Past

http://www.maths.ox.ac.uk/arg

Many large-scale optimization problems arise in the context of the discretization of infinite dimensional applications. In such cases, the description of the finite-dimensional problem is not unique, but depends on the discretization used, resulting in a natural multi-level description. How can such a problem structure be exploited, in discretized problems or more generally? The talk will focus on discussing this issue in the context of unconstrained optimization and in relation with the classical multigrid approach to elliptic systems of partial differential equations. Both theoretical convergence properties of special purpose algorithms and their numerical performances will be discussed. Perspectives will also be given.

Collaboration with S. Gratton, A. Sartenaer and M. Weber.

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There is a construction, due to Kontsevich, which produces cohomology classes in moduli spaces of Riemann surfaces from the initial data of an A-infinity algebra with an invariant inner product -- a kind of homotopy theoretic notion of a Frobenius algebra.

In this talk I will describe a version of this construction based on noncommutative symplectic geometry and use it to show that homotopy equivalent A-infinity algebras give rise to cohomologous classes. I will explain how the whole framework can be adapted to deal with Topological Conformal Field Theories in the sense of Costello, Kontsevich and Segal.

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In Dept of Statistics

http://www.maths.ox.ac.uk/arg

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In the first part of the talk I will discuss existence and regularity results for models of complex non-Newtonian fluids containing liquid crystalline polymers. The main technical tool is related to an apriori logarithmic estimate for the 2D Navier-Stokes equations.

In the second part I will consider a simplified version of the system and describe some of its qualitative properties as well as the analytical challenges posed by its study.

The first part is joint work with P Constantin, Ch. Fefferman and E Titi.

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