|Speaker :||1.Prof. Mihyun Kang (TU Graz) 2.Prof. Minking Eie (National Chung Cheng University)|
|Title :||1.Giant component, the k-core, and branching processes 2.Evaluations of some multiple zeta-star values.|
|Time :||2018-02-02 (Fri) 13:50 - 16:10|
|Place :||Seminar Room 617, Institute of Mathematics (NTU Campus)|
1. The giant component has been one of the central topics in the theory of random graphs, since the seminal work of Erd?s and Renyi five decades ago. By now, there exist extensive results on the birth, size, structure, central and local limit theorems of the giant component in the random graph. A key observation in this line of work is that the emergence of the giant component is closely related to the survival of the Galton-Watson branching process.
The k-core of a graph is the maximal subgraph of minimum degree k, which is perhaps the most natural generalisation of the giant component. Pittel, Wormald, and Spencer [JCTB 67 (1996), 111–151] were the first to determine the threshold d_k for the appearance of an extensive k-core. Recently, Coja-Oghlan, Cooley, Skubsch, and the speaker derived a multi-type Galton-Watson branching process that describes precisely how the k-core is “embedded” into the random graph G(n,p) for any k>2 and any fixed average degree d = np > d_k. In this talk we discuss some classical and recent results on the giant component, k-core, and branching processes.
2. 1735, Euler evaluated the special values at positive even integers of Riemann zeta function. He developed the infinite product formula of the sine function and discovered the evaluations of multiple zeta values with arguments 2,2,....2. This also leads to the evaluations of multiple zeta values with arguments m,m,....,m and m > 2.
Here we evaluate multiple zeta-star values with arguments r+2,2,2,....,2 through their integral representations. During the evaluations, we need some particular combinatorial identities which maybe new, but possible old. Anyway, we prove all these combinatorial identities, use them to prove identities among generating functions of multiple zeta values and obtain the evaluations in the final.