# Prof. Dr. Gisbert Wüstholz

Mathematics

External Senior Fellow

September - December 2017; January - April 2018

### CV

Gisbert Wüstholz is a German mathematician whose research interests are Algebraic Geometry, Number Theory, Diophantine Approximation, Transcendence Theory, Arithmetic Algebraic Geometry, Arakelov Theory, Periods and Hodge Theory. He was born in 1948 in Tuttlingen (Baden-Württemberg) and studied from 1967 to 1973 in Freiburg i. Br. where he finished his PhD under the supervision of Theodor Schneider in 1978. On the invitation of F. Hirzebruch he stayed for

a year as a Postdoc at Bonn and then he got a Postdoc position at the Bergische Universität Wuppertal where he worked with Walter Borho from 1979 till 1984 and then moved to Bonn to become Associate Professor at the newly founded MPI for Mathematics. From 1985 to 1987

he was full Professor for Mathematics at Wuppertal and in 1987 elected for a Chair in Mathematics at ETH Zurich and retired in 2013.

He is Member of the German National Academy of Science Leopoldina (2000), of the Berlin-Brandenburgische Akademie der Wissenschaften (2002), of the Academia Europaea (2008) where he was chairman of the Mathematics Section from 2011-2013, and of the European Academy of Arts and Science* *(2016). Since 2011 he is Senator for Mathematics at the Leopoldina Gisbert Wustholz stayed for extended periods at a number of universities and research institutes such as the University of Michigan at Ann Arbor (1984,1988), the Institute des Hautes Etudes Scientifiques in Bûres sur Yvette (1987), Kyushu University (Fukuoka). He was member of the Institute for* Advanced Studies *in Princeton (1986, 1990, 1994/95, 2011), in 1992 Visiting Fellow Commoner at* Trinity College *in Cambridge for research projects with A. Baker and visited in the following year the Mathematical Research Institute in Berkeley (1993) and frequently guest at Max Planck Institute for* Mathematics *at Bonn and the Schrödinger Institut at Vienna (2013). Since 1980 Gisbert Wüstholz has close connections to several universities in Asia: he stayed for a couple of months each at the Morningside Center of Mathematics of the Chinese Academy of Sciences at Beijing, at the Hong Kong University of Science and Technology HKUST (1996, 1997, 2006, 2010) and at the Hong Kong University HKU (1999, 2011, 2012). Several visits took him to the National University of Taiwan at Taipei (2009, 2013, 2016) and he is Honorary Advisory Professor at the TonjiUniversity, Shanghai, since 1999, and at TU Graz since 2017.

In 1986 Gisbert Wüstholz delivered an invited address at the International Congress of Mathematicians in Berkeley, in 1992 the Mordell Lecture in Cambridge, in 2001 the 13th Kuwait foundation lecture, a lecture at the Leonhard Euler Festival in St. Petersburg in 2007 on the occasion of the celebration of Leonhard Euler’s 300th birthday and the Akademie-Vorlesung der Berlin-Brandenburgischen Akademie der Wissenschaften in 2008. Highlights of his Scientific work are his Analytic Subgroup Theorem (1989) which bases on the Multiplicity Estimates on Group Varieties he published in 1989, his proof of the abelian analogue of the famous Lindemann’s theorem, the result which disproved the squaring of the circle, the joint work with Faltings on the Schmidt Subspace Theorem, the Isogeny Estimates for Abelian varieties proved jointly with Masser which furnishes an alternative approach to the Mordell conjecture and the joint work with A. Baker on logarithmic forms.

### Selected Publications

### FRIAS Project

**Cohomology in algebraic geometry and representation theory **

One of the classical mathematical subjects is integration on manifolds. This goes back at least to Leibniz in the beginnings of the 18th century. Since then a very broad theory of integration has been developed to determine the length of a path, the area of a surface and the volume of bodies in any finite dimension to give some applications.The central tools for integration are differential forms and segments of paths, surfaces or manifolds.

In the last century a very abstract theory, homology and cohomology theory has been established to formalize integration and for obtaining for example topological invariants of manifolds. Differential forms led to a cohomology theory and the geometric aspects to a homology theory. It turns out that both theories are in duality and differ only by a characteristic set of numbers called periods. The basic example of a period is pi=3.1415926… which is the circumference of a circle of diameter 1. The number-theoretical properties of periods is up to a small countable set of very explicit cases a miracle. There are many deep conjectures which seem almost intractable.

A 2000 years old problem was whether it is possible to construct from a circle of diameter 1 a square with the same area just by ruler and compass. The very famous solution was found on a walk on the Loretto Berg by Lindemann in 1882. Lindemann was professor in Freiburg from 1877 to 1893. He proved that pi is transcendental and as such not accessible by a finite number of algebraic operations like constructions with ruler and compass.

In our project we study periods at the interface between cohomology theory and transcendence theory. We are particularly interested in periods of so-called 1-motives, a rather detailed studied geometric object introduced by Deligne in the 70th which leads to interesting periods.