A (vertex) k -coloring of a graph G is a tree-coloring if each color class induces a forest, and is equitable if the sizes of any two color classes differ by at most 1. The first relative result concerning the equitable tree-coloring of graphs is due to H. Fan, H. A. Kierstead, G. Liu, T. Molla, J.-L. Wu, and X. Zhang (2011), who proved that any graph with maximum degree at most Δ has a Δ -coloring so that each color class induces a graph with maximum degree at most 1. After that, many results on this topic were published in the literature. For example, L. Esperet, L. Lemoine, and F. Maffray (2015) showed that any planar graph admits an equitable tree- k -coloring for every integer k ≥ 4 ，and G. Chen, Y. Gao, S. Shan, G. Wang, and J.-L. Wu (2017) proved that any 5-degenerate graph with maximum degree at most Δ admits an equitable tree- k -coloring for every k ≥ ⌈ Δ + 1 2 ⌉ . In this talk, we review part of the known results and the conjectures on the equitable tree-coloring of graphs, and then show the sketch proofs of our three new results as follows: (a) the vertex set of any graph G can be equitably partitioned into k subsets for any integer k ≥ max { ⌈ Δ ( G ) + 1 2 ⌉ , ⌈ | G | 4 ⌉ } so that each of them induces a linear forest; (b) any plane graph with independent crossings admits an equitable tree- k -coloring for every integer k ≥ 8 ; (c) any d -degenerate graph with maximum degree at most Δ admits an equitable tree- k -coloring for every integer k ≥ ( Δ + 1 ) / 2 provided that Δ ≥ 10 d . This is a joint work with Yuping Gao, Bi Li, Yan Li, and Bei Niu.

We investigate how minor-monotone graph parameters change if we add a few random edges to a connected graph H . Surprisingly, after adding a few random edges, its treewidth, treedepth, genus, and the size of a largest complete minor become very large regardless of the shape of H . Our results are close to best possible for various cases. We also discuss analogous results for randomly perturbed bipartite graphs as well as the size of a largest complete odd minor in randomly perturbed graphs.

Studying Galois covers of the projective line and their specializations is a central topic in inverse Galois theory, due to their connection with the inverse Galois problem. In this talk, we shall present two applications to diophantine geometry and the theory of modular forms. Firstly, we shall explain how deriving many curves over $\mathbb{Q}$ failing the Hasse principle, under the abc-conjecture. Secondly, we shall construct infinitely many non-liftable Hecke eigenforms of weight one over $\overline{\mathbb{F}_p}$ with pairwise non-isomorphic projective Galois representations, for $p \in \{3,5,7,11\}$. This talk is based on joint works with Joachim Koenig, and with Sara Arias-de-Reyna and Gabor Wiese.

In this talk, we propose an overlapping additive Schwarz method for the dual Rudin–Osher–Fatemi (ROF) model, which is a standard problem in mathematical image processing. The O(1/n)-energy convergence of the proposed method is proven, where n is the number of iterations. In addition, we introduce an interesting convergence property called pseudo-linear convergence of the proposed method; the energy of the proposed method decreases as fast as linearly convergent algorithms until it reaches a particular value. It is shown that such the particular value depends on the overlapping width, and the proposed method becomes as efficient as linearly convergent algorithms if the width is large. As the latest domain decomposition methods for the ROF model are sublinearly convergent, the proposed method outperforms them in the sense of the energy decay.

In this talk we will discuss a non-overlapping domain decomposition method for the Stokes problem with a discontinuous viscosity. There are two key ingredients in the proposed method; one is an inf-sup stable finite element for the Stokes problem and the other is a preconditioning procedure based on the FETI-DP approach. Theoretical results for the condition number estimate of the preconditioned problem will be presented along with numerical results.

We denote by $\mathcal{H}_{d,g,r}$ the Hilbert scheme of smooth curves, which is the union of components whose general point corresponds to a smooth irreducible and non-degenerate curve of degree $d$ and genus $g$ in $\mathbb{P}^r$. In this talk, we show that any non-empty $\mathcal{H}_{g+1,g,4}$ has only one component whose general element is linearly normal unless $g=9$. If $g=9$, we show that $\mathcal{H}_{10,9,4}$ is reducible with two components and a general element of each component is linearly normal. This estabilishes the validity of a certain modified version of an assertion of Severi regarding the irreducibility of $\mathcal{H}_{d,g,r}$ for the case $d=g+1$ and $r=4$. This work has been carried out by the joint work with Changho Keem.