We will discuss hormone circuits and their dynamics using new models that take into account timescales of weeks due to growth of the hormone glands. This explains some mysteries in diabetes and autoimmune disease.

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

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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https://us06web.zoom.us/j/83876347559?pwd=SVZoUzZkUkMyZno0Vk1pS0pkTEdZQT09 ID: 838 7634 7559 PW: 8XahHQ

Suppose that $E$ is a subset of $\mathbb{F}_q^n$, so that each point is contained in $E$ with probability $\theta$, independently of all other points. Then, what is the probability that there is an $m$-dimensional affine subspace that contains at least $\ell$ points of $E$? What is the probability that $E$ intersects all $m$-dimensional affine subspaces? We give Erdős-Renyi threshold functions for these properties, in some cases sharp thresholds. Our results improve previous work of Chen and Greenhill. This is joint work with Jeong Han Kim, Thang Pham, and Semin Yoo.

Morihiko Saito's theory of mixed Hodge modules is a far generalisation of classical Hodge theory, which is based on the theory of perverse sheaves, D-modules, variations of Hodge structures. One can think of mixed Hodge modules as a certain class of D-modules with Hodge structures. Naturally they are accompanied by perverse sheaves via the Riemann–Hilbert correspondence. This guide consists of about 8 talks, which may cover: review of classical Hodge theory, D-modules and filtered D-modules, nearby and vanishing cycles, etc. The main goal is to understand the notion of mixed Hodge modules and to explain two important theorems: the structure theorem and the direct image theorem. If time permits, we discuss recent applications of the theory in algebraic geometry.

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Zoom 회의 ID: 352 730 6970, 암호: 7178 대기실에서 개별 승인하오니, 실명으로 접속하시기 바랍니다.

Random simple and multigraph models based on exchangeable random measures, sometimes named graphexprocesses or generalisedgraphonmodels, have recently been proposed as a flexible class of sparse random graph models. This class of models can be seen as a generalisationof the popular graphonmodels. I will present this class of models, discuss some of their asymptotic properties, in particular the asymptotic behaviourof the degree distribution and of the clustering coefficients. I will also present some particular parametric models within this class and their use for discovering latent communities in sparse real-world networks.

In this talk, we present a short history of L^p-theories for partial differential equations. In particular, we introduce recent developments handling fractional derivatives and degenerate estimates in weighted spaces.

Random simple and multigraph models based on exchangeable random measures, sometimes named graphex processes or generalised graphon models, have recently been proposed as a flexible class of sparse random graph models. This class of models can be seen as a generalisation of the popular graphon models. I will present this class of models, discuss some of their asymptotic properties, in particular the asymptotic behaviour of the degree distribution and of the clustering coefficients. I will also present some particular parametric models within this class and their use for discovering latent communities in sparse real-world networks.

I will give an introduction to topological data analysis (TDA), in which one uses ideas from algebraic topology to study the “shape” of data. I will focus on persistent homology (PH), which is the most common approach in TDA.

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

From the venation patterns of leaves to spider webs, roads in cities, social networks, and the spread of COVID-19 infections and vaccinations, the structure of many systems is influenced significantly by space. In this talk, I will discuss the application of topological data analysis (specifically, persistent homology) to spatial systems. I will present a few examples, such as voting in presidential elections, city street networks, spatiotemporal dynamics of COVID-19 infections and vaccinations, and webs that were spun by spiders under the influence of various drugs.

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

A ruled surface is a fibration over a smooth curve with fibers being isomorphic to the projective line. If a ruled surface is assumed to not have any section with negative self-intersection, then it is known that there is no curve with negative self-intersection on the ruled surface. Moreover, if the ruled surface is “sufficiently” general in its moduli, then the surface does not admit a curve with zero self-intersection. So, it is natural to ask the questions of which ruled surface admits a curve with zero self-intersection, and how many such ruled surfaces exist in the moduli. In this talk, I will introduce some answers to the questions.

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[Zoom 링크] Zoom 회의 참가 https://kaist.zoom.us/j/2655728482?pwd=OXpJeFdDcWliSG51WUp0N1Nad2JHdz09 회의 ID: 265 572 8482 암호: 2AHRKr [Gather Town 링크] https://gather.town/app/ffr2PVibAWRIyXWO/kaistmath 두 발표 세션이 끝나면 Gather Town으로 옮겨와 대학원생들간에 자유롭게 이야기를 나누는 시간을 가질 계획입니다. Gather Town은 크롬이 깔려있는 기기(노트북, 패드, 스마트폰 등)에서 모두 접속 가능합니다. 별도의 회원가입 없이도 개별 캐릭터 설정 후 접속하면 주변의 다른 캐릭터들과 대화할 수 있는 플랫폼입니다.

Finding a given graph in a large host graph is a very essential problem in graph theory. One main variant of this is coloring edges of a host graph with many colors and trying to find a 'rainbow' subgraph, whose edges have distinct colors. I will explain some history, and introduce my recent result which searches for a rainbow color-critical graph.

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[Zoom 링크] Zoom 회의 참가 https://kaist.zoom.us/j/2655728482?pwd=OXpJeFdDcWliSG51WUp0N1Nad2JHdz09 회의 ID: 265 572 8482 암호: 2AHRKr [Gather Town 링크] https://gather.town/app/ffr2PVibAWRIyXWO/kaistmath 두 발표 세션이 끝나면 Gather Town으로 옮겨와 대학원생들간에 자유롭게 이야기를 나누는 시간을 가질 계획입니다. Gather Town은 크롬이 깔려있는 기기(노트북, 패드, 스마트폰 등)에서 모두 접속 가능합니다. 별도의 회원가입 없이도 개별 캐릭터 설정 후 접속하면 주변의 다른 캐릭터들과 대화할 수 있는 플랫폼입니다.

When a biological system is modeled using a mathematical procedure, the following step is normally to estimate the system parameters. Despite the numerous computational and statistical techniques, estimating parameters for complex systems can be a difficult task. As a result, one can think of revealing parameter-independent dynamical properties of a system. More precisely, rather than estimating parameters, one can focus on the underlying structure of a biochemical system to derive the qualitative behavior of the associated mathematical process. In this talk, we will discuss introduce reaction network theory. A reaction network is a graphical configuration of a biochemical system. One of the most important problems in this field is to relate dynamical properties and the underlying reaction network structure. When abundances of biochemical species (variables) in the system are small, then the randomness inherent in the molecular interactions is crucial to the system dynamics, and the abundances are modeled stochastically as a jump by jump fashion continuous-time Markov chain. The goal of this talk is to 1. walk you through the basic modeling aspect of the stochastically modeled reaction networks, and 2. to show how to derive stability (ergodicity) of the associated Markov process solely based on the underlying network structure.

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ZOOM Meeting ID: 868 7549 9085 Direct link: https://kaist.zoom.us/j/86875499085

In this talk we will discuss a Hirzebruch-Riemann-Roch (HRR) type theorem for matrix factorization categories of Deligne-Mumford stacks. We will first discuss a proof of a Hochschild-Kostant-Rosenberg type isomorphism and show how it can be used to define a Chern character formula which allows us to prove the HRR type theorem. This talk is based on a joint work with Dongwook Choa and Bumsig Kim.

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Zoom details: ID: 352 730 6970 Password: 1098. Please come with your real names.

The Ramsey number $R(F,H)$ is the minimum number $N$ such that any $N$-vertex graph either contains a copy of $F$ or its complement contains $H$. Burr in 1981 proved a pleasingly general result that for any graph $H$, provided $n$ is sufficiently large, a natural lower bound construction gives the correct Ramsey number involving cycles: $R(C_n,H)=(n-1)(\chi(H)-1)+\sigma(H)$, where $\sigma(H)$ is the minimum possible size of a colour class in a $\chi(H)$-colouring of $H$. Allen, Brightwell and Skokan conjectured that the same should be true already when $n\geq |H|\chi(H)$. We improve this 40-year-old result of Burr by giving quantitative bounds of the form $n\geq C|H|\log^4\chi(H)$, which is optimal up to the logarithmic factor. In particular, this proves a strengthening of the Allen-Brightwell-Skokan conjecture for all graphs $H$ with large chromatic number. This is joint work with John Haslegrave, Joseph Hyde and Hong Liu

Morihiko Saito's theory of mixed Hodge modules is a far generalisation of classical Hodge theory, which is based on the theory of perverse sheaves, D-modules, variations of Hodge structures. One can think of mixed Hodge modules as a certain class of D-modules with Hodge structures. Naturally they are accompanied by perverse sheaves via the Riemann–Hilbert correspondence. This guide consists of about 8 talks, which may cover: review of classical Hodge theory, D-modules and filtered D-modules, nearby and vanishing cycles, etc. The main goal is to understand the notion of mixed Hodge modules and to explain two important theorems: the structure theorem and the direct image theorem. If time permits, we discuss recent applications of the theory in algebraic geometry.

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Zoom 회의 ID: 352 730 6970 암호: 4114 대기실에서 개별 승인하니, 실명으로 접속하시기 바랍니다.

We prove global Holder gradient estimates for bounded positive weak solutions of fast diffusion equations in smooth bounded domains with homogeneous Dirichlet boundary condition, which then leads us to establish their optimal global regularity. It solves a problem raised by Berryman and Holland in 1980. This is joint work with Jingang Xiong.

This paper defines fair principal component analysis (PCA) as minimizing the maximum mean discrepancy (MMD) between dimensionality-reduced conditional distributions of different protected classes. The incorporation of MMD naturally leads to an exact and tractable mathematical formulation of fairness with good statistical properties. We formulate the problem of fair PCA subject to MMD constraints as a non-convex optimization over the Stiefel manifold and solve it using the Riemannian Exact Penalty Method with Smoothing (REPMS; Liu and Boumal, 2019). Importantly, we provide local optimality guarantees and explicitly show the theoretical effect of each hyperparameter in practical settings, extending previous results. Experimental comparisons based on synthetic and UCI datasets show that our approach outperforms prior work in explained variance, fairness, and runtime. This paper is accepted to the 36th AAAI Conference on Artificial Intelligence (AAAI 2022).

This paper defines fair principal component analysis (PCA) as minimizing the maximum mean discrepancy (MMD) between dimensionality-reduced conditional distributions of different protected classes. The incorporation of MMD naturally leads to an exact and tractable mathematical formulation of fairness with good statistical properties. We formulate the problem of fair PCA subject to MMD constraints as a non-convex optimization over the Stiefel manifold and solve it using the Riemannian Exact Penalty Method with Smoothing (REPMS; Liu and Boumal, 2019). Importantly, we provide local optimality guarantees and explicitly show the theoretical effect of each hyperparameter in practical settings, extending previous results. Experimental comparisons based on synthetic and UCI datasets show that our approach outperforms prior work in explained variance, fairness, and runtime. This paper is accepted to the 36th AAAI Conference on Artificial Intelligence (AAAI 2022).

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[Zoom 링크] Zoom 회의 참가 https://kaist.zoom.us/j/2655728482?pwd=OXpJeFdDcWliSG51WUp0N1Nad2JHdz09 회의 ID: 265 572 8482 암호: 2AHRKr [Gather Town 링크] https://gather.town/app/ffr2PVibAWRIyXWO/kaistmath 두 발표 세션이 끝나면 Gather Town으로 옮겨와 대학원생들간에 자유롭게 이야기를 나누는 시간을 가질 계획입니다. Gather Town은 크롬이 깔려있는 기기(노트북, 패드, 스마트폰 등)에서 모두 접속 가능합니다. 별도의 회원가입 없이도 개별 캐릭터 설정 후 접속하면 주변의 다른 캐릭터들과 대화할 수 있는 플랫폼입니다.

The Gordon-Bender-Knuth identities are determinant formulas for the sum of Schur functions of partitions with bounded length. There are interesting combinatorial consequences of the Gordon-Bender-Knuth identities, for instance, connections between standard Young tableaux of bounded height, lattice walks in a Weyl chamber, and noncrossing matchings. In this talk we prove an affine analog of the Gordon-Bender-Knuth identities and study their combinatorial properties. As a consequence we obtain an unexpected connection between cylindric standard Young tableaux and r-noncrossing and s-nonnesting matchings. This is joint work with JiSun Huh, Christian Krattenthaler, and Soichi Okada.

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ZOOM Meeting ID: 868 7549 9085 Direct link: https://kaist.zoom.us/j/86875499085

It is known that the rank- and tree-width of the random graph G(n,p) undergo a phase transition at p=1/n; whilst for subcritical p, the rank- and tree-width are bounded above by a constant, for supercritical p, both parameters are linear in n. The known proofs of these results use as a black box an important theorem of Benjamini, Kozma, and Wormald on the expansion of supercritical random graphs. We give a new, short, and direct proof of these results, leading to more explicit bounds on these parameters, and also consider the rank- and tree-width of supercritical random graphs closer to the critical point, showing that this phase transition is smooth. This is joint work with Joshua Erde and Mihyun Kang.

Morihiko Saito's theory of mixed Hodge modules is a far generalisation of classical Hodge theory, which is based on the theory of perverse sheaves, D-modules, variations of Hodge structures. One can think of mixed Hodge modules as a certain class of D-modules with Hodge structures. Naturally they are accompanied by perverse sheaves via the Riemann–Hilbert correspondence. This guide consists of about 8 talks, which may cover: review of classical Hodge theory, D-modules and filtered D-modules, nearby and vanishing cycles, etc. The main goal is to understand the notion of mixed Hodge modules and to explain two important theorems: the structure theorem and the direct image theorem. If time permits, we discuss recent applications of the theory in algebraic geometry.

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Zoom 회의 ID: 352 730 6970 암호: 0971 대기실에서 개별승인을 하오니 실명으로 접속하시기 바랍니다.

This talk follows on from the recent talk of Pascal Gollin in this seminar series, but will aim to be accessible for newcomers. 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. By relaxing `packing’ to `half-integral packing’, Reed obtained an analogous result for odd cycles, and gave a structural characterisation of when the (integral) packing version fails. We prove some far-reaching generalisations of these theorems. First, we show that 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. Similarly to Reed, we give a structural characterisation for the failure of the integral Erdős-Pósa property in this setting. This allows us to deduce the full Erdős-Pósa property for many natural classes of cycles. We will look at applications of these results to graphs embedded on surfaces, and also discuss some possibilities and obstacles for extending these results. This is joint work with Kevin Hendrey, Ken-ichi Kawarabayashi, O-joung Kwon, Sang-il Oum, and Youngho Yoo.

Morihiko Saito's theory of mixed Hodge modules is a far generalisation of classical Hodge theory, which is based on the theory of perverse sheaves, D-modules, variations of Hodge structures. One can think of mixed Hodge modules as a certain class of D-modules with Hodge structures. Naturally they are accompanied by perverse sheaves via the Riemann–Hilbert correspondence. This guide consists of about 8 talks, which may cover: review of classical Hodge theory, D-modules and filtered D-modules, nearby and vanishing cycles, etc. The main goal is to understand the notion of mixed Hodge modules and to explain two important theorems: the structure theorem and the direct image theorem. If time permits, we discuss recent applications of the theory in algebraic geometry.

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Zoom info: 352 730 6970 암호: 2032 대기실에서 개별 승인을 하니 실명으로 접속하시기 바랍니다.

Cells make fate decisions in response to dynamic environments and multicellular structure emerges from interplays among cells in space and time. The recent single-cell genomics technology provides an unprecedented opportunity to profile cells. However, those measurements are taken as snapshots for groups of individual cells with only static information. Can one infer interactions among cells from such datasets? Is it possible to recover spatial information from non-spatial datasets? How to obtain temporal relationships of cells from the static measurements? In this talk I will present our newly developed computational tools that reconstruct interactions and spatiotemporal relationships for cells using single-cell RNA-seq, ATAC-seq, and spatial transcriptomics datasets. Through applications of those methods to systems in development and regeneration, we show the discovery power of such methods and identify areas for further development in spatiotemporal reconstruction.

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

I present a version of the Moser-Trudinger inequality in the setting of complex geometry. As a very particular case, the result already gives a new Moser-Trudinger inequality for functions in the Sobolev space W1,2 of a domain in R2. As an application, we deduce a new necessary condition for the complex Monge-Ampere equation for a given measure on a compact Kahler manifold to admit a Holder continuous solution. This is a joint work with Tien-Cuong Dinh and George Marinescu.

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.

Morihiko Saito's theory of mixed Hodge modules is a far generalisation of classical Hodge theory, which is based on the theory of perverse sheaves, D-modules, variations of Hodge structures. One can think of mixed Hodge modules as a certain class of D-modules with Hodge structures. Naturally they are accompanied by perverse sheaves via the Riemann–Hilbert correspondence. This guide consists of about 8 talks, which may cover: review of classical Hodge theory, D-modules and filtered D-modules, nearby and vanishing cycles, etc. The main goal is to understand the notion of mixed Hodge modules and to explain two important theorems: the structure theorem and the direct image theorem. If time permits, we discuss recent applications of the theory in algebraic geometry.

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줌 아이디: 352 730 6970 암호 : 1541 대기실에서 개별 승인하오니 실명으로 접속하세요.

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.

Morihiko Saito's theory of mixed Hodge modules is a far generalisation of classical Hodge theory, which is based on the theory of perverse sheaves, D-modules, variations of Hodge structures. One can think of mixed Hodge modules as a certain class of D-modules with Hodge structures. Naturally they are accompanied by perverse sheaves via the Riemann–Hilbert correspondence. This guide consists of about 8 talks, which may cover: review of classical Hodge theory, D-modules and filtered D-modules, nearby and vanishing cycles, etc. The main goal is to understand the notion of mixed Hodge modules and to explain two important theorems: the structure theorem and the direct image theorem. If time permits, we discuss recent applications of the theory in algebraic geometry.

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줌 정보: 회의 ID: 352 730 6970 암호: 4401 대기실에서 개별 승인을 기다리십시오. 실명으로 들어오시기 바랍니다.

We talk about a convergence of kinetic vorticity of Boltzmann toward the vorticity of incompressible Euler in 2D. The talk would be self-contained (1) covering necessary background in basic Boltzmann theory, asymptotic expansion (Hilbert expansion). (2) When the Euler vorticity is below Yudovich, we prove a weak convergence toward Lagrangian solutions, (3) while for the Yudovich class we have a strong convergence toward a unique solution with a rate.

We talk about a convergence of kinetic vorticity of Boltzmann toward the vorticity of incompressible Euler in 2D. The talk would be self-contained (1) covering necessary background in basic Boltzmann theory, asymptotic expansion (Hilbert expansion). (2) When the Euler vorticity is below Yudovich, we prove a weak convergence toward Lagrangian solutions, (3) while for the Yudovich class we have a strong convergence toward a unique solution with a rate.

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.

We talk about a convergence of kinetic vorticity of Boltzmann toward the vorticity of incompressible Euler in 2D. The talk would be self-contained (1) covering necessary background in basic Boltzmann theory, asymptotic expansion (Hilbert expansion). (2) When the Euler vorticity is below Yudovich, we prove a weak convergence toward Lagrangian solutions, (3) while for the Yudovich class we have a strong convergence toward a unique solution with a rate.

Morihiko Saito's theory of mixed Hodge modules is a far generalisation of classical Hodge theory, which is based on the theory of perverse sheaves, D-modules, variations of Hodge structures. One can think of mixed Hodge modules as a certain class of D-modules with Hodge structures. Naturally they are accompanied by perverse sheaves via the Riemann–Hilbert correspondence. This guide consists of about 8 talks, which may cover: review of classical Hodge theory, D-modules and filtered D-modules, nearby and vanishing cycles, etc. The main goal is to understand the notion of mixed Hodge modules and to explain two important theorems: the structure theorem and the direct image theorem. If time permits, we discuss recent applications of the theory in algebraic geometry.

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zoom 방 번호 : 352 730 6970 암호: 일삼삼일 (대기실에서 승인을 기다려 주십시오.) 총 8회의 강연을 계획중이고 매주 금요일 1회씩 진행하여 3월 말- 4월 초까지 진행할 계획입니다.

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.