Georgia Institute of Technology College of Sciences

The possibility of a quantum computer has lead to much new work in theoretical physics and, naturally enough, this work has raised many new mathematical problems. What is perhaps surprising is that it has lead to interesting problems in algebraic graph theory. For example, questions about the relative power of quantum computer and classical computers lead to questions about the chromatic number of certain graphs. In my talk I will discuss some of these problems, and the progress that has been made.

A central characteristic of a computer chip is the speed at which it processes data, determined by the time it takes electrical signals to travel through the chip. A major challenge in the design of a chip is to achieve timing closure, that is to find a physical realization fulfilling the speed specifications. We give an overview over the major tasks for optimizing the performance of computer chips and present several new algorithms. For the topology generation of repeater trees, we introduce a variant of the Steiner tree problem and present fast algorithm that balances efficiently between the resource consumption and performance. Another indispensable task is gate sizing, a discrete optimization problem with nonlinear or PDE constraints, for which a fast heuristic is introduced. The effectiveness in practice is demonstrated by comparing with newly developed lower bounds for the achievable delay. We conclude with a variant of the time-cost tradeoff problem from project management. In contrast to the usual formulation cycles are allowed. We present a new method to compute the time-cost tradeoff curve in such instances using combinatorial algorithms. Several problems in chip design can be modeled as time-cost tradeoff problems, e.g. threshold voltage optimization of plane assignment.

We argue that computational models have an essential role in uncovering the principles behind a variety of biological phenomena that cannot be approached by other means. In this talk we shall focus on evolution. Living organisms function according to complex mechanisms that operate in different ways depending on conditions. Darwin's theory of evolution suggests that such mechanisms evolved through random variation guided by natural selection. However, there has existed no theory that would explain quantitatively which mechanisms can so evolve in realistic population sizes within realistic time periods, and which are too complex. Here we suggest such a theory. Evolution is treated as a form of computational learning from examples in which the course of learning depends only on the aggregate fitness of the current hypothesis on the examples, and not otherwise on individual examples. We formulate a notion of evolvability that distinguishes function classes that are evolvable with polynomially bounded resources from those that are not. For example, we can show that monotone Boolean conjunctions and disjunctions are demonstrably evolvable over the uniform distribution, while Boolean parity functions are demonstrably not. We shall discuss some broader issues in evolution and intelligence that can be addressed via such an approach.

In its simplest form, the Erdos-Ko-Rado theorem tells us that if we have a family F of subsets of size k from set of size v such that any two sets in the family have at least one point in common, then |F|<=(v-1)\choose(k-1) and, if equality holds, then F consists of all k-subsets that contain a given element of the underlying set. This theorem can also be viewed as a result in graph theory, and from this viewpoint it has many generalizations. I will outline how it can be proved using linear algebra, and then discuss how this approach can be applied in other cases.