Dublin Area Mathematics Colloquium

Benjamin McKay (UCC)
Rational Curves and Ordinary Differential Equations
Friday 5 May 2006, 4.30 pm, Salmon Theatre, Hamilton Building, TCD.

Abstract. We determine which complex analytic 2nd order ODEs have all solutions close up to become Riemann spheres. The techniques come from Penrose's twistor theory.

Martin Kolar (Brno, Czech Republic)
The local equivalence problem in complex analysis
Friday 28 April 2006, 4.30 pm, Salmon Theatre, Hamilton Building, TCD.

Abstract. H. Poincare formulated in 1907 the problem of local biholomorphic equivalence for real hypersurfaces in two-dimensional complex space. In the first part of the talk we will discuss some of its history, including Moser's solution for Levi nondegenerate hypersurfaces in terms of normal forms. In the second part we will describe some recent results in solving the problem also in the Levi degenerate case.

Brendan Guilfoyle (Tralee)
The Caratheodory Conjecture and the zeros of holomorphic polynomials
Friday 21 April 2006, 4.30 pm, Salmon Theatre, Hamilton Building, TCD.

Abstract. The Caratheodory Conjecture states that there must be at least two umbilic point on a C^2-smooth closed convex surface in Euclidean 3-space. As it stands, this Conjecture has remained unproven (and uncontradicted) for almost eighty years, with very little progress being made. In this talk, we will give the background to the Conjecture and describe its relationship with the number of zeros inside the unit circle of certain holomorphic poynomials. This we will do by giving an equivalent Conjecture on the number of complex points on a lagrangian surface in the space of oriented affine lines in R^3.

Norbert Peyerimhoff (Durham, UK)
Some Equidistribution Results in Geometry
Wednesday 19 April 2006, 4.00 pm, Maxwell Theatre, Hamilton Building, TCD.

Abstract. Beginning with Hermann Weyl's mod one-equidistribution of polynomials at integer points, we will present some results on the distribution of particular geometrically motivated subsets of non-positively curved manifolds.

David Calderbank (York, UK)
Submanifolds of celestial spheres: geometry and integrability
Friday 7 April 2006, 4.30 pm, Salmon Theatre, Hamilton Building, TCD.

Abstract. Many integrable systems arise naturally in the classical differential geometry of surfaces in ordinary Euclidean space: for example, surfaces of constant mean curvature (CMC) or K-surfaces (i.e., with constant gauss curvature) are described by integrable systems. I plan to outline a similar story for submanifold geometry in the celestial n-spheres which arise as projective light cones in Minkowski (n+1,1)-space.

Jurgen Berndt (UCC)
Projective planes and Severi varieties
Friday 10 February 2006, 4.30 pm, Salmon Theatre, Hamilton Building, TCD.

Abstract. A classical result asserts that the complex projective plane modulo complex conjugation is the 4-dimensional sphere. We generalize this result in two directions by considering the projective planes over the normed real division algebras and by considering the complexifications of these four projective planes.

Radek Erban (Oxford, UK)
(host Hyung Ju Hwang)
From Individual to Collective Behaviour in Biological Systems
Friday 3 February 2006, 4.00 pm, Salmon Theatre, Hamilton Building, TCD.

Abstract. In current complex systems modeling practice, we are often presented with a model at a fine level of description (atomistic, stochastic, individual-based), while we want to study the behavior at a macroscopic coarse-grained (continuum, population) level. A biological example is modeling of dispersal of unicellular organisms (e.g. bacterium Escherichia coli or amoeboid cell Dictyostelium discoideum) where much is known about processes at the single-cell or subcellular level. However, one is often interested in the behaviour of cellular populations, which may involve the temporal evolution of cellular density, spatial pattern formation etc. In this talk, we will present analytical (derivation of macroscopic evolution equations) and computational (macroscopic equation-free) approaches for extracting population-level behaviour from individual-based models of several biological systems, including models of biological dispersal, gene regulatory networks and other biomedical applications.

Jaroslav Zemanek (Warsaw, Poland)
Geometric interpretations of the Sendov conjecture
Wednesday 21 December 2005, 3.00 pm, Salmon Theatre, Hamilton Building, TCD.

Abstract. We intend to present various matrix formulations (in terms of unitary or similarity orbits, or companion matrices) of the 1958 Conjecture of Sendov on the distance between the roots of a polynomial and its derivative. Numerical ranges and Gerschgorin discs may also be discussed. The lecture should be clear to all (not only) mathematicians.

Bernd Kreussler (Limerick)
Semi-stability on singular elliptic curves
Friday 2 December 2005, 4.30 pm, Salmon Theatre, Hamilton Building, TCD.

Abstract. In 1957, M.F. Atiyah studied vector bundles over smooth elliptic curves. However, mainly because the category of coherent sheaves on a singular curve has infinite homological dimension, his techniques do not carry over to the singular case. In the talk, which reports about joint work with Igor Burban, I'll explain how to overcome these difficulties. At the end of the talk, a clear picture of the structure of the category of coherent sheaves in the singular case will emerge.

Gregor Fels (Tübingen, Germany)
Representations and Cycle Spaces
Friday 4 November 2005, 4.30 pm, Salmon Theatre, Hamilton Building, TCD.

Abstract. To broad classes of representations of a Lie group it is possible to assign certain geometric objects (like manifolds, vector bundles, etc) which help to understand the original representation. We explain what these geometric objects would be in the context of real semisimple Lie groups and discuss various spaces of cycles which naturally occur in this situation.

David Wraith (NUIM)
New connected sums with positive Ricci curvature
Friday 21 October 2005, 4.30 pm, Salmon Theatre, Hamilton Building, TCD.

Abstract. We explore the question of which manifolds admit Riemannian metrics of positive Ricci curvature. In particular we focus on how surgery can help in generating new families of Ricci positive manifolds. We will display some infinite families of new examples using recent surgery techniques.

Previous Seminars
Dublin Theoretical Physics Colloquium

Contact:
Dmitri Zaitsev (TCD)