Pictures of the Future    Spring 2006

Mathematics is the Key to Visualizing and Optimizing Complex Processes

Interview with Martin Grötschel

^{Prof. Martin Grötschel, 57, teaches at the Technical University in Berlin. He is Vice President of the Konrad Zuse Center for Information Technology and spokesperson of the DFG research center MATHEON, which focuses on "Mathematics for Key Technologies." An active supporter of dialogue between research and industry, Grötschel is a member of the Executive Committee of the International Mathematical Union and holder of the Leibniz Prize, the most heavily endowed German research award. He also has won the Karl Heinz Beckurts Prize, which honors innovative services to industry}

What is it about mathematics that makes it such a fascinating subject?

Grötschel: No other science is so brutally precise. The smallest mistake produces a false result. On the other hand, it’s completely wrong to equate mathematics merely with exact arithmetic and the skilful manipulation of formulas. Good mathematics also requires a lot of creativity, both for mathematical proofs and for the discovery of new structures and theorems. Moreover, it’s also fun to use the abstract world of mathematics for real applications.

Can you give us some examples?

Grötschel:In a project at the Konrad Zuse Center, we’re working on a model to generate a simple but nonetheless effective simulation of wave propagation in GSM and UMTS networks. It’s not vital here to take into account every tree or bush that might reflect the waves. In other words, we have to find out which marginal factors can be neglected without forfeiting precision. Another example is in local public transport, where we can calculate the minimum number of buses required for a network, for example, and draw up bus routes and schedules tailored to actual demand and also determine the most cost-efficient use of employees. These methods are already in use in Berlin and Bonn, as well as in the Milan subway system.

How can we increase the effectiveness of computer simulations?

Grötschel: There’s a basic choice here between modeling, simulation and optimization. Engineers often think it’s enough to repeat a simulation a few times in order to get an optimal result. But that’s not true at all. Simulation simply involves entering various parameters into a mathematical model and then running the model. Optimization, on the other hand, is all about finding the best combination of parameters. Take the example of a container terminal. A simulation won’t provide you with the optimal deployment of the containers that offer the most efficient access. What you need for that is an accurate model of the total storage area and access routes plus clearly defined performance functions. That’s what tells you how best to arrange and handle the containers.

Where can mathematics help to make improvements?

Grötschel: At MATHEON, a part of the DFG (German Research Foundation) research center, we’re optimizing a variety of real objects and processes. This ranges from risk assessment for insurers to chip manufacturing. Another complex area is route planning in the transport sector, which takes into account a host of factors. In fact, mathematics makes many fundamental contributions to the world we live and work in, although people don’t generally notice this.

To what extent do processes from nature serve as a model for mathematics?

Grötschel: If someone says that genetic and evolutionary algorithms, or ant colony optimization, particle swarm optimization or agent methods are the solution to everything, then that’s got more to do with product marketing than any serious problem-solving. Sure, nature can provide ideas, but it’s nonsense in my opinion to look to nature for ready-made theories. Ants may well be able to find the shortest way to food, but try asking an ant for the longest route. Yet that’s exactly what interests the project planner, who needs to estimate the total time it will take to implement a measure.

Given that computing power doubles every 18 months, what will mathematicians be working on in ten years?

Grötschel: The topics of the future for mathematics include optimizing how highly integrated chips function and calculating quantum effects. Another is the design of more effective medicines, an area where we are still far from an adequate understanding of all that’s involved. Likewise, mathematical models for forecasting weather and climate change are still very crude and don’t take a lot of effects sufficiently into account. Another problem is how to construct a highly efficient engine with the lowest possible emissions or an aircraft that only requires a minimal input of energy? Today, we can calculate optimal bus routes for public transport and driver schedules. But what we will need to do in the future is combine the route plan and driver schedule in a single integrated model. Today you need a supercomputer for this, but in ten years maybe a laptop will do.

                                                                                  Interview by Andreas Beuthner