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.

Courcelle’s Theorem is an influential meta-theorem published in 1990. It tells us that a property of graph can be tested in polynomial time, as long as the property can expressed in the monadic second-order logic of graphs, and as long as the input is restricted to a class of graphs with bounded tree-width. There are several properties that are NP-complete in general, but which can be expressed in monadic logic (3-colourability, Hamiltonicity…), so Courcelle’s Theorem implies that these difficult properties can be tested in polynomial time when the structural complexity of the input is limited. Matroids can be considered as a special class of hypergraphs. Any finite set of vectors over a field leads to a matroid, and such a matroid is said to be representable over that field. Hlineny produced a matroid analogue of Courcelle’s Theorem for input classes with bounded branch-width that are representable over a finite field. We have now identified the structural properties of hypergraph classes that allow a proof of Hliněný’s Theorem to go through. This means that we are able to extend his theorem to several other natural classes of matroids. This talk will contain an introduction to matroids, monadic logic, and tree-automata. This is joint work with Daryl Funk, Mike Newman, and Geoff Whittle.

What is the largest subset of $\mathbb Z_{2^n}$ that doesn’t contain a projective d-cube? In the Boolean lattice, Sperner’s, Erdos’s, Kleitman’s and Samotij’s theorems state that families that do not contain many chains must have a very specific layered structure. We show that if instead of Z n 2 we work in Z 2 n , analogous statements hold if one replaces the word k-chain by projective cube of dimension 2 k − 1 . The largest d-cube-free subset of Z 2 n , if d is not a power of two, exhibits a much more interesting behaviour. This is joint work with Jason Long.

An important family of incidence problems are discrete analogs of deep questions in geometric measure theory. Perhaps the most famous example of this is the finite field Kakeya conjecture, proved by Dvir in 2008. Dvir’s proof introduced the polynomial method to incidence geometry, which led to the solution to many long-standing problems in the area. I will talk about a generalization of the Kakeya conjecture posed by Ellenberg, Oberlin, and Tao. A ( k , m ) -Furstenberg set S in F n q has the property that, parallel to every affine k -plane V, there is a k-plane W such that | W ∩ S | > m . Using sophisticated ideas from algebraic geometry, Ellenberg and Erman showed that if S is a ( k , m ) -Furstenberg set, then | S | > c m n / k , for a constant c depending on n and k. In recent joint work with Manik Dhar and Zeev Dvir, we give simpler proofs of stronger bounds. For example, if m > 2 n + 7 q , then | S | = ( 1 − o ( 1 ) ) m q n − k , which is tight up to the o ( 1 ) term.

In many applications of machine learning, interpretable or explainable models for binary classification, such as decision trees or decision lists, are preferred over potentially more accurate but less interpretable models such as random forests or support vector machines. In this talk, we consider boolean decision rule sets (equivalent to boolean functions in disjunctive normal form) as interpretable models for binary classification. We define the complexity of a rule set to be the number of rules (clauses) plus the number of conditions (literals) across all clauses, and assume that simpler or less complex models are more interpretable. We discuss an integer programming formulation for such models that trades off classification accuracy against rule simplicity, and obtain high-quality classifiers of this type using column generation techniques. Compared to some recent alternatives, our algorithm dominates the accuracy-simplicity trade-off in 8 out of 16 datasets, and also produced the winning entry in the 2018 FICO explainable machine learning challenge. When compared to rule learning methods designed for accuracy, our algorithm sometimes finds significantly simpler solutions that are no less accurate.

It has been known that stochastic differential equations with non-degenerate diffusion admit a unique solution for sub-critical drifts. In this talk, we extend this in two different directions: the critical drift case and the degenerate diffusion case. I will briefly introduce a hypoelliptic theory, and then explain how to obtain probabilistic results on stochastic differential equations.

A classical problem in mechanics is the following: given a set of forces applied at some points in space, is there any discrete network with all the wires under tension that supports such a system of forces? In this talk I will present some recent results on the topic, obtained in collaboration with Graeme Milton, Pierre Seppecher and Guy Bouchitte. In usual wire or cable networks, such as in a bridge or bicycle wheel, one distributes the forces by adjusting the tension in the wires. Here our discrete networks provide an alternative way of distributing the forces through the geometry of the network. In particular the network can be chosen so it is uniloadable, i.e. supports only one set of forces at the terminal nodes. Such uniloadable networks are relevant as the tension in one element determines the tension in the joint wires and, by extension, uniquely determines the stress in the entire network.