Zurich colloquium in mathematics

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Date / Time

Speaker

Title

Location

21 February 2012
17:15-18:15

Prof. Dr. Richard Thomas
Imperial College, London, UK

Event Details

Title	The Göttsche conjecture
Speaker, Affiliation	Prof. Dr. Richard Thomas, Imperial College, London, UK
Date, Time	21 February 2012, 17:15-18:15
Location	KO2 F 150
Abstract	I will describe a classical problem going back to 1848 (Steiner, Cayley, Salmon,...) and a solution using simple techniques that one would never have thought of without ideas coming from string theory (Gromov-Witten invariants, BPS states) and modern geometry (the Maulik-Nekrasov-Okounkov-Pandharipande conjecture). In generic families of curves C on a complex surface S, nodal curves - those with the simplest possible singularities - appear in codimension 1. More generally those with d nodes occur in codimension d. In particular a d-dimensional linear family of curves should contain a finite number of such d-nodal curves. The classical problem - at least in the case of S being the projective plane - is to determine this number. The Göttsche conjecture states that the answer should be topological, given by a universal degree d polynomial in the four numbers C.C, c_1(S).C, c_1(S)^2 and c_2(S). There are now proofs in various settings; a completely algebraic proof was found recently by Tzeng. I will explain a simpler proof which was joint work with Martijn Kool and Vivek Shende.

The Göttsche conjectureread_more

KO2 F 150

6 March 2012
17:15-18:15

Prof. Dr. Burkhard Wilking
Universität Münster, Deutschland

Event Details

Title	Structure of fundamental groups of manifolds with lower Ricci curvature bounds
Speaker, Affiliation	Prof. Dr. Burkhard Wilking, Universität Münster, Deutschland
Date, Time	6 March 2012, 17:15-18:15
Location	KO2 F 150

Structure of fundamental groups of manifolds with lower Ricci curvature bounds

KO2 F 150

3 April 2012
17:15-18:15

Prof. Dr. Emmanuel Hebey
Université de Cergy-Pontoise, Paris

Event Details

Title	Blow-up theory and elliptic stability
Speaker, Affiliation	Prof. Dr. Emmanuel Hebey, Université de Cergy-Pontoise, Paris
Date, Time	3 April 2012, 17:15-18:15
Location	KO2 F 150

Blow-up theory and elliptic stability

KO2 F 150

24 April 2012
17:15-18:15

Prof. Dr. Endre Süli
University of Oxford, UK

Event Details

Title	Navier-Stokes-Fokker-Planck systems: analysis and approximation
Speaker, Affiliation	Prof. Dr. Endre Süli, University of Oxford, UK
Date, Time	24 April 2012, 17:15-18:15
Location	KO2 F 150
Abstract	The talk will survey recent developments concerning the existence and the approximation of global weak solutions to a general class of coupled microscopic-macroscopic bead-spring chain models that arise in the kinetic theory of dilute solutions of polymeric liquids with noninteracting polymer chains. The class of models involves the unsteady incompressible Navier-Stokes equations in a bounded domain for the velocity and the pressure of the fluid, with an elastic extra-stress tensor appearing on the right-hand side of the momentum equation. The extra-stress tensor stems from the random movement of the polymer chains and is defined by the Kramers expression through the associated probability density function that satisfies a Fokker-Planck type parabolic equation. Models of this kind were proposed in work of Hans Kramers in the early 1940's, and the existence of global weak solutions to the model has been a long-standing question in the mathematical analysis of kinetic models of dilute polymers. We also discuss computational challenges associated with the numerical approximation of the high-dimensional Fokker-Planck equation featuring in the model.

Navier-Stokes-Fokker-Planck systems: analysis and approximationread_more

KO2 F 150

22 May 2012
17:15-18:15

Brendan Hassett
Rice University, Houston, TX, USA

Event Details

Title	Fibrations in rational surfaces and their sections
Speaker, Affiliation	Brendan Hassett, Rice University, Houston, TX, USA
Date, Time	22 May 2012, 17:15-18:15
Location	KO2 F 150
Abstract	A fibration is a surjective morphism from a smooth projective variety to a smooth curve, all defined over a field k. Assume the fibers are rational surfaces. Then every fibration admits a section, when k is algebraically closed. There is a conjectural framework for deciding whether there is a section when k is finite, expressed in terms of the Brauer group and the existence of local sections. Our approach to these questions hinges on understanding the geometry of the scheme parametrizing all sections of our fibration, especially in contexts where the rational surfaces are relatively simple, e.g., quadric surfaces and intersections of two quadric hypersurfaces. The main application is the existence of sections provided the fibration is sufficiently general, in a sense that can be made precise. (Joint with Yuri Tschinkel)

Fibrations in rational surfaces and their sectionsread_more

KO2 F 150

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