Algebra in Computational Complexity

At its core, much of Computational Complexity is concerned with combinatorial objects and structures. But it has often proven true that the best way to prove things about these combinatorial objects is by establishing a connection to a more well-behaved algebraic setting. Indeed, many of the deepest and most powerful results in Computational Complexity rely on algebraic proof techniques. The Razborov-Smolensky polynomial-approximation method for proving constant-depth circuit lower bounds, the PCP characterization of NP, and the Agrawal-Kayal-Saxena polynomial-time primality test are some of the most prominent examples. The algebraic theme continues in some of the most exciting recent progress in computational complexity. There have been significant recent advances in algebraic circuit lower bounds, and the so-called "chasm at depth 4" suggests that the restricted models now being considered are not so far from ones that would lead to a general result. There have been similar successes concerning the related problems of polynomial identity testing and circuit reconstruction in the algebraic model, and these are tied to central questions regarding the power of randomness in computation. Representation theory has emerged as an important tool in three separate lines of work: the "Geometric Complexity Theory" approach to P vs. NP and circuit lower bounds, the effort to resolve the complexity of matrix multiplication, and a framework for constructing locally testable codes. Coding theory has seen several algebraic innovations in recent years, including multiplicity codes, and new lower bounds. This seminar brought together researchers who are using a diverse array of algebraic methods in a variety of settings. It plays an important role in educating a diverse community about the latest new techniques, spurring further progress

Darwin and Fisher meet at biotech : on the potential of computational molecular evolution in industry

Today computational molecular evolution is a vibrant research field that benefits from the availability of large and complex new generation sequencing data - ranging from full genomes and proteomes to microbiomes, metabolomes and epigenomes. The grounds for this progress were established long before the discovery of the DNA structure. Specifically, Darwin's theory of evolution by means of natural selection not only remains relevant today, but also provides a solid basis for computational research with a variety of applications. But a long-term progress in biology was ensured by the mathematical sciences, as exemplified by Sir R. Fisher in early 20th century. Now this is true more than ever: The data size and its complexity require biologists to work in close collaboration with experts in computational sciences, modeling and statistics

Randomness Extraction in AC0 and with Small Locality

Randomness extractors, which extract high quality (almost-uniform) random bits from biased random sources, are important objects both in theory and in practice. While there have been significant progress in obtaining near optimal constructions of randomness extractors in various settings, the computational complexity of randomness extractors is still much less studied. In particular, it is not clear whether randomness extractors with good parameters can be computed in several interesting complexity classes that are much weaker than P. In this paper we study randomness extractors in the following two models of computation: (1) constant-depth circuits (AC0), and (2) the local computation model. Previous work in these models, such as [Vio05a], [GVW15] and [BG13], only achieve constructions with weak parameters. In this work we give explicit constructions of randomness extractors with much better parameters. As an application, we use our AC0 extractors to study pseudorandom generators in AC0, and show that we can construct both cryptographic pseudorandom generators (under reasonable computational assumptions) and unconditional pseudorandom generators for space bounded computation with very good parameters. Our constructions combine several previous techniques in randomness extractors, as well as introduce new techniques to reduce or preserve the complexity of extractors, which may be of independent interest. These include (1) a general way to reduce the error of strong seeded extractors while preserving the AC0 property and small locality, and (2) a seeded randomness condenser with small locality.Comment: 62 page

Scientific Progress on the Semantic View : An Account of Scientific Progress as Objective Logical and Empirical Strength Increments

The aim of this master thesis is to make a convincing argument that scientific progress can be spoken of in objective terms. In order to make this argument I will propose a philosophical theory of scientific progress. Two concepts will be constructed with this aim in mind, both which are types of strength measures on scientific theories. The first concept, that of logical strength, pertains to the way a theory may exclude, or permit less, model classes compared to another theory. The second concept, that of empirical strength, pertains to an objective measure of the informational content of data models, defined in terms of Kolmogorov complexity. This latter idea stems from communication and computational theory. Scientific progress is then defined as the interaction, or the stepwise increases, of these two strength measures. Central for the conception of a scientific theory is the philosophical framework known as The Semantic View of Scientific Theories. This view can briefly be characterized as an empirical extension of Tarskian model-theory. Another central notion for this theory of scientific progress is the philosophical or metaphysical thesis called structural realism. Both will accordingly be explained and argued for. Finally, as a test on this proposed theory of scientific progress, it will be applied to two examples of theory transition from the history of physical theory. I conclude that the proposed theory handles these two cases well