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 Finally, we also have a sneak peek on the 69-conjecture, part of an unpublished work on an inverse problem on the number of independent sets.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.

TBA

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

Fano varieties are algebraic varieties with positive curvature; they are basic building blocks of algebraic varieties. Great progress has been recently made by Xu et al. to construct moduli spaces of Fano varieties by using K-stability (which is related to the existence of Kähler-Einstein metrics). These moduli spaces are called K-moduli. In this talk I will explain how to easily deduce some geometric properties of K-moduli by using toric geometry and deformation theory. In particular, I will show how to construct a 1-dimensional component of K-moduli which parametrises certain K-polystable del Pezzo surfaces. * ZOOM information will not be provided. Please send an email to Jinhyung Park if you are interested in.

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심사위원장: 백형렬, 심사위원:최서영, 응우옌 응옥 쿠옹, 김상현(KIAS), 이계선(서울대학교)

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심사위원장: 백형렬, 심사위원:최서영, 응우옌 응옥 쿠옹, 김상현(KIAS), 이계선(서울대학교)

Deep generative models (DGM) have been an intersection between the probabilistic modeling and the machine learning communities. Particularly, DGM has impacted the field by introducing VAE, GAN, Flow, and recently Diffusion models with its capability to learn the data density of datasets. While there are many model variations in DGM, there are also common fundamental theories, assumptions and limitations to study from the theoretic perspectives. This seminar provides such general and fundamental challenges in DGMs, and later we particularly focus on the key developments in diffusion models and their mathematical properties in detail.

Machine learning (ML) has achieved unprecedented empirical success in diverse applications. It now has been applied to solve scientific problems, which has become an emerging field, Scientific Machine Learning (SciML). Many ML techniques, however, are very complex and sophisticated, commonly requiring many trial-and-error and tricks. These result in a lack of robustness and interpretability, which are critical factors for scientific applications. This talk centers around mathematical approaches for SciML, promoting trustworthiness. The first part is about how to embed physics into neural networks (NNs). I will present a general framework for designing NNs that obey the first and second laws of thermodynamics. The framework not only provides flexible ways of leveraging available physics information but also results in expressive NN architectures. The second part is about the training of NNs, one of the biggest challenges in ML. I will present an efficient training method for NNs - Active Neuron Least Squares (ANLS). ANLS is developed from the insight gained from the analysis of gradient descent training.

TBD

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

Let $E$ be a number field and $X$ a smooth geometrically connected variety defined over a characteristic $p$ finite field. Given an $n$-dimensional pure $E$-compatible system of semisimple $\lambda$-adic representations of the \'etale fundamental group of $X$ with connected algebraic monodromy groups $\bG_\lambda$, we construct a common $E$-form $\bG$ of all the groups $\bG_\lambda$ and in the absolutely irreducible case, a common $E$-form $\bG\hookrightarrow\GL_{n,E}$ of all the tautological representations $\bG_\lambda\hookrightarrow\GL_{n,E_\lambda}$. Analogous rationality results in characteristic $p$ assuming the existence of crystalline companions in $\mathrm{\textbf{F-Isoc}}^{\dagger}(X)\otimes E_{v}$ for all $v|p$ and in characteristic zero assuming ordinariness are also obtained. Applications include a construction of $\bG$-compatible system from some $\GL_n$-compatible system and some results predicted by the Mumford-Tate conjecture. (If you would like to join this seminar please contact Bo-Hae Im to get the zoom link.)

A linear $3$-graph is called a (3-)hypertree if there exists exactly one path between each pair of two distinct vertices.  A linear $3$-graph is called a Steiner triple system if each pair of two distinct vertices belong to a unique edge. A simple greedy algorithm shows that every $n$-vertex Steiner triple system $G$ contains all hypertrees $T$ of order at most $\frac{n+3}{2}$. On the other hand, it is not immediately clear whether one can always find larger hypertrees in $G$. In 2011, Goodall and de Mier proved that a Steiner triple system $G$ contains at least one spanning tree. However, one cannot expect the Steiner triple system to contain all possible spanning trees, as there are many Steiner triple systems that avoid numerous spanning trees as subgraphs. Hence it is natural to wonder how much one can improve the bound from the greedy algorithm. Indeed, Elliott and Rödl conjectured that an $n$-vertex Steiner triple system $G$ contains all hypertrees of order at most $(1-o(1))n$. We prove the conjecture by Elliott and Rödl. This is joint work with Jaehoon Kim, Joonkyung Lee, and Abhishek Methuku.

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서울대학교 산업공학과 학사 행정고시 37회 재경직 합격 KDI 국제정책대학원 석사 서울지방국세청 조사과장 국세청 기획재정담당관

TBA

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

The development and analysis of efficient methods and techniques for solving an inverse scattering problem are of great interest due to their potential in various applications, such as nondestructive testing, biomedical imaging, radar imaging, and structural imaging, among others. Sampling-type imaging methods allow us to non-iteratively retrieve the support of (possibly multiconnected) targets with low computational cost, assuming no a priori information about the targets. A sampling method tests a region of interest with its associated indicator function; the indicator function blows up if a test location is in support of inhomogeneities. Even though the sampling methods show promising results in ideal (multistatic, full-aperture, sufficiently many receivers) measurement configuration, one can frequently encounter limited measurement cases in practical applications. This presentation introduces the sampling-type imaging methods in two-dimensional limited-aperture, monostatic, and bistatic measurement cases. We identify the asymptotic structure of imaging methods to explore the applicability and intrinsic properties.

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

Geometric group theory concerns about how to see geometric properties in finitely generated groups. Defining Cayley graph of a finitely generated group with respect to finite generating set gives a perspective to describe geometric properties of finitely generated groups. Once we get a geometric perspective, we can classify finitely generated groups via quasi-isometry, since two Cayley graphs are quasi-isometric. In this talk, we will explain some basic notions appeared in geometric group theory (for example, quasi-isometry, hyperbolic groups, Švarc–Milnor lemma) and some theorems related to (relative) hyperbolicity of groups.

In this talk, we will introduce the absolute coregularity of Fano varieties. The coregularity measures the singularities of the anti-pluricanonical sections. Philosophically, most Fano varieties have coregularity 0. In the talk, we will explain some theorems that support this philosophy. We will show that a Fano variety of coregularity 0 admits a non-trivial section in |-2K_X|, independently of the dimension of X. This is joint work with Fernando Figueroa, Stefano Filipazzo, and Junyao Peng. * ZOOM information will not be provided. Please send an email to Jinhyung Park if you are interested in.

By a seminal result of Valiant, computing the permanent of (0, 1)-matrices is, in general, #P-hard. In 1913 Pólya asked for which (0, 1)-matrices A it is possible to change some signs such that the permanent of A equals the determinant of the resulting matrix. In 1975, Little showed these matrices to be exactly the biadjacency matrices of bipartite graphs excluding $K_{3,3}$ as a matching minor. This was turned into a polynomial time algorithm by McCuaig, Robertson, Seymour, and Thomas in 1999. However, the relation between the exclusion of some matching minor in a bipartite graph and the tractability of the permanent extends beyond K3,3. Recently it was shown that the exclusion of any planar bipartite graph as a matching minor yields a class of bipartite graphs on which the permanent of the corresponding (0, 1)-matrices can be computed efficiently. In this paper we unify the two results above into a single, more general result in the style of the celebrated structure theorem for single-crossing minor-free graphs. We identify a class of bipartite graphs strictly generalising planar bipartite graphs and $K_{3,3}$ which includes infinitely many non-Pfaffian graphs. The exclusion of any member of this class as a matching minor yields a structure that allows for the efficient evaluation of the permanent. Moreover, we show that the evaluation of the permanent remains #P-hard on bipartite graphs which exclude $K_{5,5}$ as a matching minor. This establishes a first computational lower bound for the problem of counting perfect matchings on matching minor closed classes. As another application of our structure theorem, we obtain a strict generalisation of the algorithm for the k-vertex disjoint directed paths problem on digraphs of bounded directed treewidth. This is joint work with Archontia Giannopoulou and Dimitrios Thilikos.

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서울대학교 산업공학과 학/석사 삼성전자 연구원 조지아텍 박사 삼성전자 종합기술원 전문연구원 현) 삼성전자 종합기술원 머신러닝 랩장

Stochastic finite-sum optimization problems are ubiquitous in many areas such as machine learning, and stochastic optimization algorithms to solve these finite-sum problems are actively studied in the literature. However, there is a major gap between practice and theory: practical algorithms shuffle and iterate through component indices, while most theoretical analyses of these algorithms assume uniformly sampling the indices. In this talk, we talk about recent research efforts to close this theory-practice gap. We will discuss recent developments in the theoretical convergence analysis of shuffling-based optimization methods. We will first consider minimization algorithms, mainly focusing on stochastic gradient descent (SGD) with shuffling; we will then briefly talk about some new progress on minimax optimization methods.

Neural networks (NNs) are currently changing the computational paradigm on how to combine data with mathematical laws in physics and engineering in a profound way, tackling challenging inverse and ill-posed problems not solvable with traditional methods. However, quantifying errors and uncertainties in NN-based inference is more complicated than in traditional methods. Although there are some recent works on uncertainty quantification (UQ) in NNs, there is no systematic investigation of suitable methods towards quantifying the total uncertainty effectively and efficiently even for function approximation, and there is even less work on solving partial differential equations and learning operator mappings between infinite-dimensional function spaces using NNs. In this talk, we will present a comprehensive framework that includes uncertainty modeling, new and existing solution methods, as well as evaluation metrics and post-hoc improvement approaches. To demonstrate the applicability and reliability of our framework, we will also present an extensive comparative study in which various methods are tested on prototype problems, including problems with mixed input-output data, and stochastic problems in high dimensions.

In this talk, we discuss the Cauchy problem for the Vlasov-Riesz system, which is a Vlasov equation featuring interaction potentials generalizing various previously studied cases, including the Coulomb and Manev potentials. For the first time, we extend the local theory of classical solutions to interaction potentials which are more singular than that for the Manev. Then, we obtain finite-time singularity formation for solutions with various attractive interaction potentials, extending the well-known singularity formation result for attractive Vlasov-Poisson. Our local well-posedness and singularity formation results extend to cases with linear diffusion and damping in velocity.

In recent years, community detection has been an active research area in various fields including machine learning and statistics. While a plethora of works has been published over the past few years, most of the existing methods depend on a predetermined number of communities. Given the situation, determining the proper number of communities is directly related to the performance of these methods. Currently, there does not exist a golden rule for choosing the ideal number, and people usually rely on their background knowledge of the domain to make their choices. To address this issue, we propose a community detection method that is equipped with data-adaptive methods of finding the number of the underlying communities. Central to our method is fused l-1 penalty applied on an induced graph from the given data. The proposed method shows promising results.

A theorem of Khare-Wintenberger and Kisin (once Serre’s modularity conjecture) says that every two-dimensional odd absolutely irreducible representation \bar\rho of the Galois group of the rationals over a finite field comes from a modular form f, that is, \bar\rho ~ \bar\rho_f. The conjecture even provides a recipe for the weight, level and character of f, but does not give any information about the slope of f. In this talk, based on joint work with Kumar, we provide conditions on f - the main one being that the weight of f is close to 0 - which guarantee that the slope of a modular form g giving rise to the twist of \bar\rho_f by the cyclotomic character has slope one more than the slope of f. This provides a global explanation of some local patterns mentioned in our first talk. The proof uses the theta operator and Coleman-Hida families of overconvergent forms. (This is the second of the two KAIX Invited Lectures.)

Wearable analytics hold far more potential than sleep tracking or step counting. In recent years, a number of applications have emerged which leverage the massive quantities of data being amassed by wearables around the world, such as real-time mood detection, advanced COVID screening, and heart rate variability analysis. Yet packaging insights from research for success in the consumer market means prioritizing design and understandability, while also seamlessly managing the sometimes-unreliable stream of data from the device. In this presentation, I will discuss my own experiences building apps which interface with wearable data and process the data using mathematical modeling, as well as recent work extending to other wearable streams and environmental controls.

Cells make individual fate decisions through linear and nonlinear regulation of gene network, generating diverse dynamics from a single reaction pathway. In this colloquium, I will present two topics of our recent work on signaling dynamics at cellular and patient levels. The first example is about the initial value of the model, as a mechanism to generate different dynamics from a single pathway in cancer and the use of the dynamics for stratification of the patients [1-3]. Models of ErbB receptor signaling have been widely used in prediction of drug sensitivity for many types of cancers. We trained the ErbB model with the data obtained from cancer cell lines and predicted the common parameters of the model. By simulation of the ErbB model with those parameters and individual patient transcriptome data as initial values, we were able to classify the prognosis of breast cancer patients and drug sensitivity based on their in silico signaling dynamics. This result raises the question whether gene expression levels, rather than genetic mutations, might be better suited to classify the disease. Another example is about the regulation of transcription factors, the recipients of signal dynamics, for target gene expression [4-6]. By focusing on the NFkB transcription factor, we found that the opening and closing of chromatin at the DNA regions of the putative transcription factor binding sites and the cooperativity in their interaction significantly influenced the cell-to cell heterogeneity in gene expression levels. This study indicates that the noise in gene expression is rather strongly regulated by the DNA side, even though the signals are similarly regulated in a cell population. Overall these mechanisms are important in our understanding the cell as a system for encoding and decoding signals for fate decisions and its application to human diseases.

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

The zig-zag conjecture predicts that the reductions of two-dimensional irreducible p-adic crystalline representations of half-integral slope and exceptional weights - weights which are two more than twice the slope modulo (p-1) - have reductions which are given by an alternating sequence of irreducible and reducible representations. Some partial progress was made towards this conjecture over the years by Buzzard-Gee (slope 1/2), Bhattacharya-G-Rozensztajn (slope 1) and G-Rai (slope 3/2). In this talk I shall use work of Breuil-Mézard and Guerberoff-Park in the semi-stable case and a limiting argument connecting crystalline and semi-stable representations due to Chitrao-G-Yasuda to show that zig-zag holds for half-integal slopes bounded by (p-1)/2, at least on the inertia subgroup, if the weight is sufficiently close to a weight bounded by p+1. (This is the first of the two KAIX Invited Lectures.)

Shift workers experience profound circadian disruption due to the nature of their work, which often has them working at times when their internal clock is sending a strong signal for sleep. Mathematical models can be used to generate recommendations for shift workers that shift their body’s clock to better align with their work schedules, to help them sleep, feel, and perform better. In this talk, I will discuss our recent mobile app, Shift, which pulls wearable data from user’s devices and generates personalized recommendations to help them manage shift work schedules. I will also discuss how this product was designed, how it can interface with Internet of Things devices, and how its insights can be useful for other groups beyond shift workers.

Shift workers experience profound circadian disruption due to the nature of their work, which often has them working at times when their internal clock is sending a strong signal for sleep. Mathematical models can be used to generate recommendations for shift workers that shift their body’s clock to better align with their work schedules, to help them sleep, feel, and perform better. In this talk, I will discuss our recent mobile app, Shift, which pulls wearable data from user’s devices and generates personalized recommendations to help them manage shift work schedules. I will also discuss how this product was designed, how it can interface with Internet of Things devices, and how its insights can be useful for other groups beyond shift workers.

Let $\mathcal{F}$ be a family of graphs, and let $p$ and $r$ be nonnegative integers. The $(p,r,\mathcal{F})$-Covering problem asks whether for a graph $G$ and an integer $k$, there exists a set $D$ of at most $k$ vertices in $G$ such that $G^p\setminus N_G^r[D]$ has no induced subgraph isomorphic to a graph in $\mathcal{F}$, where $G^p$ is the $p$-th power of $G$ and $N^r_G[D]$ is the set of all vertices in $G$ at distance at most $r$ from $D$ in $G$. The $(p,r,\mathcal{F})$-Packing problem asks whether for a graph $G$ and an integer $k$, $G^p$ has $k$ induced subgraphs $H_1,\ldots,H_k$ such that each $H_i$ is isomorphic to a graph in $\mathcal{F}$, and for distinct $i,j\in \{1, \ldots, k\}$, the distance between $V(H_i)$ and $V(H_j)$ in $G$ is larger than $r$. The $(p,r,\mathcal{F})$-Covering problem generalizes Distance-$r$ Dominating Set and Distance-$r$ Vertex Cover, and the $(p,r,\mathcal{F})$-Packing problem generalizes Distance-$r$ Independent Set and Distance-$r$ Matching. By taking $(p',r',\mathcal{F}')=(pt, rt, \mathcal{F})$, we may formulate the $(p,r,\mathcal{F})$-Covering and $(p, r, \mathcal{F})$-Packing problems on the $t$-th power of a graph. Moreover, $(1,0,\mathcal{F})$-Covering is the $\mathcal{F}$-Free Vertex Deletion problem, and $(1,0,\mathcal{F})$-Packing is the Induced-$\mathcal{F}$-Packing problem. We show that for every fixed nonnegative integers $p,r$ and every fixed nonempty finite family $\mathcal{F}$ of connected graphs, the $(p,r,\mathcal{F})$-Covering problem with $p\leq2r+1$ and the $(p,r,\mathcal{F})$-Packing problem with $p\leq2\lfloor r/2\rfloor+1$ admit almost linear kernels on every nowhere dense class of graphs, and admit linear kernels on every class of graphs with bounded expansion, parameterized by the solution size $k$. We obtain the same kernels for their annotated variants. As corollaries, we prove that Distance-$r$ Vertex Cover, Distance-$r$ Matching, $\mathcal{F}$-Free Vertex Deletion, and Induced-$\mathcal{F}$-Packing for any fixed finite family $\mathcal{F}$ of connected graphs admit almost linear kernels on every nowhere dense class of graphs and linear kernels on every class of graphs with bounded expansion. Our results extend the results for Distance-$r$ Dominating Set by Drange et al. (STACS 2016) and Eickmeyer et al. (ICALP 2017), and the result for Distance-$r$ Independent Set by Pilipczuk and Siebertz (EJC 2021). This is joint work with Jinha Kim and O-joung Kwon.

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온라인(ZOOM) https://kaist.zoom.us/j/7011567816

In extremal graph theory, one big question is finding a condition of the number of edges that guarantees the existence of a particular substructure in a graph. In the first half of this talk, I'll talk about the history of such problems, especially focusing on clique subdivisions. In the last half of the talk, I'll introduce my recent result with Jaehoon Kim, Younjin Kim, and Hong Liu, which states that if a graph G has no dense small subgraph, then G has a clique subdivision of size almost linear in its average degree and discuss some applications and further open questions.

The classification of terminal Fano 3-folds has been tackled from different directions: for instance, using the Minimal Model Program, via explicit Birational Geometry, and via Graded Rings methods. In this talk I would like to introduce the Graded Ring Database - an upper bound to the numerics of Fano 3-folds - and discuss the role it plays in the classification and construction of codimension 4 Fano 3-folds having Fano index 2.

Castelnuovo-Mumford regularity, simply regularity, is one of the most interesting invariants in projective algebraic geometry, and the regularity conjecture due to Eisenbud and Goto says that the regularity can be controlled by the degree for any projective variety. But counterexamples to the conjecture have been constructed by some methods. In this talk we review the counterexample constructions including the Rees-like algebra method by McCullough and Peeva and the unprojection method.