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This series of talks is intended to be a gentle introduction to the random walk theory on infinite groups and hyperbolic spaces. We will touch upon keywords including hyperbolicity, stationary measure, boundaries and limit laws. Those who are interested in geometric group theory or random walks are welcomed to join.
This is a casual seminar among TARGET students, but other graduate students are also welcomed.
한국어 (필요한 경우 영어 가능) ( )     2022-09-29 19:59:13
Compositional data analysis with a high proportion of zeros has gained increasing popularity, especially in chemometrics and human gut microbiomes research. Statistical analyses of this type of data are typically carried out via a log-ratio transformation after replacing zeros with small positive values. We should note, however, that this procedure is geometrically improper, as it causes anomalous distortions through the transformation. We propose a radial transformation that does not require zero substitutions and more importantly results in essential equivalence between domains before and after the transformation. We show that a rich class of kernels on hyperspheres can successfully define a kernel embedding for compositional data based on this equivalence. The applicability of the proposed approach is demonstrated with kernel principal component analysis.
Host: 김범호, 김영종, 안정호,     Contact: 김영종 (+821094985488)     한국어 (필요한 경우 영어 가능) ( )     2022-09-05 15:18:35
We consider a deep generative model for nonparametric distribution estimation problems. The true data-generating distribution is assumed to possess a certain low-dimensional structure. Under this assumption, we study convergence rates of estimators obtained by likelihood approaches and generative adversarial networks (GAN). The convergence rate depends only on the noise level, intrinsic dimension and smoothness of the underlying structure. The true distribution may or may not possess the Lebesgue density, depending on the underlying structure. For the singular case (no Lebesgue density), the convergence rate of GAN is strictly better than that of the likelihood approaches. Our lower bound of the minimax optimal rates shows that the convergence rate of GAN is close to the optimal rate. If the true distribution allows a smooth Lebesgue density, an estimator obtained by a likelihood approach achieves the minimax optimal rate.
Host: Cheolwoo Park     미정     2022-08-19 10:43:13
What allowed for many developments in algebraic geometry and commutative algebra was a discovery of the notion of a Frobenius splitting, which, briefly speaking, detects how pathological positive characteristic Fano and Calabi-Yau varieties can be. Recently, Yobuko introduced a more general concept, a quasi-F-splitting, which captures much more refined arithmetic invariants. In my talk, I will discuss on-going projects in which we develop the theory of quasi-F-splittings in the context of birational geometry and derive applications, for example, to liftability of singularities. This is joint work with Tatsuro Kawakami, Hiromu Tanaka, Teppei Takamatsu, Fuetaro Yobuko, and Shou Yoshikawa. * Zoom information will not be provided. Please send an email to Jinhyung Park if you are interested in.
Host: DongSeon Hwang     Contact: Jinhyung Park (042-350-2747)     영어     2022-09-22 13:39:01
In real world, people are interested in causality rather than association. For example, pharmaceutical companies want to know effectiveness of their new drugs against diseases. South Korea Government officials are concerned about the effects of recent regulation with respect to an electric car subsidy from United States. Due to this reason, causal inference has been received much attention in decades and it is now a big research field in statistics. In this seminar, I will talk about basic idea and theory in the causal inference. Real data examples will be discussed.
Host: Jae Kyoung Kim     미정     2022-09-26 10:09:52
Van der Waerden's theorem states that any coloring of $\mathbb{N}$ with a finite number of colors will contain arbitrarily long monochromatic arithmetic progressions. This motivates the definition of the van der Waerden number $W(r,k)$ which is the smallest $n$ such that any $r$-coloring of $\{1,2,\cdots,n\}$ guarantees the presence of a monochromatic arithmetic progression of length $k$. It is natural to ask what other arithmetic structures exhibit van der Waerden-type results. One notion, introduced by Landman and Robertson, is that of a $D$-diffsequence, which is an increasing sequence $a_1 Host: Sang-il Oum     영어     2022-09-02 18:06:28
In this talk, we present a short history of Lp theories for (stochastic) partial differential equations. In particular, we introduce recent developments handling degenerate equations in weighted Sobolev spaces. It is well known that there exist probabilistic representations of solutions to second order (stochastic) partial equations, which enables us to use many interesting probabilistic theories to investigate solutions. Recently, by applying probabilistic tools, we obtain interesting new type weighted estimates to second order degenerate (stochastic) partial differential equations.
Host: 확률 해석 및 응용 연구센터     Contact: 확률 해석 및 응용 연구센터 (042-350-8111/8117)     미정     2022-09-16 17:31:07
A family of surfaces is called mean curvature flow (MCF) if the velocity of surface is equal to the mean curvature of the surface at that point. Even starting from smooth surface, the MCF typically encounters some singularities and various generalized notions of MCF have been proposed to extend the existence past singularities. They are level set flow, Brakke flow and BV flow, just to name a few. In my talk I explain a recent global-in-time existence result of a particular generalized solution which has some desirable properties. I describe a basic outline of how to construct the solution.
Contact: 강문진 ()     영어     2022-09-12 23:17:14
A family of surfaces is called mean curvature flow (MCF) if the velocity of surface is equal to the mean curvature of the surface at that point. Even starting from smooth surface, the MCF typically encounters some singularities and various generalized notions of MCF have been proposed to extend the existence past singularities. They are level set flow, Brakke flow and BV flow, just to name a few. In my talk I explain a recent global-in-time existence result of a particular generalized solution which has some desirable properties. I describe a basic outline of how to construct the solution.
Host: 확률 해석 및 응용 연구센터     Contact: 확률 해석 및 응용 연구센터 (042-350-8111/8117)     미정     2022-09-16 13:38:16
Over recent years, data science and machine learning have been the center of attention in both the scientific community and the general public. Closely tied to the ‘AI-hype’, these fields are enjoying expanding scientific influence as well as a booming job market. In this talk, I will first discuss why mathematical knowledge is important for becoming a good machine learner and/or data scientist, by covering various topics in modern deep learning research. I will then introduce my recent efforts in utilizing various deep learning methods for statistical analysis of mathematical simulations and observational data, including surrogate modeling, parameter estimation, and long-term trend reconstruction. Various scientific application examples will be discussed, including ocean diffusivity estimation, WRF-hydro calibration, AMOC reconstruction, and SIR calibration.
Host: jaekyoung kim     미정     2022-08-19 10:41:55
Polarization is a technique in algebra which provides combinatorial tools to study algebraic invariants of monomial ideals. Depolarization of a square free monomial ideal is a monomial ideal whose polarization is the original ideal. In this talk, we briefly introduce the depolarization and related problems and introduce the new method using hyper graph coloring.
Host: 곽시종     Contact: 김윤옥 (5745)     미정     2022-09-19 15:36:36
It is challenging to perform a multiscale analysis of mesoscopic systems exhibiting singularities at the macroscopic scale. In this paper, we study the hydrodynamic limit of the Boltzmann equations \begin{equation} \mathrm{St} \partial_t F + v \cdot \nabla_x F = \frac{1}{\mathrm{Kn} } Q(F, F) \end{equation} toward the singular solutions of 2D incompressible Euler equations whose vorticity is unbounded \begin{equation} \partial_t u + u \cdot \nabla_x u + \nabla_x p = 0, \quad \mathrm{div} u = 0. \end{equation} We obtain a microscopic description of the singularity through the so-called kinetic vorticity and understand its behavior in the vicinity of the macroscopic singularity. As a consequence of our new analysis, we settle affirmatively an open problem of convergence toward Lagrangian solutions of the 2D incompressible Euler equation whose vorticity is unbounded ($\omega \in L^{\mathfrak{p} }$ for any fixed $1 \le \mathfrak{p} < \infty$). Moreover, we prove the convergence of kinetic vorticities toward the vorticity of the Lagrangian solution of the Euler equation. In particular, we obtain the rate of convergence when the vorticity blows up moderately in $L^{\mathfrak{p} }$ as $\mathfrak{p} \rightarrow \infty$ (localized Yudovich class).
Host: 확률 해석 및 응용 연구센터     Contact: 확률 해석 및 응용 연구센터 (042-350-8111/8117)     미정     2022-09-16 13:24:57
We will discuss on large time behavior of the one dimensional barotropic compressible Navier-Stokes equations with initial data connecting two different constant states. When the two constant states are prescribed by the Riemann data of the associated Euler equations, the Navier-Stokes flow would converge to a viscous counterpart of Riemann solution. This talk will present the latest result on the cases where the Riemann solution consist of two shocks, and introduce the main idea for using to prove.
Host: 김범호, 김영종, 안정호,     Contact: 김영종 (+821094985488)     한국어     2022-09-05 15:13:15
Deep neural networks have proven to work very well on many complicated tasks. However, theoretical explanations on why deep networks are very good at such tasks are yet to come. To give a satisfactory mathematical explanation, one recently developed theory considers an idealized network where it has infinitely many nodes on each layer and an infinitesimal learning rate. This simplifies the stochastic behavior of the whole network at initialization and during the training. This way, it is possible to answer, at least partly, why the initialization and training of such a network is good at particular tasks, in terms of other statistical tools that have been previously developed. In this talk, we consider the limiting behavior of a deep feed-forward network and its training dynamics, under the setting where the width tends to infinity. Then we see that the limiting behaviors can be related to Bayesian posterior inference and kernel methods. If time allows, we will also introduce a particular way to encode heavy-tailed behaviors into the network, as there are some empirical evidences that some neural networks exhibit heavy-tailed distributions.
Host: 김범호, 김영종, 안정호,     Contact: 김영종 (+821094985488)     한국어 (필요한 경우 영어 가능) ( )     2022-09-05 15:15:18
Several years ago, Chi Li introduced the local volume of a klt singularity in his work on K-stability. The local-global analogy between klt singularities and Fano varieties, together with recent study in K-stability lead to the conjecture that klt singularities whose local volumes are bounded away from zero are bounded up to special degeneration. In this talk, I will discuss some recent work on this conjecture through the minimal log discrepancies of Kollár components. * Zoom information will not be provided. Please send an email to Jinhyung Park if you are interested in.
Host: DongSeon Hwang (IBS-CCG)     Contact: Jinhyung Park (042-350-2747)     영어     2022-09-06 16:25:30
We introduce homotopy coherent nerves of Kan-enriched categories. We discuss homotopy theory of Kan complexes and how composition is performed inside infinity-categories. For this, we introduce the
Host: 김완수     미정     2022-09-06 17:37:37
The Structural Theorem of the Graph Minors series of Robertson and Seymour asserts that, for every $t\in\mathbb{N},$ there exists some constant $c_{t}$ such that every $K_{t}$-minor-free graph admits a tree decomposition whose torsos can be transformed, by the removal of at most $c_{t}$ vertices, to graphs that can be seen as the union of some graph that is embeddable to some surface of Euler genus at most $c_{t}$ and "at most $c_{t}$ vortices of depth $c_{t}$". Our main combinatorial result is a "vortex-free" refinement of the above structural theorem as follows: we identify a (parameterized) graph $H_{t}$, called shallow vortex grid, and we prove that if in the above structural theorem we replace $K_{t}$ by $H_{t},$ then the resulting decomposition becomes "vortex-free". Up to now, the most general classes of graphs admitting such a result were either bounded Euler genus graphs or the so called single-crossing minor-free graphs. Our result is tight in the sense that, whenever we minor-exclude a graph that is not a minor of some $H_{t},$ the appearance of vortices is unavoidable. Using the above decomposition theorem, we design an algorithm that, given an $H_{t}$-minor-free graph $G$, computes the generating function of all perfect matchings of $G$ in polynomial time. This algorithm yields, on $H_{t}$-minor-free graphs, polynomial algorithms for computational problems such as the {dimer problem, the exact matching problem}, and the computation of the permanent. Our results, combined with known complexity results, imply a complete characterization of minor-closed graphs classes where the number of perfect matchings is polynomially computable: They are exactly those graph classes that do not contain every $H_{t}$ as a minor. This provides a sharp complexity dichotomy for the problem of counting perfect matchings in minor-closed classes. This is joint work with Dimitrios M. Thilikos.
Host: Sang-il Oum     영어     2022-07-20 19:55:23
We define the notion of infinity-categories and Kan complex using observations from the previous talk. A process, called the nerve construction, producing infinity-categories from usual categories will be introduced and we will set dictionaries between them. Infinity-categories of functors will be introduced as well.
Host: 김완수     미정     2022-08-26 10:17:15
Katona's intersection theorem states that every intersecting family $\mathcal F\subseteq[n]^{(k)}$ satisfies $\vert\partial\mathcal F\vert\geq\vert\mathcal F\vert$, where $\partial\mathcal F=\{F\setminus x:x\in F\in\mathcal F\}$ is the shadow of $\mathcal F$. Frankl conjectured that for $n>2k$ and every intersecting family $\mathcal F\subseteq [n]^{(k)}$, there is some $i\in[n]$ such that $\vert \partial \mathcal F(i)\vert\geq \vert\mathcal F(i)\vert$, where $\mathcal F(i)=\{F\setminus i:i\in F\in\mathcal F\}$ is the link of $\mathcal F$ at $i$. Here, we prove this conjecture in a very strong form for $n> \binom{k+1}{2}$. In particular, our result implies that for any $j\in[k]$, there is a $j$-set $\{a_1,\dots,a_j\}\in[n]^{(j)}$ such that \[ \vert \partial \mathcal F(a_1,\dots,a_j)\vert\geq \vert\mathcal F(a_1,\dots,a_j)\vert.\]A similar statement is also obtained for cross-intersecting families.
Host: Sang-il Oum     영어     2022-08-28 08:33:58
The activation of Ras depends upon the translocation of its guanine nucleotide exchange factor, Sos, to the plasma membrane. Moreover, artificially inducing Sos to translocate to the plasma membrane is sufficient to bring about Ras activation and activation of Ras’s targets. There are many other examples of signaling proteins that must translocate to the membrane in order to relay a signal. One attractive idea is that translocation promotes signaling by bringing a protein closer to its target. However, proteins that are anchored to the membrane diffuse more slowly than cytosolic proteins do, and it is not clear whether the concentration effect or the diffusion effect would be expected to dominate. Here we have used a reconstituted, controllable system to measure the association rate for the same binding reaction in 3D vs. 2D to see whether association is promoted, and, if so, how.
This talk will be presented online. ZOOM ID: 997 8258 4700 (Biomedical Mathematics Online Colloquium), (pw: 1234)
Host: Jae Kyoung Kim     영어     2022-08-29 14:48:21
This talk is the very first talk for a semester long series on higher algebra. Higher algerbra is the study of algebraic objects in spaces, correcting abnormal behavior of classical homological algebra and encorporating spheres into algebra. In this talk, we will see a concerete example of a higher-algebraic structure on singular cochains, and what structures are needed to formalize such structures with concrete diagrams. We assume familiarity with algebraic topology, category theory and homological algebra (at the first year graduate level).
Host: 김완수     미정     2022-08-26 10:14:13
We prove that for $n>k\geq 3$, if $G$ is an $n$-vertex graph with chromatic number $k$ but any its proper subgraph has smaller chromatic number, then $G$ contains at most $n-k+3$ copies of cliques of size $k-1$. This answers a problem of Abbott and Zhou and provides a tight bound on a conjecture of Gallai. This is joint work with Jie Ma.
Host: Sang-il Oum     영어     2022-08-07 22:29:47
A temporal graph is a graph whose edges are available only at specific times. In this scenario, the only valid walks are the ones traversing adjacent edges respecting their availability, i.e. sequence of adjacent edges whose appearing times are non-decreasing. Given a graph G and vertices s and t of G, Menger’s Theorem states that the maximum number of (internally) vertex disjoint s,t-paths is equal to the minimum size of a subset X for which G-X contains no s,t-path. This is a classical result in Graph Theory, taught in most basic Graph Theory courses, and it holds also when G is directed and when edge disjoint paths and edge cuts are considered instead. A direct translation of Menger’s Theorem to the temporal context has been known not to hold since an example was shown in the seminal paper by Kempe, Kleinberg and Kumar (STOC’00). In this talk, an overview of possible temporal versions of Menger’s Theorem will be discussed, as well as the complexity of the related problems.
Host: Sang-il Oum     영어     2022-08-05 11:22:11
In this talk, I will mostly discuss the singularity formation of Burgers equation. It is well-known that, when the initial data has negative gradient at some point, the solutions blow up in a finite time. We shall study the properties of the blow-up profile of Burgers equation by introducing the self-similar variables and the modulations, which can be used to study the blow-up for general nonlinear hyperbolic systems. If time permits, I will also discuss the singularity formation for the 1D compressible Euler equations and the related open questions.
Host: 강문진     한국어     2022-08-18 00:07:14
In this talk, I will mostly discuss the singularity formation of Burgers equation. It is well-known that, when the initial data has negative gradient at some point, the solutions blow up in a finite time. We shall study the properties of the blow-up profile of Burgers equation by introducing the self-similar variables and the modulations, which can be used to study the blow-up for general nonlinear hyperbolic systems. If time permits, I will also discuss the singularity formation for the 1D compressible Euler equations and the related open questions.
Host: 강문진     한국어     2022-08-18 00:05:47
In this talk, we present a Weisfielier-Leman Isomorphism test algorithm of featured graphs and how it can be used to extract representing features of nodes or entire graphs. This leads to a message passing framework of Aggregate-Combine of node-features which is one of the fundamental procedures to currently uesd graph neural networks. We proceed by showing various basic examples arised in real-world non-standard datasets like social network, knowledge graph and chemical compounds.
Host: 곽시종     Contact: 김윤옥 (5745)     미정     2022-08-13 17:52:56
In this talk, I will mostly discuss the singularity formation of Burgers equation. It is well-known that, when the initial data has negative gradient at some point, the solutions blow up in a finite time. We shall study the properties of the blow-up profile of Burgers equation by introducing the self-similar variables and the modulations, which can be used to study the blow-up for general nonlinear hyperbolic systems. If time permits, I will also discuss the singularity formation for the 1D compressible Euler equations and the related open questions.
Host: 강문진     미정     2022-08-16 17:24:20
Various plasma phenomena will be discussed using a fundamentalfluid model for plasmas, called the Euler-Poisson system. These include plasma sheaths and plasma soliton. First we will briefly introduce recent results on the stability of plasma sheath solutions, and the quasi-neutral limit of the Euler-Poisson system in the presence of plasma sheaths. Another example of ourinterest is plasma solitary waves, for which we discuss existence, stability, and the time-asymptotic behavior. To study the nonlinear stability of solitary waves, the global existence of smooth solutions must be established, which is completelyopen. As a negative answer for global existence, we look into the finite-time blow-up results for the Euler-Poisson system, and discuss the related open questions.
Host: 확률 해석 및 응용 연구센터     Contact: 확률 해석 및 응용 연구센터 (042-350-8111/8117)     미정     2022-08-16 09:03:48
Abstract: Let S:=S(a_{1}, ..., a_{n}) \subset P^{n} be a smooth rational normal n-fold scroll. Then the dimension of the projective automophism group {rm Aut}(S,``VecP^ {N} ) of S is \dim(Aut(S, P^{N})) = 2+ \frac{n(n+1)}{2}-(n+1)(N-n+1)+2 sum _{ n} ^{ j=1} ja_j + #{(i,j)|i Host: 곽시종     Contact: 김윤옥 (5745)     미정     2022-08-13 17:47:33
In this talk, we propose the Landau-Lifshitz type system augmented with Chern-Simons gauge terms, which can be considered as the geometric analog of so-called the Chern-Simons-Schrodinger equations. We first derive its self-dual equations through the energy minimization so that we can provide $N$-equivariant solitons. We next deliver basic ideas of constructing $N$-equivariant solitary waves for non-self-dual cases and investigating their qualitative properties.
Host: 확률 해석 및 응용 연구센터     Contact: 확률 해석 및 응용 연구센터 (042-350-8111/8117)     미정     2022-08-16 09:01:57
It has been told that deep learning is a black box. The universal approximation theorem was the key theorem which makes the stories going on. On the other hand, in the perspective of the function classes generated by deep neural network, it can be analyzed by in terms of the choice of the various activation functions. The piecewise linear functions, fourier series, wavelets and many other classes would be considered for the purpose of the tasks such as classification, prediction and generation models which heavily depend on the data sets. It might be a challenging problem for mathematicians to develop a new optimization theory depending on the various function classes.
Host: 곽시종     Contact: 김윤옥 (5745)     미정     2022-08-13 17:50:14
Property testers are probabilistic algorithms aiming to solve a decision problem efficiently in the context of big-data. A property tester for a property P has to decide (with high probability correctly) whether a given input graph has property P or is far from having property P while having local access to the graph. We study property testing of properties that are definable in first-order logic (FO) in the bounded-degree model. We show that any FO property that is defined by a formula with quantifier prefix ∃*∀* is testable, while there exists an FO property that is expressible by a formula with quantifier prefix ∀*∃* that is not testable. In the dense graph model, a similar picture is long known (Alon, Fischer, Krivelevich, Szegedy, Combinatorica 2000), despite the very different nature of the two models. In particular, we obtain our lower bound by a first-order formula that defines a class of bounded-degree expanders, based on zig-zag products of graphs. This is joint work with Isolde Adler and Pan Peng.
Host: Sang-il Oum     영어     2022-07-20 19:56:58
Global wellposedness and asymptotic stability of the Boltzmann equation with specular reflection boundary condition in 3D non-convex domain is an outstanding open problem in kinetic theory. Motivated by Guo’s L^2-L^\infty theory, the problem was completely solved for general C^3 domain, but it is still widely open for general non-convex domains. The problem was solved in cylindrical domain with analytic non-convex cross section. Generalizing previous work, we study the problem in general solid torus, a solid torus with general analytic convex cross-section. This is the first results for the domain which contains essentially 3D non-convex structure. This is a joint work with Chanwoo Kim and Gyeonghun Ko.
Host: 강문진     한국어     2022-08-11 09:21:00
The past decade has witnessed a great advancement on the Tate conjecture for varieties with Hodge number $h^{2,0} = 1$. Charles, Madapusi-Pera and Maulik completely settled the conjecture for K3 surfaces over finite fields, and Moonen proved the Mumford-Tate (and hence also Tate) conjecture for more or less arbitrary $h^{2,0} = 1$ varieties in characteristic $0$. In this talk, I will explain that the Tate conjecture is true for mod $p$ reductions of complex projective $h^{2,0} = 1$ varieties when $p \gg 0$, under a mild assumption on moduli. By refining this general result, we prove that in characteristic $p \geq 5$ the BSD conjecture holds for a height $1$ elliptic curve $E$ over a function field of genus 1, as long as $E$ is subject to the generic condition that all singular fibers in its minimal compactification are irreducible. We also prove the Tate conjecture over finite fields for a class of surfaces of general type and a class of Fano varieties. The overall philosophy is that the connection between the Tate conjecture over finite fields and the Lefschetz $(1,1)$-theorem over $C$ is very robust for $h^{2,0} = 1$ varieties, and works well beyond the hyperk\”{a}hler world. This is based on joint work with Paul Hamacher and Xiaolei Zhao.
Please contact Wansu Kim at for Zoom meeting info or any inquiry.
미정     2022-08-04 14:23:39
The past decade has witnessed a great advancement on the Tate conjecture for varieties with Hodge number $h^{2,0} = 1$. Charles, Madapusi-Pera and Maulik completely settled the conjecture for K3 surfaces over finite fields, and Moonen proved the Mumford-Tate (and hence also Tate) conjecture for more or less arbitrary $h^{2,0} = 1$ varieties in characteristic $0$. In this talk, I will explain that the Tate conjecture is true for mod $p$ reductions of complex projective $h^{2,0} = 1$ varieties when $p \gg 0$, under a mild assumption on moduli. By refining this general result, we prove that in characteristic $p \geq 5$ the BSD conjecture holds for a height $1$ elliptic curve $E$ over a function field of genus 1, as long as $E$ is subject to the generic condition that all singular fibers in its minimal compactification are irreducible. We also prove the Tate conjecture over finite fields for a class of surfaces of general type and a class of Fano varieties. The overall philosophy is that the connection between the Tate conjecture over finite fields and the Lefschetz $(1,1)$-theorem over $C$ is very robust for $h^{2,0} = 1$ varieties, and works well beyond the hyperk\”{a}hler world. This is based on joint work with Paul Hamacher and Xiaolei Zhao.
Please contact Wansu Kim at for Zoom meeting info or any inquiry.
미정     2022-08-04 14:25:45
(학사과정 학생 개별연구 결과 발표 세미나) Čech cohomology is the direct limit of cohomology taken from the cochain complex obtained by an open cover and a sheaf. In this talk we will derive some important results about Riemann surfaces such as Riemann-Roch theorem and Serre Duality, regarding low level Čech cohomologies. We will also discuss some basic structure and properties of Riemann surfaces using these results, focusing on genus and the embeddings.
Host: 박진현     Contact: 박진현 (2734)     한국어     2022-06-30 16:51:44
We show a flow-augmentation algorithm in directed graphs: There exists a polynomial-time algorithm that, given a directed graph G, two integers $s,t\in V(G)$, and an integer $k$, adds (randomly) to $G$ a number of arcs such that for every minimal st-cut $Z$ in $G$ of size at most $k$, with probability $2^{−\operatorname{poly}(k)}$ the set $Z$ becomes a minimum $st$-cut in the resulting graph. The directed flow-augmentation tool allows us to prove fixed-parameter tractability of a number of problems parameterized by the cardinality of the deletion set, whose parameterized complexity status was repeatedly posed as open problems: (1) Chain SAT, defined by Chitnis, Egri, and Marx [ESA'13, Algorithmica'17], (2) a number of weighted variants of classic directed cut problems, such as Weighted st-Cut, Weighted Directed Feedback Vertex Set, or Weighted Almost 2-SAT. By proving that Chain SAT is FPT, we confirm a conjecture of Chitnis, Egri, and Marx that, for any graph H, if the List H-Coloring problem is polynomial-time solvable, then the corresponding vertex-deletion problem is fixed-parameter tractable. Joint work with Stefan Kratsch, Marcin Pilipczuk, Magnus Wahlström.
Host: Sang-il Oum     영어     2022-06-20 14:22:21
For a given stable subalgebra of the formal power series ring, its Laurent extension, or others, we define an operator algebra over the subalgebra. One of the important operator algebras is the Weyl algebra or its generalization. We define generalized radical Weyl algebras (GRWA) and define the generalized radical Weyl algebra modules, we prove that the algebras and modules are simple. An automorphism of the GRWA define a twisted simple module as well. Since GRWA is an associative algebra, it has an ${\Bbb F}$-subalgebra which is a Lie algebra with respect to the commutator and we show that the Lie algebra is simple. We consider some other generalized Weyl algebra and its descended consequences as well.
Host: 백상훈     Contact: 윤상영 (350-2704)     미정     2022-07-14 11:44:17
We call an order type inscribable if it is realized by a point configuration where all extreme points are all on a circle. In this talk, we investigate inscribability of order types. We first show that every simple order type with at most 2 interior points is inscribable, and that the number of such order types is $\Theta(\frac{4^n}{n^{3/2}})$. We further construct an infinite family of minimally uninscribable order types. The proof of uninscribability mainly uses Möbius transformations. We also suggest open problems around inscribability. This is a joint work with Michael Gene Dobbins.
Host: Sang-il Oum     영어     2022-06-22 06:16:23