Hausdorff Kolloquium 2017/18

Date: November 22, 2017 - January 24, 2018

Venue: Mathematik-Zentrum, Lipschitz Lecture Hall, Endenicher Allee 60, Bonn

Abstracts:

Mark Gross (Cambridge): A general mirror symmetry construction

Mirror symmetry is an area of geometry which had its beginnings in string theory around 1989, and has led to a great deal of interesting mathematics. It was noticed in many examples that certain kinds of complex manifolds called Calabi-Yau manifolds came in pairs, and there was a subtle duality between the geometry of the members of the pairs, allowing for previously inaccessible algebro-geometric calculations to be carried out. I will survey some of the main ideas of mirror symmetry since its origins around 1990, leading to recent work joint with Siebert giving a general construction of mirror pairs.

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Frances Kirwan (Oxford): Moduli spaces of unstable curves

Moduli spaces arise naturally in classification problems in geometry. The construction of the moduli spaces of stable curves of fixed genus is one of the classical applications of Mumford's geometric invariant theory (GIT), developed in the 1960s.  Here a projective curve is stable if it has only nodes as singularities and its automorphism group is finite. The aim of this talk is to describe these moduli spaces and outline their GIT construction, and then explain how recent methods from non-reductive GIT can help us try to classify the singularities of unstable curves in such a way that we can construct moduli spaces of unstable curves (of fixed singularity type).

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Andrea Malchiodi (Pisa): Variational structure of Liouville equations

Liouville equations arise when trying to uniformize curvatures or to extremise energies depending on spectra of surfaces. We consider cases when Gauss-Bonnet integrals are "large", as it might happen in presence of conical singularities. We prove existence of solutions combining geometric functional inequalities and a micro/macroscopic study of conformal volume distribution.

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Susanna Terracini (Torino): Spiralling and other solutions in limiting profiles of competition-diffusion systems

Several physical phenomena can be described by a certain number of densities (of mass, population, probability, ...) distributed in a domain and subject to laws of diffusion, reaction, and competitive interaction. Whenever the competitive inter- action is the prevailing phenomenon, the several densities can not coexist and tend to segregate, hence determining a partition of the domain (Gause’s experimental principle of competitive exclusion (1932)). As a model problem, we consider the system of stationary equations:

                 −∆ui= fi(ui) − βuij≠i gij (uj )

                 ui > 0 .

The cases gij (s) = βij s (Lotka-Volterra competitive interactions) and gij (s) = βij s2 (gradient system for Gross-Pitaevskii energies) are of particular interest in the applications to population dynamics and theoretical physics respectively.

We will undertake the analysis of qualitative properties of solutions to sys- tems of semilinear elliptic equations, whenever the parameter β, accounting for the competitive interactions, diverges to infinity. At the limit, when the minimal interspecific competition rate β = minij βij diverges to infinity, we find a vector U = (u1 , · · · , uh ) of functions with mutually disjoint supports: the segregated states: ui · uj ≡ 0, for i ≠ j, satisfying

                 −∆ui = fi (x, ui)                   whenever ui ≠ 0 , i = 1, . . . , h,

We will review the known results and focus on spiralling solutions in the non symmetrical case: (βij ≠ βji ).

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