The Erdős–Faber–Lovász conjecture (posed in 1972) states that the chromatic index of any linear hypergraph on n vertices is at most n. In this talk, I will sketch a proof of this conjecture for every large n. Joint work with D.Kang, T. Kelly, D. Kühn and D. Osthus.

Complexity of the cellular organization of the tumor microenvironment as an ecosystem remains to be fully appreciated. Here, for a comprehensive investigation of tumor ecosystems across a wide variety of cancer types, we performed integrative transcriptome analyses of 4.4 million single cells from 978 tumor and 474 normal samples in combination with 9,510 TCGA and 1,339 checkpoint inhibitor-treated bulk tumors. Our analysis enabled us to define 28 different epithelial cell states, some of which had prognostic effects in cancers of relevant origin. Malignant fibroblast signatures defined according to the organ of origin demonstrated prognostic significance across diverse cancer types and revealed FN1, BGN, THBS2, and CTHRC1 as common cancer-associated fibroblast genes. Novel associations were revealed between the AKR1C1+ inflammatory fibroblast and myeloid-derived PRR-induced activation states and between the CXCL10+ fibroblast and squamous/LAMP3+ DC/SPP1+ macrophage states. We discovered tumor-specific rewiring of the tertiary lymphoid structure (TLS) network, involving previously unappreciated DC1, and pDC.. Along with other TLS component states, the tumor-associated germinal center B cell state identified from adjacent normal tissues was able to predict responses to checkpoint immunotherapy. Distinct groups of pan-cancer ecosystems were identified and characterized along the axis of immunotherapy responses. Our systematic, high-resolution dissection of tumor ecosystems provides a deeper understanding of inter- and intra-tumoral heterogeneity.

We introduce the notion of delineation. A graph class $\mathcal C$ is said delineated by twin-width (or simply, delineated) if for every hereditary closure $\mathcal D$ of a subclass of $\mathcal C$, it holds that $\mathcal D$ has bounded twin-width if and only if $\mathcal D$ is monadically dependent. An effective strengthening of delineation for a class $\mathcal C$ implies that tractable FO model checking on $\mathcal C$ is perfectly understood: On hereditary closures of subclasses $\mathcal D$ of $\mathcal C$, FO model checking on $\mathcal D$ is fixed-parameter tractable (FPT) exactly when $\mathcal D$ has bounded twin-width. Ordered graphs [BGOdMSTT, STOC ’22] and permutation graphs [BKTW, JACM ’22] are effectively delineated, while subcubic graphs are not. On the one hand, we prove that interval graphs, and even, rooted directed path graphs are delineated. On the other hand, we observe or show that segment graphs, directed path graphs (with arbitrarily many roots), and visibility graphs of simple polygons are not delineated. In an effort to draw the delineation frontier between interval graphs (that are delineated) and axis-parallel two-lengthed segment graphs (that are not), we investigate the twin-width of restricted segment intersection classes. It was known that (triangle-free) pure axis-parallel unit segment graphs have unbounded twin-width [BGKTW, SODA ’21]. We show that $K_{t,t}$-free segment graphs, and axis-parallel $H_t$-free unit segment graphs have bounded twin-width, where $H_t$ is the half-graph or ladder of height $t$. In contrast, axis-parallel $H_4$-free two-lengthed segment graphs have unbounded twin-width. We leave as an open question whether unit segment graphs are delineated. More broadly, we explore which structures (large bicliques, half-graphs, or independent sets) are responsible for making the twin-width large on the main classes of intersection and visibility graphs. Our new results, combined with the FPT algorithm for first-order model checking on graphs given with $O(1)$-sequences [BKTW, JACM ’22], give rise to a variety of algorithmic win-win arguments. They all fall in the same framework: If $p$ is an FO definable graph parameter that effectively functionally upperbounds twin-width on a class C, then $p(G) \ge k$ can be decided in FPT time $f(k)\cdot |V (G)|O(1)$. For instance, we readily derive FPT algorithms for k-Ladder on visibility graphs of 1.5D terrains, and k-Independent Set on visibility graphs of simple polygons. This showcases that the theory of twin-width can serve outside of classes of bounded twin-width. Joint work with Édouard Bonnet, Dibyayan Chakraborty, Eun Jung Kim, Raul Lopes and Stéphan Thomassé.

The Graphic Travelling Salesman Problem is the problem of finding a spanning closed walk (a TSP walk) of minimum length in a given connected graph. The special case of the Graphic TSP on subcubic graphs has been studied extensively due to their worst-case behaviour in the famous $\frac{4}{3}$-integrality-gap conjecture on the "subtour elimination" linear programming relaxation of the Metric TSP. We prove that every simple 2-connected subcubic graph on $n$ vertices with $n_2$ vertices of degree 2 has a TSP walk of length at most $\frac{5n+n_2}{4}-1$, confirming a conjecture of Dvořák, Král', and Mohar. This bound is best possible and we characterize the extremal subcubic examples meeting this bound. We also give a quadratic time combinatorial algorithm to find such a TSP walk. In particular, we obtain a $\frac{5}{4}$-approximation algorithm for the Graphic TSP on cubic graphs. Joint work with Michael Wigal and Xingxing Yu.

사람이 어떻게 만들어지고 각 기관이 어떻게 발달하는지에 대한 질문은 아주 오래전부터 있었습니다. 체외수정(IVF)의 고유의 장점으로 인해 과학자들이 수정란을 외부에서 관찰할 수 있게 되었습니다. 하지만, 1979년도에 제정된 14일 규정(the 14-day rule)으로 인해, 수정 후 최대 14일까지의 배아 만의 연구가 가능합니다. 따라서, 이 14일 규정은 발생 생물학자들이 사람 발생학 연구에 있어서 수정 후 2주 이상(신경계 발달, 기관 형성 등)에 나타나는 현상을 연구하고자 할 경우 다른 방향을 모색할 수밖에 없게 되었습니다. 본 연구는 이 지점에서부터 시작합니다. 연구진들은 세포 분열 때 우연히 발생하는 생리학적 체세포 변이(Post-zygotic Variants)를 추적하여 각 세포들의 운명을 재구성하였습니다. 특히 사망 후 기증된 시신에서 단일 세포를 배양하고, 최근 개발된 차세대 염기서열 분석 기술을 사용하여 인간 발생 연구의 후향적 혈통 추적(Retrospective Lineage Tracing)을 수행하는 과정을 발표하고자 합니다. 이번 발표를 통해서 이런 방법론이 어떻게 가능했는지에 대한 생물학적 및 과학적 배경과 인간 발생학의 미래에서 해결해야 할 과제와 가설을 강조할 예정입니다. 추가로, 이 과정에서 필요한 수학적인 해석이 필요한 질문들에 대해서도 논의할 예정입니다. 여러분들의 참신한 시각과 질문을 크게 환영합니다. 1) Park, S., Mali, N.M., Kim, R. et al. Clonal dynamics in early human embryogenesis inferred from somatic mutation. Nature 597, 393–397 (2021). https://doi.org/10.1038/s41586-021-03786-8 2) Kwon, S.G., Bae, G.H., Choi, J.H. et al. Asymmetric Contribution of Blastomere Lineages of First Division of the Zygote to Entire Human Body Using Post-Zygotic Variants. Tissue Eng Regen Med 19, 809–821 (2022). https://doi.org/10.1007/s13770-022-00443-7

Various types of independent sets have been studied for decades. As an example, the minimum number of maximal independent sets in a connected graph of given order is easy to determine (hint; the answer is written in the stars). When considering this question for twin-free graphs, it becomes less trivial and one discovers some surprising behaviour. The minimum number of maximal independent sets turns out to be; logarithmic in the number of vertices for arbitrary graphs, linear for bipartite graphs and exponential for trees. Finally, we also have a sneak peek on the 69-conjecture, part of an unpublished work on an inverse problem on the number of independent sets. In this talk, we will focus on the basic concepts, the intuition behind the statements and sketch some proof ideas. The talk is based on joint work with Stephan Wagner, with the main chunk being available at arXiv:2211.04357.

Absolute Concentration Robustness (ACR) was introduced by Shinar and Feinberg (Science 327:1389-1391, 2010) as robustness of equilibrium species concentration in a mass action dynamical system. Their aim was to devise a mathematical condition that will ensure robustness in the function of the biological system being modeled. The robustness of function rests on what we refer to as empirical robustness — the concentration of a species remains unvarying, when measured in the long run, across arbitrary initial conditions. Even simple examples show that the ACR notion introduced in Shinar and Feinberg (here referred to as static ACR) is neither necessary nor sufficient for empirical robustness. To make a stronger connection with empirical robustness, we define dynamic ACR, a property related to long-term, global dynamics, rather than only to equilibrium behavior. We discuss general dynamical systems with dynamic ACR properties as well as parametrized families of dynamical systems related to reaction networks. In particular, we find necessary and sufficient conditions for dynamic ACR in complex balanced reaction networks, a class of networks that is central to the theory of reaction networks.This is joint work with Badal Joshi (CSUSM)

TBA

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ZOOM ID: 997 8258 4700 (Biomedical Mathematics Online Colloquium), (pw: 1234)

The disjoint paths logic, FOL+DP,  is an extension of First Order Logic (FOL) with the extra atomic predicate $\mathsf{dp}_k(x_1,y_1,\ldots,x_k,y_k),$ expressing the existence of internally vertex-disjoint paths between $x_i$ and $y_i,$ for $i\in \{1,\ldots, k\}$. This logic can express a wide variety of problems that escape the expressibility potential of FOL. We prove that for every minor-closed graph class, model-checking for FOL+DP can be done in quadratic time. We also introduce an extension of FOL+DP, namely the scattered disjoint paths logic, FOL+SDP, where we further consider the atomic predicate $\mathsf{s-sdp}_k(x_1,y_1,\ldots,x_k,y_k),$ demanding that the disjoint paths are within distance bigger than some fixed value $s$. Using the same technique we prove that model-checking for FOL+SDP can be done in quadratic time on classes of graphs with bounded Euler genus. Joint work with Petr A. Golovach and Dimitrios M. Thilikos.

The ability to reliably engineer the mammalian cell will impact a variety of applications in a disruptive way, including cell fate control and reprogramming, targeted drug delivery, and regenerative medicine. However, our current ability to engineer mammalian genetic circuits that behave as predicted remains limited. These circuits depend on the intra and extra cellular environment in ways that are difficult to anticipate, and this fact often hampers genetic circuit performance. This lack of robustness to poorly known and often variable cellular environment is the subject of this talk. Specifically, I will describe control engineering approaches that make the performance of genetic devices robust to context. I will show a feedforward controller that makes gene expression robust to variability in cellular resources and, more generally, to changes in intra-cellular context linked to differences in cell type. I will then show a feedback controller that uses bacterial two component signaling systems to create a quasi-integral controller that makes the input/output response of a genetic device robust to a variety of perturbations that affect gene expression. These solutions support rational and modular design of sophisticated genetic circuits and can serve for engineering biological circuits that are more robust and predictable across changing contexts.

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ZOOM ID: 997 8258 4700 (Biomedical Mathematics Online Colloquium), (pw: 1234)