## Mai 2018

More information you can find on the conference homepage:

https://www.mis.mpg.de/calendar/conferences/2018/combinatorics2018/index.html

We study the regularity for finite modular lattices which are not distributive. In particular, we show that the join-meet ideal of such a lattice does not have a linear resolution. Joint work with V. Ene and T. Hibi.

Abstract. Given two convex polyhedra, we intend to compute, for instance, their Minkowski sum, intersection or convex hull of the union. Another basic problem is to compute the polar of a polyhedron. The talk surveys practically relevant methods for such operations. One method is based on a recent result saying that multiple objective linear programming is equivalent to the projection of a convex polyhedron into a lower dimensional space. We show how a multiple objection linear programming solver can be utilized for polyhedral calculus. Finally we discuss a method to represent polyhedral convex functions in a very general way. Given two polyhedral convex functions, we provide a practically relevant method to compute, for instance, their infimal convolution, pointwise maximum or lower convex envelope. We compute a representation of the conjugate of a polyhedral convex function. The talk also discusses modelling techniques for optimization problems involving polyhedral convex functions.

Wente’s $L^infty$-estimate is a fundamental example of a ‘gain’ of regularity due to the special structure of Jacobian determinants. It concerns the the following Dirichlet problem: let $V in H^1(D,R^2)$

begin{equation*}label{eq:Dirichlet}left{begin{aligned}

-Delta u &= det(

abla V) %= star dv^1wedge dv^2

&&text{ in } D

u &= 0 &&text{ on } partial D.

end{aligned}right.

end{equation*}

Wente’s theorem states that the solution $uin W^{1,1}_0(D,

R)$ to the above Dirichlet problem is in the space $L^infty(D) cap H^1_0(D)$. This estimate found many applications in geometric analysis, for instance in the existence of immersed surfaces with constant mean curvature.

It is natural to ask whether a similar estimate holds true for the Neumann problem:

begin{equation*}label{eq:Neumann}left{begin{aligned}

-Delta u &= det(

abla V) % = star dv^1wedge dv^2

&&text{ in } D

frac{partial u}{partial

u} &= frac{1}{2pi} int_{D} det(

abla V) &&text{ on } partial D

end{aligned}right.end{equation*}

The aim of this talk will be to present a counterexample. We will present at first a possible motivation for studying the Neumann problem. Thereafter we will try to sketch the ideas of the proof.

History and Philosophy of Science (HPS) defends ideas about the methods of science based on historical evidence. The received HPS (propagated for instance by Popper, Kuhn, and Lakatos) supported their ideas by each selecting favorable historical episodes of science which they reinterpreted according to their own views. Because of biased case selections and very liberal interpretations, they lost historical evidence. In contrast, evidence-based HPS transforms the ideas of speculative HPS into operational categories and draws random samples from the history of science for statistical analysis, largely applying empirical methods of the social sciences. I illustrate the approach by an analysis of synthetic chemistry, a field that has remained incomprehensible and neglected by the received HPS, despite its all-dominating size.

References

Schummer: J.: “Scientometric Studies on Chemistry I: The Exponential Growth of Chemical Substances, 1800-1995”, Scientometrics, 39 (1997), 107-123.

Schummer: J.: “Scientometric Studies on Chemistry II: Aims and Methods of Producing new Chemical Substances”, Scientometrics, 39 (1997), 125-140.

Schummer: J.: “Why do Chemists Perform Experiments?”, in: D. Sobczynska, P. Zeidler & E. Zielonacka-Lis (eds.), Chemistry in the Philosophical Melting Pot, Frankfurt: Peter Lang, 2004, pp. 395-410.

Vortrag in der Reihe: Arbeitsgemeinschaft ANGEWANDTE ANALYSIS

In this talk, I consider periodic homogenization of non-convex integral functionals that are motivated by non-linear elasticity. In this situation long wavelength buckling can occur which mathematically implies that the homogenized integrand is given by an asymptotic multi-cell formula. From this formula it is difficult to deduce qualitative or quantitative properties of the effective energy. Under suitable assumptions, in particular that the integrand has a single, non-degenerate, energy well at the set of rotations, we show that the multi-cell formula reduces to a much simpler single-cell formula in a neighbourhood of the rotations. This allows for a more refined, corrector based, analysis. In particular, for small data, we obtain a quantitative two-scale expansion and uniform Lipschitz estimates for energy minimizer. This is joint work with Stefan Neukamm (Dresden).

Vortrag in der Reihe: Oberseminar ANALYSIS – PROBABILITY

In this talk I will describe some results obtained in collaboration with Sergio Conti (Bonn), Gilles Francfort (Paris Nord), Flaviana Iurlano (Paris) and Vito Crismale (Palaiseau) on the existence and properties of minimizers of energies which arise in the variational approach to fracture. I will recall the basics of the theory, discuss some mathematical difficulties and describe some approaches to address them.

Vortrag in der Reihe: Oberseminar ANALYSIS – PROBABILITY

We discuss the long-time transport properties of linear wave equations in heterogeneous media that are small random perturbations of periodic media. Although periodic Bloch waves cannot be simply deformed into an exact diagonalisation of the perturbed operator, we construct a natural approximate diagonalisation that leads to a very precise description of the solution as long as the transport remains ballistic. In the case of a quasiperiodic perturbation, this approach establishes ballistic transport up to exponential time scales.