Abstract

Abstract: Cocycle categories give a simple, flexible way to describe morphisms in a homotopy category, provided that the underlying model structure is sufficiently well behaved. "Well behaved" model structures include simplicial sets, spectra, simplicial presheaves and presheaves of spectra, together with all good localizations such as the motivic model structure of Morel and Voevodsky

Abstract

Abstract: Suppose that H is an algebraic group which is defined over a field k, and let L be the algebraic closure of k. The canonical stalk for the etale topology on k induces a simplicial set map from the classifying space B(H-tors) of the groupoid of H-torsors (aka. principal H-bundles) to the space BH(L). The homotopy fibres of this map are groupoids of pointed torsors, suitably defined. These fibres can be analyzed with cocycle techniques: their path components are representations of the "absolute Galois groupoid"  in H, and each path component is contractible. The arguments for these results are relatively simple, and applications will be displayed.

Abstract

Abstract:  Conjecture:  For any map f: E \to S^1 from a closed 4-manifold to a circle whose homotopy fiber has the homotopy type of a 3-manifold, there exists a fiber bundle \bar f : \bar E \to S^1 where \bar E is a 4-manifold homotopy equivalent to E.

Theorem (joint with Shmuel Weinberger)  The conjecture is true when the 3-manifold is a lens space with odd order fundamental group.

The proof involves a surgery theoretic argument which involves a lemma of Gauss used in his third proof of the law of quadratic reciprocity.

This theorem answers a question of Jonathan Hillman, asked in the context of 4-dimensional geometries:

Theorem:  Any 4-manifold with Euler characteristic zero and fundamental group a semidirect product where Z acts on Z/odd is homotopy equivalent to a self isometry of a lens space.

Abstract

Nonlinear dynamical systems are prevalent in systems biology, where they are often used to represent a biological system. Its dynamical behaviour is often impossible to understand by intuition alone without such mathematical models. Ideas and methods from systems and control engineering can help us to understand how the pathway architecture and parameter choices produce the desired performance and robustness in the observed dynamics. In this talk, we first show the direct interaction of a theoretical analysis with efficiently setting up experiments. We present the application of tools from engineering for designing biological experiments to elucidate the signalling pathway in the chemotactic system of /Rhodobacter sphaeroides/. In the second part, we focus on the problem of finding experimental setups that allow for full state observability and parameter identifiability of a nonlinear dynamical system; that is, whether the values of system states and parameters can be deduced from output data (experimental observations). This is an important question to answer as often observability and identifiability are assumed, which might lead to costly repetitions of experiments. We present a novel approach to check a priori for parameter identifiability and use new, state of the art computational tools for the implementation. Examples from biology are used to illustrate our method.