Curves, their jacobians and endomorphisms

Speaker: 

Professor Yuri Zarhin

Institution: 

Pennsylvania State University

Time: 

Wednesday, November 28, 2007 - 4:00pm

Location: 

MSTB 254

A smooth plane projective cubic curve (also known as an elliptic curve or a curve of genus 1) carries a natural structure of a commutative group: the addition is defined geometrically by the "chord and tangent method". An attempt "to add" points on a curve of arbitrary positive genus g leads to the notion of the jacobian of the curve. This jacobian is a g-dimensional commutative algebraic group that is a projective algebraic variety; in particular, it cannot be realized as a matrix group. Geometric properties of jacobians play a crucial role in the study of arithmetic and geometric properties of curves involved. One of the most important geometric invariants of a jacobian is its endomorphism ring.

We discuss how to compute explicitly endomorphism rings of jacobians for certain interesting classes of curves that may be viewed as natural (and useful) generalizations of elliptic curves.

Generalized theta functions

Speaker: 

Szego Assistant Professor Dragos Oprea

Institution: 

Stanford University

Time: 

Monday, November 26, 2007 - 11:00am

Location: 

MSTB 254

The Jacobian of any compact Riemann surface carries a natural theta divisor, which can be defined as the zero locus of an explicit function, the Riemann theta function. I will describe a generalization of this idea, which starts by replacing the Jacobian with the moduli space of sheaves over a Riemann surface or a higher dimensional base. These moduli spaces also carry theta divisors, described as zero loci of "generalized" theta functions. I will discuss recent progress in the study of generalized theta functions. In particular, I will emphasize an unexpected geometric duality between spaces of generalized theta functions, as well as its geometric consequences for the study of the moduli spaces of sheaves.

Proving projective determinacy

Speaker: 

Professor Ralf Schindler

Institution: 

UC Berkeley and Universitaet Muenster, Germany

Time: 

Monday, November 26, 2007 - 2:00pm

Location: 

MSTB 254

The principle of projective determinacy, being independent from the standard axiom system of set theory, produces a fairly complete picture of the theory of "definable" sets of reals. It is an amazing fact that projective determinacy is implied by many apparently entirely unrelated statements. One has to go through inner model theory in order to prove such implications.

On the regularity of weak solutions of the 3D Navier-Stokes equations in the largest critical space.

Speaker: 

L.E. Dickson Instructor Alexey Cheskidov

Institution: 

University of Chicago

Time: 

Tuesday, November 27, 2007 - 3:00pm

Location: 

MSTB 254

Even though the regularity problem for the 3D Navier-Stokes equations is far from been solved, numerous regularity criteria have been proved since the work of Leray. We will discuss some classical results as well as their extensions in Besov spaces.

Nonlinear water waves over strongly varying bottom topography

Speaker: 

Professor John Grue

Institution: 

University of Oslo, Norway

Time: 

Tuesday, November 27, 2007 - 2:00pm

Location: 

MSTB 254

A fully nonlinear time-stepping model for water wave motion over strongly varying topography
in three dimensions is presented. The modl is fully dispersive, fully nonlinear and, and also very rapid. The kinematic and dynamic boundary
condition at the free surface are used to derive the prognostic equations. Conservation of mass yields two integral equations for the normal velocity at the free surface and the wave potential at the sea floor. These are inverted analytically be means of Fourier transform. Various levels of nonlinearity of the equations are derived. A highly efficient computational scheme is obtained by the FFT-part of the formulation. Computations exemplify how a very long tsunami with leading depression running into very shallow water develop very short waves, that in the beginning are linear, developing then into a train of solitary waves of
large amplitude. Numerical examples on the formation of very strong ocean surface waves - rogue waves - are given.

Higher dimensional Schrodinger flow into Grassmannians

Speaker: 

Professor Bo Dai

Institution: 

Peking University

Time: 

Tuesday, January 22, 2008 - 4:00pm

Location: 

MSTB 254

Schrodinger flow is the Hamiltonian flow for energy functional on the space of maps from a Riemannian manifold into a Kahler manifold. I'll talk about some background on this flow, then focus on the special case of maps from a Euclidean space into the complex Grassmannian Gr(k,C^n). Terng and Uhlenbeck proved that Schrodinger flow of maps from R^1 into complex Grassmannian is gauge equivalent to the matrix nonlinear Schrodinger equation. Using this gauge equivalence and the result of Beals and Coifman, they obtained the global existence of Schrodinger flow with rapidly decay initial data. Applying the method of Terng and Uhlenbeck, we will see that Schrodinger flow of radial maps from R^m into the complex Grassmannian is gauge equivalent to a generalized matrix nonlinear Schrodinger equation. When the target is the 2-sphere, the gauge equivalence was studied by Lakshmanan and his colleagues by different method. They also observed that if the domain is R^2, then the corresponding matrix nonlinear Schrodinger equation is an integrable system.

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