Complex Algebraic Geometry

Schedule

Titles and abstracts

Hélène Esnault, Coherent D-modules over the complex numbers and in characteristic p>0.

Bernd Kreussler, Moishezon twistor spaces -- the past two decades
Abstract: After briefly recalling twistor spaces, I shall give a survey of results about compact Moishezon twistor spaces of complex dimension three. This includes an answer to a question raised by Herbert Kurke in 1991.

Daniel Huybrechts, Chow groups of K3 surfaces

Motohico Mulase, The Laplace transform of Hurwitz numbers
Abstract: Laplace transformation changes a function defined on positive integers (or more generally, on positive real numbers) into a holomorphic function. While Fourier transformation is a duality, the nature of Laplace transform is quite different. For a fixed genus, the Hurwitz number is a function of a partition. What happens if we apply the Laplace transform to the Hurwitz numbers? Surprisingly, the result turns out to be a simple polynomial. These polynomials are easy to calculate, since they satisfy an effective recursion formula. When we restrict the formula to the highest degree terms of the polynomial, we recover the Witten-Kontsevich theorem. And when restricted to the lowest degree terms, the recursion gives the lambda-g formula of Faber and Pandharipande. The recursion also proves the Bouchard-Marino conjecture on Hurwitz numbers, which is a special case of the Eynard-Orantin formula that is originated in random matrix theory and is implemented to topological string theory by Bouchard-Klamm-Marino-Pasquetti and Dijkgraaf-Vafa. I will review a mathematical side of the developments on these subjects. This talk is based on my recent work with Borot, Eynard, Safnuk, and Zhang.

Le Dung Trang, Bifurcation diagrams and Morse theory
Abstract: We shall introduce bifurcation diagrams of complex functions and show how they contribute to understand Morse theory associated to singular spaces.

Gerhard Pfister, 25 Years of SINGULAR
Abstract: The computer algebra system SINGULAR has its 25th birthday. The talk will explain the interplay between computer algebra and its applications and illustrate the development during the last 25 years. We will see that needs in algebraic geometry and singularity theory always forced the development of the system. Deep theorems in group theory could be solved using the system. New applications to electrical engineering will be explained. We will also see applications to coding theory, partial differential equations. SINGULAR can also solve a Sudoku.

Eckart Viehweg, Algebraically integrable foliations of general type
Abstract: A foliation on a non-singular projective variety is algebraically integrable if all leaves are algebraic subvarieties. A non-singular hypersurface X in a non-singular projective variety M equipped with a symplectic form has a naturally defined foliation, called the characteristic foliation on X. We show that if X is of general type and if the dimension of M is larger than or equal to 4, then the characteristic foliation on X cannot be algebraically integrable. This is a consequence of a more general result on Iitaka dimensions of certain invertible sheaves associated with algebraically integrable foliations by curves. Joint work with Jun-Muk Hwang.