Every minor-closed class of matroids of bounded branch-width can be characterized by a minimal list of excluded minors, but unlike graphs, this list could be infinite in general. However, for each fixed finite field $\mathbb F$, the list contains only finitely many $\mathbb F$-representable matroids, due to the well-quasi-ordering of $\mathbb F$-representable matroids of bounded branch-width under taking matroid minors [J. F. Geelen, A. M. H. Gerards, and G. Whittle (2002)]. But this proof is non-constructive and does not provide any algorithm for computing these $\mathbb F$-representable excluded minors in general. We consider the class of matroids of path-width at most $k$ for fixed $k$. We prove that for a finite field $\mathbb F$, every $\mathbb F$-representable excluded minor for the class of matroids of path-width at most~$k$ has at most $2^{|\mathbb{F}|^{O(k^2)}}$ elements. We can therefore compute, for any integer $k$ and a fixed finite field $\mathbb F$, the set of $\mathbb F$-representable excluded minors for the class of matroids of path-width $k$, and this gives as a corollary a polynomial-time algorithm for checking whether the path-width of an $\mathbb F$-represented matroid is at most $k$. We also prove that every excluded pivot-minor for the class of graphs having linear rank-width at most $k$ has at most $2^{2^{O(k^2)}}$ vertices, which also results in a similar algorithmic consequence for linear rank-width of graphs. This is joint work with Mamadou M. Kanté, Eun Jung Kim, and O-joung Kwon.

Tutte (1961) proved that every simple $3$-connected graph $G$ has an edge $e$ such that $G \setminus e$ or $G / e$ is simple $3$-connected, unless $G$ is isomorphic to a wheel. We call such an edge non-essential. Oxley and Wu (2000) proved that every simple $3$-connected graph has at least $2$ non-essential edges unless it is isomorphic to a wheel. Moreover, they proved that every simple $3$-connected graph has at least $3$ non-essential edges if and only if it is isomorphic to neither a twisted wheel nor a $k$-dimensional wheel with $k\geq2$. We prove analogous results for graphs with vertex-minors. For a vertex $v$ of a graph $G$, let $G*v$ be the graph obtained from $G$ by deleting all edges joining two neighbors of $v$ and adding edges joining non-adjacent pairs of two neighbors of $v$. This operation is called the local complementation at $v$, and we say two graphs are locally equivalent if one can be obtained from the other by applying a sequence of local complementations. A graph $H$ is a vertex-minor of a graph $G$ if $H$ is an induced subgraph of a graph locally equivalent to $G$. A split of a graph is a partition $(A,B)$ of its vertex set such that $|A|,|B| \geq 2$ and for some $A'\subseteq A$ and $B'\subseteq B$, two vertices $x\in A$ and $y\in B$ are adjacent if and only if $x\in A'$ and $y\in B’$. A graph is prime if it has no split. A vertex $v$ of a graph is non-essential if at least two of three kinds of vertex-minor reductions at $v$ result in prime graphs. We prove that every prime graph with at least $5$ vertices has at least two non-essential vertices unless it is locally equivalent to a cycle. It is stronger than a theorem proved by Allys (1994), which states that every prime graph with at least $5$ vertices has a non-essential vertex unless it is locally equivalent to a cycle. As a corollary of our result, one can obtain the first result of Oxley and Wu. Furthermore, we show that every prime graph with at least $5$ vertices has at least $3$ non-essential vertices if and only if it is not locally equivalent to a graph with two specified vertices $x$ and $y$ consisting of at least two internally-disjoint paths from $x$ to $y$ in which $x$ and $y$ have no common neighbor. This is joint work with Sang-il Oum.

An independent dominating set of a graph, also known as a maximal independent set, is a set $S$ of pairwise non-adjacent vertices such that every vertex not in $S$ is adjacent to some vertex in $S$. We prove that for $\Delta=4$ or $\Delta\ge 6$, every connected $n$-vertex graph of maximum degree at most $\Delta$ has an independent dominating set of size at most $(1-\frac{\Delta}{ \lfloor\Delta^2/4\rfloor+\Delta })(n-1)+1$. In addition, we characterize all connected graphs having the equality and we show that other connected graphs have an independent dominating set of size at most $(1-\frac{\Delta}{ \lfloor\Delta^2/4\rfloor+\Delta })n$. This is joint work with Eun-Kyung Cho, Minki Kim, and Sang-il Oum.

Erdős and Pósa proved in 1965 that there is a duality between the maximum size of a packing of cycles and the minimum size of a vertex set hitting all cycles. We therefore say that cycles satisfy the Erdős-Pósa property. However, while odd cycles do not satisfy the Erdős-Pósa property, Reed proved in 1999 an analogue by relaxing packing to half-integral packing, where each vertex is allowed to be contained in at most two such cycles. Moreover, he gave a structural characterisation for when the Erdős-Pósa property for odd cycles fails. We prove a far-reaching generalisation of the theorem of Reed; if the edges of a graph are labelled by finitely many abelian groups, then the cycles whose values avoid a fixed finite set for each abelian group satisfy the half-integral Erdős-Pósa property, and we similarly give a structural characterisation for the failure of the Erdős-Pósa property. A multitude of natural properties of cycles can be encoded in this setting. For example, we show that the cycles of length $\ell$ modulo $m$ satisfy the half-integral Erdős-Pósa property, and we characterise for which values of $\ell$ and $m$ these cycles satisfy the Erdős-Pósa property. This is joint work with Kevin Hendrey, Ken-ichi Kawarabayashi, O-joung Kwon, Sang-il Oum, and Youngho Yoo.

Variable selection is an approach to identifying a set of covariates that are truly important to explain the feature of a response variable. It is closely connected or belongs to model selection approaches. This talk provides a gentle introduction to Bayesian variable selection methods. The basic notion of variable selection is introduced, followed by several Bayesian approaches with a simple application example.

In a reduction sequence of a graph, vertices are successively identified until the graph has one vertex. At each step, when identifying $u$ and $v$, each edge incident to exactly one of $u$ and $v$ is coloured red. Bonnet, Kim, Thomassé, and Watrigant [FOCS 2020] defined the twin-width of a graph $G$ to be the minimum integer $k$ such that there is a reduction sequence of $G$ in which every red graph has maximum degree at most $k$. For any graph parameter $f$, we define the reduced-$f$ of a graph $G$ to be the minimum integer $k$ such that there is a reduction sequence of $G$ in which every red graph has $f$ at most $k$. Our focus is on graph classes with bounded reduced-bandwidth, which implies and is stronger than bounded twin-width (reduced-maximum-degree). We show that every proper minor-closed class has bounded reduced-bandwidth, which is qualitatively stronger than a result of Bonnet et al. for bounded twin-width. In many instances, we also make quantitative improvements. For example, all previous upper bounds on the twin-width of planar graphs were at least $2^{1000}$. We show that planar graphs have reduced-bandwidth at most $466$ and twin-width at most $583$; moreover, the witnessing reduction sequence can be constructed in polynomial time. We show that $d$-powers of graphs in a proper minor-closed class have bounded reduced-bandwidth (irrespective of the degree of the vertices). This is joint work with Édouard bonnet and David Wood.

본 발표에서는 다양한 기초 생명-의학 분야에서 생성되고 있는 오믹스 자료의 연구 개발 현황에 대해서 다룰 예정이다. 보다 큰 규모로, 보다 빠르게, 보다 정확하게, 보다 정밀하게 라는 궁극적인 목표하에 이뤄지고 있는 오믹스 자료의 진화에 발맞춰, 이를 분석하는 수리통계적 모형 역시 진화하고 있다. 그 중, 이번 발표에서는 미국의 초 대형 정밀의료 프로젝트인 TopMed에서 진행하고 있는 COPD에 관한 다중 오믹스 자료의 통합 분석 방법 및 결과에 대해서 자세히 다룰 예정이다. 아울러 정밀의료라는 목표를 달성하기 위해 반드시 모형에서 고려해야 하는 “환경 특이적 효과”에 대해 강연할 예정이다.

For given $k$ graphs $G_1,\dots, G_k$ over a common vertex set of size $n$, what conditions on $G_i$ ensures a 'colorful' copy of $H$, i.e. a copy of $H$ containing at most one edge from each $G_i$? Keevash, Saks, Sudakov, and Verstraëte defined $\operatorname{ex}_k(n,H)$ to be the maximum total number of edges of the graphs $G_1,\dots, G_k$ on a common vertex set of size $n$ having no colorful copy of $H$. They completely determined $\operatorname{ex}_k(n,K_r)$ for large $n$ by showing that, depending on the value of $k$, one of the two natural constructions is always the extremal construction. Moreover, they conjectured the same holds for every color-critical graphs and proved it for $3$-color-critical graphs. We prove their conjecture for $4$-color-critical graphs and for almost all $r$-color-critical graphs when $r > 4$. Moreover, we show that for every non-color-critical non-bipartite graphs, none of the two natural constructions is extremal for certain values of $k$. This is a joint work with Debsoumya Chakraborti, Jaehoon Kim, Hyunwoo Lee, and Hong Liu.

A geometric transversal to a family of convex sets in $\mathbb R^d$ is an affine flat that intersects the members of the family. While there exists a far-reaching theory concerning 0-dimensional transversals (intersection patterns of convex sets), much less is known when it comes to higher dimensional transversals. In this talk I will present some new and old results and problems regarding geometric transversals, based on joint work with Otfried Cheong and Xavier Goaoc.

Inside living cells, chemical reactions form a large web of networks. Understanding the behavior of those complex reaction networks is an important and challenging problem. In many situations, it is hard to identify the details of the reactions, such as the reaction kinetics and parameter values. It would be good if we can clarify what we can say about the behavior of reaction systems, when we know the structure of reaction networks but reaction kinetics is unknown. In these talks, I plan to introduce two approaches in this spirit. Firstly, we will discuss the sensitivity analysis of reaction systems based on the structural information of reaction networks [1]. I will give an introduction to the method of identifying subnetworks inside which the effects of the perturbation of reaction parameters are confined. Secondly, I will introduce the reduction method that we developed recently [2]. In those two methods, an integer determined by the topology of a subnetwork, which we call an influence index, plays a crucial role. [1] T. Okada, A. Mochizuki, “Law of Localization in Chemical Reaction Networks,” Phys. Rev. Lett. 117, 048101 (2016); T. Okada, A. Mochizuki, “Sensitivity and network topology in chemical reaction systems,” Phys. Rev. E 96, 022322 (2017). [2] Y. Hirono, T. Okada, H. Miyazaki, Y. Hidaka, “Structural reduction of chemical reaction networks based on topology”, Phys. Rev. Research 3, 043123 (2021).

Abstract: Inside living cells, chemical reactions form a large web of networks. Understanding the behavior of those complex reaction networks is an important and challenging problem. In many situations, it is hard to identify the details of the reactions, such as the reaction kinetics and parameter values. It would be good if we can clarify what we can say about the behavior of reaction systems, when we know the structure of reaction networks but reaction kinetics is unknown. In these talks, I plan to introduce two approaches in this spirit. Firstly, we will discuss the sensitivity analysis of reaction systems based on the structural information of reaction networks [1]. I will give an introduction to the method of identifying subnetworks inside which the effects of the perturbation of reaction parameters are confined. Secondly, I will introduce the reduction method that we developed recently [2]. In those two methods, an integer determined by the topology of a subnetwork, which we call an influence index, plays a crucial role. References [1] T. Okada, A. Mochizuki, “Law of Localization in Chemical Reaction Networks,” Phys. Rev. Lett. 117, 048101 (2016); T. Okada, A. Mochizuki, “Sensitivity and network topology in chemical reaction systems,” Phys. Rev. E 96, 022322 (2017). [2] Y. Hirono, T. Okada, H. Miyazaki, Y. Hidaka, “Structural reduction of chemical reaction networks based on topology”, Phys. Rev. Research 3, 043123 (2021).

In adult tissues, stem cells undergo clonal competition because they proliferate while the stem cell niche provides limited space. This clonal competition allows the selection of healthy stem cells over time as unfit stem cells tend to lose from the competition. It could also be a mechanism to select a supercompetitor with tumorigenic mutations, which may lead to tumorigenesis. I am going to explain general concepts of clonal competition and how a simple model can explain the behaviour of adult stem cells. I will also show how geometric constraints affect the overall dynamics of stem cell competition.

Motivated from the surrounding property of a point set in $\mathbb{R}^d$ introduced by Holmsen, Pach and Tverberg, we consider the transversal number and chromatic number of a simplicial sphere. As an attempt to give a lower bound for the maximum transversal ratio of simplicial $d$-spheres, we provide two infinite constructions. The first construction gives infinitely many $(d+1)$-dimensional simplicial polytopes with the transversal ratio exactly $\frac{2}{d+2}$ for every $d\geq 2$. In the case of $d=2$, this meets the previously well-known upper bound $1/2$ tightly. The second gives infinitely many simplicial 3-spheres with the transversal ratio greater than $1/2$. This was unexpected from what was previously known about the surrounding property. Moreover, we show that, for $d\geq 3$, the facet hypergraph $\mathcal{F}(\mathbf{P})$ of a $(d+1)$-dimensional simplicial polytope $\mathbf{P}$ has the chromatic number $\chi(\mathcal{F}(\mathbf{P})) \in O(n^{\frac{\lceil d/2\rceil-1}{d}})$, where $n$ is the number of vertices of $\mathbf{P}$. This slightly improves the upper bound previously obtained by Heise, Panagiotou, Pikhurko, and Taraz. This is a joint work with Joseph Briggs and Michael Gene Dobbins.