Given a set $E$ and a point $y$ in a vector space over a finite field, the radial projection $\pi_y(E)$ of $E$ from $y$ is the set of lines that through $y$ and points of $E$. Clearly, $|pi_y(E)|$ is at most the minimum of the number of lines through $y$ and $|E|$. I will discuss several results on the general question: For how many points $y$ can $|\pi_y(E)|$ be much smaller than this maximum? This is motivated by an analogous question in fractal geometry. The Hausdorff dimension of a radial projection of a set $E$ in $n$ dimensional real space will typically be the minimum of $n-1$ and the Hausdorff dimension of $E$. Several recent papers by authors including Matilla, Orponen, Liu, Shmerikin, and Wang consider the question: How large can the set of points with small radial projections be? This body of work has several important applications, including recent progress on the Falconer distance conjecture. This is joint with Thang Pham and Vu Thi Huong Thu.

Over the recent years, various methods based on deep neural networks have been developed and utilized in a wide range of scientific fields. Deep neural networks are highly suitable for analyzing time series or spatial data with complicated dependence structures, making them particularly useful for environmental sciences and biosciences where such type of simulation model output and observations are prevalent. In this talk, I will introduce my recent efforts in utilizing various deep learning methods for statistical analysis of mathematical simulations and observational data in those areas, including surrogate modeling, parameter estimation, and long-term trend reconstruction. Various scientific application examples will also be discussed, including ocean diffusivity estimation, WRF-hydro calibration, AMOC reconstruction, and SIR calibration.

In addition to diffusive signals, cells in tissue also communicate via long, thin cellular protrusions, such as airinemes in zebrafish. Before establishing communication, cellular protrusions must find their target cell. In this talk, we demonstrate that the shapes of airinemes in zebrafish are consistent with a persistent random walk model. The probability of contacting the target cell is maximized for a balance between ballistic search (straight) and diffusive search (highly curved, random). We find that the curvature of airinemes in zebrafish, extracted from live cell microscopy, is approximately the same value as the optimum in the simple persistent random walk model. We also explore the ability of the target cell to infer direction of the airineme’s source, finding that there is a theoretical trade-off between search optimality and directional information. This provides a framework to characterize the shape, and performance objectives, of non-canonical cellular protrusions in general.

The connection between deep neural networks and ordinary differential equations (ODEs) is an active field of research in machine learning. In this talk, we view the hidden states of a neural network as a continuous object governed by a dynamical system. The underlying vector field is written using a dictionary representation motivated by the equation discovery method. Within this framework, we develop models for two particular machine learning tasks: time-series classification and dimension reduction. We train the parameters in the models by minimizing a loss, which is defined using the solution to the governing ODE. To attain a regular vector field, we introduce a regularization term measuring the mean total kinetic energy of the flow, which is motivated by optimal transportation theory. We solve the optimization problem using a gradient-based method where the gradients are computed via the adjoint method from optimal control theory. Through various experiments on synthetic and real-world datasets, we demonstrate the performance of the proposed models. We also interpret the learned models by visualizing the phase plots of the underlying vector field and solution trajectories.

Twin-width is a recently introduced graph parameter based on vertex contraction sequences. On classes of bounded twin-width, problems expressible in FO logic can be solved in FPT time when provided with a sequence witnessing the bound. Classes of bounded twin-width are very diverse, notably including bounded rank-width, $\Omega ( \log (n) )$-subdivisions of graphs of size $n$, and proper minor closed classes. In this talk, we look at developing a structural understanding of twin-width in terms of induced subdivisions. Structural characterizations of graph parameters have mostly looked at graph minors, for instance, bounded tree-width graphs are exactly those forbidding a large wall minor. An analogue in terms of induced subgraphs could be that, for sparse graphs, large treewidth implies the existence of an induced subdivision of a large wall. However, Sintiari and Trotignon have ruled out such a characterization by showing the existence of graphs with arbitrarily large girth avoiding any induced subdivision of a theta ($K_{2,3}$). Abrishami, Chudnovsky, Hajebi and Spirkl have recently shown that such (theta, triangle)-free classes have nevertheless logarithmic treewidth. After an introduction to twin-width and its ties to vertex orderings, we show that theta-free graphs of girth at least 5 have bounded twin-width. Joint work with Édouard Bonnet, Eun Jung Kim, Stéphan Thomassé and Rémi Watrigant.

Circadian rhythm is a robust internal 24 hours timekeeping mechanism maintained by the master circadian pacemaker Suprachiasmatic Nuclei (SCN). Numerous mathematical models have been proposed to capture SCN’s timekeeping mechanism and predict the circadian phase. There has been an increased demand for applying these models to the various unexplored data sets. One potential application is on data from commercially available wearable devices, which provide the noninvasive measurements of physiological proxies, such as activity and heart rate. Using these physiological proxies, we can estimate the circadian phase of the central and peripheral circadian pacemakers. Here, we propose a new framework for estimating the circadian phase using wearable data and the Level Set Kalman Filter on the nonlinear state-space model of the human circadian pacemaker. Analysis of over 200,000 days of wearable data from over 3,000 subjects using our framework successfully identified misalignment in central and peripheral pacemakers with a significantly smaller uncertainty than previous methods.

논문은 저자와 독자 사이의 학문적 소통을 위한 논리적인 글이다. 이번 강연을 통해 논문을 작성하기 위한 기본 원리를 배울 수 있다. 특히, 연구가 거의 마무리되는 시점에 (After completing your research), 연구 결과를 그림과 표로 잘 정리한 다음에 (Based on well-organized figures and tables), 본격적으로 논문 작성을 시작하는 (Compose your manuscript from a title to a conclusion) ‘ABC 논문 작성법’을 소개한다. 논문 작성의 준비 과정으로 (A와 B의 과정), 연구 노트 작성 방법, 저널 클럽 운영 방법, 한 페이지 활용 방법을 설명한다. 논문 작성 준비가 완료되면, 제목부터 결론까지 순서대 로 논문 원고를 작성할 수 있다 (C의 과정). 이렇게 하면, 단기간에 집중하여 효율적으 로 논문을 작성할 수 있다. *참고: 원병묵 교수의 과학 논문 쓰는 법

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2022년 6월 3일 오후 4시 – 5시, 성균관대 신소재공학부의 원병묵 교수님의 논문 글쓰기 워크샵 강의가 있습니다.

이 ‘영어 논문 쓰기’ 워크숍은 현재의 영어 수준이나 과거의 영어 학습 경험과 상관 없이 누구나 영어 논문 쓰기를 시작할 수 있는 방법에 관한 것입니다. 우선 영어에 대한 막연한 두려움, 과거 경험으로 인한 자신감 부족, 게다가 ‘쓰기’라는 쉽지 않은 인지 활동, 심지어 논문이라는 큰 벽, 혹은 섣부른 자신감 등 ‘영어 논문 쓰기’를 방해하는 요인을 생각해 봅니다. 이런 요인을 자세하게 들여다보면 각각의 걸림돌을 넘어갈 방법도 명쾌하게 발견할 수 있습니다. 이 세미나에서 다룰 구체적 내용은 다음과 같습니다. - 한국인들이 ‘영어’와 ‘영어 쓰기’를 어렵게 느끼는 원인 이해하고 극복하기 - ‘읽기’와 ‘쓰기’의 서로 다른 두 가지 인지활동의 관련성 이해하고 적용하기 - 논문에 적합한 단어, 시제, 구두점 선택하고 사용하기 - 영문초록, 제목, 소제목 스타일 이해하고 작성하기 - 영어논문 쓰기를 돕는 디지털 도구 선택하고 활용하기 이 워크숍에서 안내될 몇 가지 방법은 영어 논문 쓰기가 더 이상 두려운 것이 아닌 연구자로서의 목표를 이루는 디딤돌이 되도록 할 것입니다.

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2022년 6월 3일 오후 5시 – 6시, 부산교육정책연구소 박영민 연구원님의 논문 글쓰기 워크샵 강의가 있습니다.

Quantitative characterization of biomolecular networks is important for the analysis and design of network functionality. Reliable models of such networks need to account for intrinsic and extrinsic noise present in the cellular environment. Stochastic kinetic models provide a principled framework for developing quantitatively predictive tools in this scenario. Calibration of such models requires an experimental setup capable of monitoring a large number of individual cells over time, automatic extraction of fluorescence levels for each cell and a scalable inference approach. In the first part of the talk we will cover our microfluidic setup and a deep-learning based approach to cell segmentation and data extraction. The second part will introduce moment-based variational inference as a scalable framework for approximate inference of kinetic models based on single cell data.

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This talk will be presented online. ZOOM ID: 997 8258 4700 (Biomedical Mathematics Online Colloquium), (pw: 1234)

SATNet is a differentiable constraint solver with a custom backpropagation algorithm, which can be used as a layer in a deep-learning system. It is a promising proposal for bridging deep learning and logical reasoning. In fact, SATNet has been successfully applied to learn, among others, the rules of a complex logical puzzle, such as Sudoku, just from input and output pairs where inputs are given as images. In this paper, we show how to improve the learning of SATNet by exploiting symmetries in the target rules of a given but unknown logical puzzle or more generally a logical formula. We present SymSATNet, a variant of SATNet that translates the given symmetries of the target rules to a condition on the parameters of SATNet and requires that the parameters should have a particular parametric form that guarantees the condition. The requirement dramatically reduces the number of parameters to learn for the rules with enough symmetries, and makes the parameter learning of SymSATNet much easier than that of SATNet. We also describe a technique for automatically discovering symmetries of the target rules from examples. Our experiments with Sudoku and Rubik’s cube show the substantial improvement of SymSATNet over the baseline SATNet. This is joint work with Sangho Lim and Eungyeol Oh.

We shall give an explicit estimate of the lower bound of the Bergman kernel associated to a positive line bundle. In the compact Riemann surface case, our result can be seen as an explicit version of Tian’s partial C0-estimate.

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https://zoom.us/j/98194255942?pwd=cWpLM0c1T2U2OG9MR0VJNHpOTFBrdz09 아이디: 981 9425 5942 비밀번호: 373452

I will introduce mathematical and computational methods for spatio-temporal modelling in molecular and cell biology, including all-atom and coarse-grained molecular dynamics (MD), Brownian dynamics (BD), stochastic reaction-diffusion models and macroscopic mean-field equations. Microscopic (BD, MD) models are based on the simulation of trajectories of individual molecules and their localized interactions (for example, reactions). Mesoscopic (lattice-based) stochastic reaction-diffusion approaches divide the computational domain into a finite number of compartments and simulate the time evolution of the numbers of molecules in each compartment, while macroscopic models are often written in terms of mean-field reaction-diffusion partial differential equations for spatially varying concentrations.

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This talk will be presented online. ZOOM ID: 997 8258 4700 (Biomedical Mathematics Online Colloquium), (pw: 1234)

I will discuss the development, analysis and applications of multi-resolution methods for spatio-temporal modelling of intracellular processes, which use (detailed) Brownian dynamics or molecular dynamics simulations in localized regions of particular interest (in which accuracy and microscopic details are important) and a (less-detailed) coarser model in other regions in which accuracy may be traded for simulation efficiency. I will discuss the error analysis and convergence properties of the developed multi-resolution methods, their software implementation and applications of these multiscale methodologies to modelling of intracellular calcium dynamics, actin dynamics and DNA dynamics. I will also discuss the development of multiscale methods which couple molecular dynamics and coarser stochastic models in the same dynamic simulation.

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This talk will be presented online. ZOOM ID: 997 8258 4700 (Biomedical Mathematics Online Colloquium), (pw: 1234)

Hadwiger’s transversal theorem gives necessary and sufficient conditions for the existence of a line transversal to a family of pairwise disjoint convex sets in the plane. These conditions were subsequently generalized to hyperplane transversals in R d by Goodman, Pollack, and Wenger. Here we establish a colorful extension of their theorem, which proves a conjecture of Arocha, Bracho, and Montejano. The proof uses topological methods, in particular the Borsuk-Ulam theorem. The same methods also allow us to generalize some colorful transversal theorems of Montejano and Karasev.

The idea of balance between excitation and inhibition is central in the theory of biological neural networks. I will give a brief introduction to the concept of such balance, and an overview of the mathematical ideas that can be used to study it.

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This talk will be presented online. ZOOM ID: 997 8258 4700 (Biomedical Mathematics Online Colloquium), (pw: 1234)

I will first describe how to extend the theory of balanced networks to account for synaptic plasticity. This theory can be used to show when a plastic network will maintain balance, and when it will be driven into an unbalanced state. I will next discuss how this approach provides evidence for a novel form of rapid compensatory inhibitory plasticity. Experimental evidence for such plasticity comes from optogenetic activation of excitatory neurons in primate visual cortex (area V1) which induces a population-wide dynamic reduction in the strength of neuronal interactions over the timescale of minutes during the awake state, but not during rest. I will shift gears in the final part of the talk, and discuss how community detection algorithms can help uncover the large scale organization of neuronal networks from connectome data.

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This talk will be presented online. ZOOM ID: 997 8258 4700 (Biomedical Mathematics Online Colloquium), (pw: 1234)

A subset V of a domain Ω has the extension property if for every holomorphic function p on V there is a bounded holomorphic function φ on Ω that agrees with p on V and whose sup-norm on Ω equals the sup-norm of p on V. Within the talk, we shall study mutual relations between extension property and interpolations problems.

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Zoom link https://zoom.us/j/95969402165?pwd=MUhVTUQ5azZSOW1EdkRMMFRVM1R4QT09 ID: 959 6940 2165 PW: 962685

The upper tail problem for subgraph counts in the Erdos-Renyi graph, introduced by Janson-Ruciński, has attracted a lot of attention. There is a class of Gibbs measures associated with subgraph counts, called exponential random graph model (ERGM). Despite its importance, lots of fundamental questions have remained unanswered owing to the lack of exact solvability. In this talk, I will talk about a brief overview on the upper tail problem and the concentration of measure results for the ERGM. Joint work with Shirshendu Ganguly and Ella Hiesmayr.

We investigate in depth the behaviour of Monge-Ampère volumes of quasi-psh functions on a given compact hermitian manifold. We prove that the property for these Monge-Ampère volumes to stay bounded away from zero or infinity is a bimeromorphic invariant. We show in particular that a conjecture of Demailly-Paun holds true if and only if such Monge-Ampère volumes stay bounded away from infinity. This is a joint work with Vincent Guedj.

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Zoom link https://zoom.us/j/95121528401?pwd=SXd2bnVFNW1veEJEZUJuNUhVaUdJZz09 ID: 951 2152 8401 PW: 641513

In this talk, we introduce homomorphisms between binary matroids that generalize graph homomorphisms. For a binary matroid $N$, we prove a complexity dichotomy for the problem $\rm{Hom}_\mathbb{M}(N)$ of deciding if a binary matroid $M$ admits a homomorphism to $N$. The problem is polynomial-time solvable if $N$ has a loop or has no circuits of odd length, and is otherwise $\rm{NP}$-complete. We also get dichotomies for the list, extension, and retraction versions of the problem. This is joint work with Hyobin Kim and Mark Siggers at Kyungpook National University.