Hodge Conjecture: Evening Event

Mathematicians and their problems

Speaker: Jürgen Richter-Gebert

Real mathematicians are not afraid of challenges. On the contrary: the more difficult a problem appears, the more attractive it becomes. Because then it is even more appealing to try to find the solution. This is not only the case for brain-teasers, riddles or puzzles, but also for solving the unanswered questions in fundamental mathematical research. 

This lecture illustrates how and why the really difficult problems are often a motor for new developments in mathematics, and indeed sometimes captivate whole fields of research over many generations. The listeners will then be taken on a light and colorful journey through some of the other seven Millennium Problems - for which a reward of 1 million dollar for the solution of each has been offered - and on to the classic Hilbert problems of 1900. Along the way, Richter-Gebert will show how different some of the solutions are, how some problems stubbornly persist in eluding all attempts to solve them, the human tragedies behind the search for these solutions and how the problems nonetheless effectuate other advances in mathematics as a whole. The talk will be illustrated by visualizations and software demonstrations. 

The Hodge conjecture

Speaker: Christian Liedtke

How much more complicated is a random curve in comparison to a line? In the twentieth century, mathematics developed techniques in order to come closer to answering exactly this question: every geometric object is assigned to a linear object - a curve to a straight line, or even to a plane. With this process of linearization, large amounts of geometric information are lost. Or are they? Is perhaps only insignificant information lost?

The Hodge conjecture implies that the loss of geometric information should be minimal in a certain sense. The conjecture is named after the British mathematician William Hodge (1903 -1975). His conjecture – should it at some time be proved to be correct – has deep and extensive consequences for mathematics, both in the field of geometry and with regard to the still mysterious theory of the category of motives. 

In his talk, Christian Liedtke presents spatial curves and surfaces and the linear objects to which they are assigned. He explains what the Hodge conjecture predicts for these objects, and how they are correct for these examples. In this way, it becomes evident, that it is not so easy to explicitly verify the conjecture, nor to see specific patterns which could maybe point towards a general proof of the theory.