orbifold corresponding to a given symmetry group.

More than ten years ago, John Conway and William Thurston worked out an approach towards naming and classifying the three-dimensional crystallographic space groups based on describing features of fibrations of the corresponding three-dimensional orbifolds. In recent joint work we have now completed this investigation.

Of the 219 isomorphism types of three-dimensional space groups, 184 are reducible and can be fibered over one of the 17 plane groups. A ``fibrifold'' name is given to a fibered space group by ``embellishing'' or adding information to the name of the plane group that it fibers over to indicatehow horizontal symmetries are coupled to vertical ones.

This leads to a new and simple proof of the classification of three-dimensional space groups: One can obtain a classification of all fibered space groups by systematically enumerating all valid embellishments of each of the 17 plane groups. (Any reducible space group can have up to three different fibrations, but one can detect whether two different fibrifolds correspond to the same group). Moreover, the classification of the remaining 35 irreducible groups can be solved using a different - but equally simple - approach proposed by John Conway.

Topic: Rational versus real cohomology algebras of low-dimensional toric varieties April 29

Presenter: Eva Feichtner, IAS

Abstract: The rational cohomology groups of a compact quasi-smooth toric variety are completely determined by the combinatorial data of its defining simplicial fan. We will review results on the cohomology algebras of toric varieties, and present examples which show that the ring structure is not combinatorially determined in general.

In low dimension, the situation turns out to be surprisingly intricate: We show that the real cohomology algebra of a compact toric variety in complex dimension~$2$ is completely determined by the combinatorial data of its defining fan. This is a consequence of special properties of the Hodge decomposition for the cohomology of quasi-smooth, projective toric varieties. For rational coefficients, the cohomology algebras are no longer combinatorially determined, even in complex dimension~$2$. Moreover, our examples show that neither the real nor the complex cohomology algebras of compact quasi-smooth toric varieties are combinatorial invariants in general.

Topic: The Theta divisor and three-manifold invariants April 29

Presenter: Peter Ozsvath, Princeton University

Topic: On the rank of elliptic curves April 29

Presenter: Joseph Silverman, Brown University

Topic: Stable minimal surfaces and holomorphic curves in April 30

Kahler-Einstein manifolds

Presenter: Claudio Arezzo, MIT

Time: 1:30 p.m.

Topic: Some symmetry and monotonicity results for elliptic PDE on manifolds

Presenter: Luis Almeida, ENS Cachan, France

Time: 2:30 p.m.

Topic: The what and the why of quantum mechanics April 30

Presenter: Sheldon Goldstein, Rutgers University

Topic: Large Amplitude Nonlinear Acoustic Waves without Shocks, May 3

Co-authors Michael Chefter & Dmitriy Vaynblat

Presenter: Rodolfo R. Rosales, MIT

Tea will be served in the Common Room, Fine 317 at 3:45 p.m.

Abstract: A new class of solutions for the inviscid Euler equations of Gas Dynamics in a bounded domain is studied. These solutions do not present the usual wave breaking leading to shock formation, even though they have highly nontrivial acoustic components and operate in the nonlinear regime. Furthermore, these ``Non Breaking for All Times'' (NBAT) solutions are globally attracting for the long time evolution of the equations and are quasiperiodic in time, with two periods.

NBAT solutions are possible due to the cumulative effect of nonlinear resonances between the genuinely nonlinear sound waves and a third ``passive'' wave such as, for example, entropy variations. This is possible only in a bounded domain where the waves interact with each other repeatedly and creates an effective dispersion on the acoustic field --- which is responsible for stopping the shocks from forming. Extremely large amplitudes are possible even with rather small amplitudes in the passive wave.

Topic: Complexity and Generation of Information in Molecular Evolution May 5

Presenter: Manfred Eigen, Max-Planck-Institut Fur Biophysikalische Chemie

Abstract: Combinatorial versus dynamical complexity. Evolving molecular complexity: NP-Complete? An experiment demonstrating evolving ccomplexity. Darwinian algorithms: The generation of information, What is life?

Topic: Lanford's Program May 6

Presenter: Michael Yampolsky, Yale University

Topic: Triholomorphic curves and complex anti-self dual connections May 14

Presenter: JingYi Chen, MIT