Christian Heron

Christian Reiher 2012

Christian Reiher (born April 19, 1984 in Starnberg ) is a German mathematician.

Heron won gold medals four times in a row at the International Mathematical Olympiad from 2000 to 2003 . He studied at the Ludwig Maximilians University in Munich and received his doctorate in 2010 from the University of Rostock under Hans-Dietrich Gronau ( A proof of the theorem according to which every prime number possesses property B ). He is a lecturer at the University of Hamburg .

In 2003 he proved Arnfried Kemnitz's conjecture : Let S be the set of lattice points in the plane for a natural number , then there is a subset T of S with n points, so that its centroid is also a lattice point. ${\displaystyle 4n-3}$ ${\displaystyle n}$

Kemnitz's conjecture generalizes a theorem by Erdös, Ginzburg and Ziv (1961) in the one-dimensional case (every set of integers has a subset with n integers whose mean is an integer). Another formulation of Kemnitz's conjecture asks the smallest number so that every set of grid points in k-dimensional Euclidean space contains a subset S of the cardinality , so that the sum of the elements of S is divisible by . Then according to Erdös u. a. and according to Kemnitz's assumption . ${\displaystyle 2n-1}$ ${\displaystyle f(n,k)}$ ${\displaystyle f}$ ${\displaystyle f}$ ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle f(n,1)=2n-1}$ ${\displaystyle f(n,2)=4n-3}$

Reiher used Chevalley and Warning's theorem for his proof .

In 2017 he received the European Prize in Combinatorics in particular for his solution to the Kemnitz conjecture and the clique density problem by Lovász-Simonovits. The clique-density conjecture by László Lovász and Miklós Simonovits was proven by Reiher in 2016 based on partial results from Razborov (r = 3) and Vladimir Nikoforov (r = 4). The theorem is based on Turán's theorem of extremal graph theory, which makes a statement about the minimum number of edges that a graph with a given number of nodes must have in order for the existence of an r- clique to be ensured. Lovasz and Simonovits hypothesized in the 1970s that a graph with n nodes and at least edges ( ) asymptotically contains at least r-cliques with one constant . They further assumed that the graph extremal with regard to this problem is given by the complete multipartite graph with this number of edges and nodes, in which all partition classes are of the same size except possibly for one that can be smaller. ${\displaystyle r\geq 3}$ ${\displaystyle \gamma n^{2}}$ ${\displaystyle \gamma \in [0,{\frac {1}{2}})}$ ${\displaystyle F_{r}(\gamma )n^{r}+O(n^{r-2})}$ ${\displaystyle F_{r}(\gamma )}$

Individual evidence

1. ↑ IMO website on herons
2. ↑ Christian Reiher in the Mathematics Genealogy Project (English)
3. ^ A. Kemnitz, On a lattice point problem, Ars Combinatoria, Volume 16b, 1983, pp. 151-160
4. ^ Reiher, On Kemnitz 'conjecture concerning lattice-points in the plane , The Ramanujan Journal, Volume 13, 2007, pp. 333–337
5. ↑ Erdös, Ginzburg, Ziv, A theorem in the additive number theory, Bull. Research Council Israel, Volume 10 F, 1961, pp. 41-43
6. ↑ European Prize in Combinatorics 2017
7. ^ Reiher, The Clique density theorem, Annals of Mathematics, Volume 184, 2016, pp. 683-707, Arxiv