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.

The Discrete Gaussian model is a type of integer-valued random height function. In the 2D setting, it exhibits a phase transition between a localised phase and a delocalised phase. This phenomenon is also called the Kosterlitz-Thouless phase transition, whose terminology originates from its dual counterpart, the planar XY model. Motivation for studying the Discrete Gaussian model is multifold. Due to its duality relations with a number of 2D mathematical physics models, such as the XY model or the Coulomb gas, studies on integer-value height functions are capable of proving a number of conjectures usually not accessible using classical methods. Other discrete height functions also have dualities with a number of different interesting models, so it will be of vast interest to develop a general framework that deals with discrete height functions. Also, discrete height functions are considered to be appropriate test cases for recently developed techniques from probability theory. In this talk, we discuss a particular method called the renormalisation group method, which is believed to serve as a general framework for studying random fields. We also discuss briefly how the renormalisation group method can be used to prove that the scaling limit of the 2D Discrete Gaussian model is a 2D Gaussian free field.

Prismatic cohomology, which is recently developed by Bhatt and Scholze, is a p-adic cohomology theory unifying etale, de Rham, and crystalline cohomology. In this series of two talks, we will discuss its central object of study called prismatic F-crystals, and some applications to studying p-adic Galois representations. The first part will be mainly devoted to explaining motivational background on the topic. Then we will discuss the relation between prismatic F-crystals and crystalline local systems on p-adic formal scheme, and talk about applications on purity of crystalline local system and on crystalline deformation ring. If time permits, we will also discuss recent work in progress on log prismatic F-crystals and semistable local systems. A part of the results is based on joint work with Du, Liu, Shimizu.

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.

Prismatic cohomology, which is recently developed by Bhatt and Scholze, is a p-adic cohomology theory unifying etale, de Rham, and crystalline cohomology. In this series of two talks, we will discuss its central object of study called prismatic F-crystals, and some applications to studying p-adic Galois representations. The first part will be mainly devoted to explaining motivational background on the topic. Then we will discuss the relation between prismatic F-crystals and crystalline local systems on p-adic formal scheme, and talk about applications on purity of crystalline local system and on crystalline deformation ring. If time permits, we will also discuss recent work in progress on log prismatic F-crystals and semistable local systems. A part of the results is based on joint work with Du, Liu, Shimizu.

Affine Deligne-Lusztig varieties show up naturally in the study of Shimura varieties, Rapoport-Zink spaces, and moduli spaces of local shtukas. Among various questions on its geometric properties, the question on the connected components turns out to be a fairly important problem. For example, Kisin, in his proof of the Langlands-Rapoport conjecture (in a weak sense) for abelian type Shimura variety with the hyperspecial level structure, crucially used the description of the set of connected components. Since then, many authors have answered this question in various restricted cases. I will first discuss what is the conjectural description of the connected components and related previous works. Then, I will explain my new result (joint work with Ian Gleason and Yujie Xu) which finishes the question in the mixed characteristic case and, if time permits, new ingredients.

Affine Deligne-Lusztig varieties are first defined by Rapoport as the (conjectural) p-part of the so-called Langlands-Rapoport conjecture. It can be understood as a p-adic generalization of the classical Deligne-Lusztig varieties. One of the most basic questions is 'when they are nonempty'. For a certain union, the nonemptiness criterion is completely known (by the so-called Mazur's inequality or B(G,μ)). However, the question about the "individual" ones is moderately open (with no general conjecture). I will discuss old and new nonemptiness results and suggest a new conjecture, for the individual ones, in the basic case. As an application, I will briefly mention a new explicit dimension formula in the rank 2 case (for which no conjectural formula was stated before).

The converse theorem for automorphic forms has a long history beginning with the work of Hecke (1936) and a work of Weil (1967): relating the automorphy relations satisfied by classical modular forms to analytic properties of their L-functions and the L-functions twisted by Dirichlet characters. The classical converse theorems were reformulated and generalised in the setting of automorphic representations for GL(2) by Jacquet and Langlands (1970). Since then, the converse theorem has been a cornerstone of the theory of automorphic representations. Venkatesh (2002), in his thesis, gave new proof of the classical converse theorem for modular forms of level 1 in the context of Langlands’ “Beyond Endoscopy”. In this talk, we extend Venkatesh’s proof of the converse theorem to forms of arbitrary levels and characters with the gamma factors of the Selberg class type. This is joint work with Andrew R. Booker and Michael Farmer.

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)

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심사위원장: 백형렬, 심사위원: 남경식, 최서영, Kasra Rafi(University of Toronto), Giulio Tiozzo(University of Toronto)

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심사위원장: 이창옥, 심사위원: 김동환, 임미경, 예종철(겸임교수), 한송희(삼성전자)

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심사위원장: 김재경, 심사위원: 김용정, 정연승, 황강욱, 이승규(경상대학교)

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심사위원장: 김용정, 심사위원: 권순식, 강문진, 김재경, 윤창욱(충남대학교)

TBA

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

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심사위원장: 안드레아스 홈슨, 심사위원: 김동수, 김재훈, 엄상일, 김민기(광주과학기술원)

In this talk we shall first review our recent results about the equivalence of non-linear Fokker-Planck equations and McKean Vlasov SDEs. Then we shall recall our results on existence of weak solutions to both such equations in the singular case, where the measure dependence of the coefficients are of Nemytskii-type. The main new results to be presented are about weak uniqueness of solutions to both nonlinear Fokker-Planck equations and the corresponding McKean-Vlasov SDEs in the case of (possibly) degenerate diffusion coefficients . As a consequence of this and one obtains that the laws on path space of the solutions to the McKean-Vlasov SDEs form a nonlinear Markov process in the sense of McKean.

This study is concerned with multivariate approximation by non-polynomial functions with internal shape parameters. The main topics of this presentation are two folds. First, interpolation by radial basis function (RBF) is considered. We especially discuss the convergence behavior of the RBF interpolants when the basis function is scaled to be increasingly flat. Moreover, we investigate the advantages of interpolation methods based on exponential polynomials. The second topic of this presentation is the approximation method based on sparse grids in $[0,1]^d \subset \RR^d$. The goal of sparse grid methods is to approximate high dimensional functions with good accuracy using as few grid points as possible. In this study, we present a new class of quasi-interpolation schemes for the approximation of multivariate functions on sparse grids. Each scheme in this class is based on shifts of kernels constructed from one-dimensional RBFs such as multiquadrics. The kernels are modified near the boundaries to prevent deterioration of the fidelity of the approximation. We show that our methods provide significantly better rates of approximation, compared to another quasi-interpolation scheme in the literature based on the Gaussian kernel using the multilevel technique. Some numerical results are presented to demonstrate the performance of the proposed schemes.

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Online: https://kaist.zoom.us/j/81807153144

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심사위원장: 엄상일, 심사위원: 안드레아스 홈슨, 김재훈, 권오정(한양대학교), Hong Liu(기초과학연구원)

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)

Metal artifact reduction has become a challenging issue for practical X-ray CT applications since metal artifacts severely cause contrast degradation and the misinterpretation of information about the property and structure of a scanned object. In this talk, we propose a methodology to reduce metal artifacts by extending the method proposed by Jeon and Lee (2018) to a three-dimensional industrial cone beam CT system. We develop a registration technique managing the three dimensional data in order to find accurate segmentation regions needed for the proposed algorithm. Through various simulations and experiments, we verify that the proposed algorithm reduces metal artifacts successfully.

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(Online participation) Zoom Link: https://kaist.zoom.us/j/87958862292

In a region closer to the boundary compared to Prandtl layer, an inviscid disturbance can be manifested by the interaction with viscous mode via the no-slip boundary condition due to resonance. In some unstable range of parameters, this leads to instability in the transition regime from laminar flow to turbulence. This instability phenomenon was observed by physicists long time ago, such as Heisenberg, Tollmien and C.C. Lin, etc. And it was justified rigorously in mathematics by Grenier-Guo-Nguyen using the incompressible Navier-Stokes equation. In this talk, we will present some results on this phenomenon in some other physical situations in which the governing system is either MHD or compressible Navier-Stokes equation. The talk is based on some recent joint work with Chengjie Liu and Zhu Zhang.

In this talk, I will give a brief introduction of what a linear algebraic group is and how it is structured. Then I will talk about the Galois descent related to linear algebraic groups. At last, I will explain what a torsor is and how it is related to other algebraic structures.