Ulm University, Germany
Spectral clustering of combinatorial fullerene isomers based on their facet graph structure
After Curl, Kroto and Smalley were awarded 1996 the Nobel Prize in chemistry, fullerenes have been subject of much research. One part of that research is the prediction of a fullerene’s stability using topological descriptors. It was mainly done by considering the distribution of the twelve pentagonal facets on its surface, calculations mostly were performed on all isomers of $C_{40}$, $C_{60}$ and $C_{80}$. We suggest a novel method for the classification of combinatorial fullerene isomers using spectral graph theory. The classification presupposes an invariant scheme for the facets based on the Schlegel diagram. The main idea is to find clusters of isomers by analyzing their graph structure of hexagonal facets only. We also show that our classification scheme can serve as a formal stability criterion, which became evident from a comparison of our results with recent quantum chemical calculations.
We apply our method to classify all isomers of $C_{60}$ and give an example of two different cospectral isomers of $C_{44}$. The only input for our algorithms is the vector of positions of pentagons in the facet spiral.
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Université Paris-Est Créteil, France
The Functional Form of Mahler’s Conjecture for Even Log-Concave Functions in Dimension 2
Mahler conjectured that the minimum of the volume product among all symmetric convex bodies in $\mathbb{R}^n$ should be achieved for the cube. We present a functional version of the conjecture which involves even log-concave functions and their Legendre transform. Finally, we give the sharp lower bound of the functional volume product and characterize the equality case of this conjecture for functions defined on $\mathbb{R}^2$.
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Universite Gustave Eiffel, France
Criteria for positive entropic curvature on discrete spaces
According to the works of Lott-Sturm-Villani, the curvature $\kappa$ of a manifold can be expressed in terms of the $\kappa$-convexity of the relative entropy along Wasserstein geodesics. Such an analogous property on graphs has been first proposed by M. Erbar and J. Maas.
Recently, Paul-Marie Samson has proposed another definition of curvature for graphs based on convexity of rel- ative entropy along the Schrödinger bridges which will be the core of this poster. After explaining all these terms, we will address the following question: How can this curvature be calculated locally on graphs and what kind of consequences globally does it has?
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Technische Universität München, Germany
Mean inequalities for symmetrizations of convex sets
The arithmetic-harmonic mean inequality can be generalized for convex sets, considering the intersection, the harmonic and the arithmetic mean, as well as the convex hull of two convex sets. We study those relations of symmetrization of convex sets, i.e., dealing with the means of some convex set $C$ and $-C$. We determine the dilatation factors, depending on the asymmetry of $C$, to reverse the containments between any of those symmetrizations, and tighten the relations proven by Firey and show a stability result concerning those factors near the simplex.
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