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2019 Intensive Lectures by Yujiro Kawamata
Intensive Lectures on Semi-orthogonal decomposition of a derived category of an algebraic variety IV Yujiro Kawamata (Univ. of Tokyo, Japan) 2019-01-24
I will start with the definition of a derived category of an algebraic variety. The derived category contains much essential information of the algebraic variety.
Then I consider a decomposition of the derived category into simpler components called a semi-orthogonal decomposition (SOD) and explain examples.
I will explain SOD's related to the minimal model program in birational geometry. Some SOD's are obtained by looking at so-called exceptional objects or their deformed variants called relative exceptional objects. I will explain examples of SOD's for some singular surfaces.
The 5th Korea Toric Topology Winter Workshop
GKM-theoretical description of double coinvariant rings of pseudo-reflection groups Takashi Sato (Osaka City University) 2019-01-23
The double coinvariant ring of a pseudo-reflection group is a quotient ring
of the tensor product of two copies of the ring of polynomial functions on
the vector space on which the pseudo-reflection group acts. It is an analogue
of the equivariant cohomology rings of flag varieties. There is a natural
homomorphism from the double coinvariant ring to the direct product of copies of the ring of polynomial functions, and McDaniel shows that its image is described in terms of pseudo-reflections analogously to the GKM theory. I will introduce his work and give other description which is more suitable for combinatorics.
Grossberg-Karshon twisted cubes and connected walks Eunjeong Lee (IBS-CGP) 2019-01-23
Let G be a complex semisimple simply connected algebraic group of rank
r. Let i = (i1; i2; : : : ; in) 2 [r]n be a word decomposition, and let ` =
(`1; : : : ; `n) be a sequence of non-negative integers. Grossberg and Karshon
introduced a virtual lattice polytope associated to i and `, called a twisted
cube, whose lattice points encode characters of representations of B. More
precisely, counted lattice points in twisted cube with sign according to a
density function, one get the character of the generalized Demazure module
associated to i and `. We introduce the notion of hesitant connected `-
walks and then prove that the associated Grossberg{Karshon twisted cube
is a closed convex polytope precisely when i is a hesitant-connected-`-walkavoiding.
On orbit braid groups Zhi Lu (Fudan University) 2019-01-23
Let M be a (topological) connected manifold with an action of a finite group G. We establish the notion of (pure) orbit braid groups for the G-manifold. We will see that this notion is not only the generalization to the classical (pure) braid groups dened by the fundamental groups 1(F(M; n)) and 1(F(M; n)=) of the conguration space F(M; n) of M, but also the classical (pure) braid groups can be regarded as subgroups of the orbit braid group dened by us. In addition, we also study the relation between the orbit braid group and the fundamental group of the orbit conguration space FG(M; n).We try to establish a theory of orbit braid group. This is a joint work with Hao Li and Fengling Li.
Toric Richardson varieties and Bruhat interval polytopes Park, Seonjeong (Ajou University) 2019-01-23
Given v;w 2 Sn with v  w, the Richardson variety Xvw is the intersection
of the Schubert varietyXw and the opposite Schubert variety Xv. A Bruhat
interval polytope Qv;w is the convex hull of all permutation vectors x =
(x(1); x(2); :::; x(n)) with v  x  w. It is known that Qv
α-invariant of submanifolds of generalised Bott towers Hao Li (Fudan University) 2019-01-23
Stolz theorem states that for a n-dimensional simple connected spin manifold
where n  5, it admits a positive scalar curvature if and only if the α-invariant is zero. Weiping Zhang gave an equation to compute the α-invariant of spin complex hypersurfaces. In this talk, I use these tools to construct a family of submanifolds of generalised Bott towers which admit positive scalar curvatures.
Cohomology of the graphical real toric manifold Park, Hanchul (Jeju National University) 2019-01-22
The topology of the real toric manifold MR has been less known than that
of its complex counterpart. In 1985, Jurkiewicz gave the formula for the
Z2-cohomology ring of MR. Its Q-Betti numbers were calculated by Suciu
and Trevisan in their unpublished paper, and the result was strengthened for
coecient ring R in which 2 is a unit by Suyoung Choi and the speaker. Recently,
the cup product for H(MR;R) is computed by Choi and the speaker.
For any simple graph G, the graph associahedron 4G and the graph
cubeahedron G are simple polytopes which support natural projective toric
manifolds XG and YG respectively. For connected G, they are obtained by
cutting faces of the simplex and the cube respectively. In this talk, we describe
the structure of H(XR G; F) and H(Y R G ; F) in terms of the graph G,
where F is a eld with characteristic other than 2. Note that their Q-Betti
numbers can be calculated using the graph invariants known as the a-number
and b-number of G respectively. A weird thing about XR G and Y R G is that it
seems that a slight generalization of them would make the computation of
the cohomology too dicult. For example, nestohedra or simple generalized
permutohedra dene real toric manifolds, but in general their cohomology
rings are very dicult to compute.
It is suspected that H(XR G; F) determines G, and in particular, the family
of XR G is Q-cohomologically rigid.
Fundamental groups of small covers Lisu Wu (Fudan University) 2019-01-22
Davis and Januszkiewicz [Duke Math.J. 62(1911), pp. 417-451] proved that
the fundamental group of a small cover is isomorphic to the kernel of a map from the associated Coxeter group to Zn2 , such description may not be sucient in some specic problems. In this talk, I will describe this relation more explicitly based on the presentation of fundamental group we have calculated. Furthermore, our main result shows that the 1-injectivity of a facial submanifold of a small cover only depends on the local combinatorial structure of the corresponding face. This is a joint work with Li Yu.
Intensive Lectures on Semi-orthogonal decomposition of a derived category of an algebraic variety III Yujiro Kawamata (Univ. of Tokyo, Japan) 2019-01-22
Towards transverse toric geometry Hiroaki Ishida (Osaka City University) 2019-01-22
We say that an eective action of a compact torus G on a connected smooth
manifold M is maximal if there exists a point x 2 M such that dimG +
dimGx = dimM. On the other hand, a compact connected complex manifold
equipped with a compact torus action has a holomorphic foliation coming
from the torus action. In this talk, we discuss a classication of compact
connected complex manifolds with maximal torus actions up to transversely
equivalence. If time permits, we also discuss the basic cohomology and basic
Dolbeault cohomology of such manifolds.
This talk is based on a joint work with Roman Krutovsky and Taras
Panov.
Classification of semifree monotone Hamiltonian $S^1$ -manifolds Cho, Yunhyung (Sungkyunkwan University) 2019-01-21
In this talk, we consider a six dimensional monotone symplectic manifold M
and show that if M admits a semifree Hamiltonian circle action and has an
isolated minimum, then M is Fano.
Cohomology rings of a class of torus manifolds Soumen Sarkar (IIT-Madras) 2019-01-21
Torus manifolds are topological generalization of smooth projective toric manifolds. In this talk, I will compute the rational cohomology ring of a class of smooth locally standard torus manifolds whose orbit space is a connected sum of simple polytopes. This is a joint work with Donald Stanley.
Equivariant cohomology of torus orbifolds Darby, Alastair (Xi'an Jiaotong-Liverpool University) 2019-01-21
We calculate the integral cohomology, in terms of generators and relations,
of locally standard torus orbifolds whose odd degree cohomology vanishes.
This involves extending the notion of a GKM-graph to the orbifold case and considering certain integrality conditions. This is joint work with Shintaro Kuroki and Jongbaek Song.
On lower bounds of the sum of multigraded Betti numbers of simplicial complexes Li Yu (Nanjing University) 2019-01-21
We nd some general lower bounds of the sum of certain families of multigraded
Betti numbers of any simplicial complex K with a vertex coloring.
These multigraded Betti numbers of K are intimately related to some quotient
spaces of the moment-angle complex of K under some free torus actions.
Intensive Lectures on Semi-orthogonal decomposition of a derived category of an algebraic variety II Yujiro Kawamata (Univ. of Tokyo, Japan) 2019-01-18
Intensive Lectures on Semi-orthogonal decomposition of a derived category of an algebraic variety I Yujiro Kawamata (Univ. of Tokyo, Japan) 2019-01-16
2018 KAIST 세미나
Group actions in low dimensions - Part 3-1 Kathryn Mann (Brown University) 2018-12-11
This series of lecture will introduce the study of groups acting on the circle and the line, the moduli spaces of such actions, and the role of these spaces in questions of geometric topology, dynamics, and foliation theory. I will focus on new rigidity results in low regularity, surveying important techniques, geometrically motivated examples, and open problems.
Syzygies and Koszul modules Marian Aprodu (University of Bucharest) 2018-12-07
I will recall some basic facts in the syzygy theory, with emphasis on connections with Koszul cohomology, and I will present a vanishing result for Koszul modules.
Syzygy schemes Edoardo Sernesi (Universita Di Roma) 2018-12-07
I will introduce the syzygy schemes of a canonical curve and compute a few examples.
2018 KAIST Math. Colloquium
Orderable groups in topology and dynamics Kathryn Mann (Brown University) 2018-12-06
Given a group G and a manifold M, can one describe all the actions of G on M? This is a basic, natural question in geometric topology, but also a very difficult one -- even in the case where M is 1-dimensional, and G is a familiar, finitely generated group.
This talk will introduce the theory of groups acting on 1-manifolds, through the study of orderable groups. I will describe some connections between this theory and themes in topology and dynamics (like rigidity and foliation theory ), some current open problems, and indicate new approaches coming from recent joint work with C. Rivas.
2018 Discrete Math 세미나
Largest 2-regular subgraphs in 3-regular graphs Choi, Ilkyoo (Hankuk University of Foreign Studies) 2018-11-26
For a graph G, let f2(G) denote the largest number of vertices in a 2-regular subgraph of G. We determine the minimum of f2(G) over 3-regular n-vertex simple graphs G. To do this, we prove that every 3-regular multigraph with exactly c cut-edges has a 2-regular subgraph that omits at most max{0,⎣(c-1)/2⎦} vertices.
More generally, every n-vertex multigraph with maximum degree 3 and m edges has a 2-regular subgraph that omits at most max{0, ⎣(3n-2m+c-1)/2⎦} vertices. More generally, every n-vertex multigraph with maximum degree 3 and m edges has a 2-regular subgraph that omits at most max{0, ⎣(3n-2m+c-1)/2⎦} vertices.
Pareto distributions in economics 김지희 (KAIST) 2018-11-22
This talk introduces one of surprising empirical regularities observed in economics: Pareto distributions are everywhere. Wealth and income, the size of cities and firms, stock market returns, to list but a few, are all known to follow a Pareto distribution. I first highlight key empirical facts and describe some economic theories that have been proposed to explain the regularity. To be more specific, I will put emphasis on inequality in income distributions. A simple mechanism as well as more complex random growth models that give rise to Pareto distributions will be discussed to explore dynamics of income inequality.
2018 KAIST-HKUST-NUS Joint Workshop in Mathematics : Analysis, PDE and Probability
Green's functions and Well-posedness of Compressible Navier-Stokes equation Shih-Hsien Yu (National University of Singapore) 2018-11-17
A class of decomposition of Green's functions for the compressilbe Navier-Stokes linearized around a constant state is introduced. The singular structures of the Green's functions are developed as essential devices to use the nonlinearity directly to covert the 2nd order quasi-linear PDE into a system of zero-th order integral equation with regular integral kernels. The system of integrable equations allows a wider class of functions such as BV solutions. We have shown global existence and well-posedness of the compressible Navier-Stokes equation for isentropic gas with the gas constant $gamma in (0,e)$ in the Lagrangian coordinate for the class of the BV functions and point wise $L^infty$ around a constant state; and the underline pointwise structure of the solutions is constructed. It is also shown that for the class of BV solution the solution is at most piecewise $C^2$-solution even though the initial data is piecewise $C^infty$.
Local single ring theorem on optimal scale Zhigang Bao (HKUST) 2018-11-17
The celebrated single ring theorem asserts that the eigenvalues of a non-Hermitian random matrix with given singular values are asymptotically distributed on a single ring in the complex plane. In this talk, I will present a local version of this theorem on the optimal scale, which implies the optimal convergence rate of the single ring law. This is a joint work with Laszlo Erdos and Kevin Schnelli.
Scattering of the defocusing generalized Benjamin-Ono equations. Soonsik Kwon (KAIST) 2018-11-17
I present my recent work with Kihyun Kim on a scattering problem of nonlinear dispersive equations, namely generalized Benjamin-Ono equations. Due to its dispersion relation, linear Benjamin-Ono flow travels the right and the higher frequency travel faster. This give rise to a unidirectional propagation of waves. We prove the phenomena still hold true for defocusing BO. More precisely, we show that the center of energy moves faster than the center of mass to the right. This type of monotonicity was first observed by Tao(2007) in the defocusing gKdV equations. We use the monotonicity formula as a non-rigidity theorem, in the setting of compactness-contradiction argument to prove the large data scattering in the energy space. In the talk, I will first explain the whole scheme of concentration-compactness, and the heuristics of monotonicity. Then I will discuss obstacles arising in the Benjamin-Ono setting.
Emergence of Black Holes in General Relativity An, Xinliang (NUS) 2018-11-16
Black holes are predicted by Einstein's theory of general relativity, and now we have ample observational evidence for their existence. However theoretically there are many unanswered questions about how black holes come into being. In this talk, with tools from hyperbolic PDE, quasilinear elliptic equations, geometric analysis and dynamical systems, we will prove that, through a nonlinear focusing effect, initially low-amplitude and diffused gravitational waves can give birth to a black hole region in our universe. This result extends the 1965 Penrose's singularity theorem and it also proves a conjecture of Ashtekar on black-hole thermodynamics.
Levy-Khintchine Random Matrices Paul Jung (KAIST) 2018-11-16
We study a class of Hermitian random matrices which includes Wigner matrices, heavy-tailed random matrices, and sparse random matrices such as adjacency matrices of ErdHos-R'enyi random graphs with $p_nsimfrac 1 n$. Our $ntimes n$ random matrices have real entries which are i.i.d. up to symmetry. The distribution of entries depends on $n$, and we require row sums to converge in distribution; it is then well-known that the limit distribution must be infinitely divisible.We show that a limiting empirical spectral distribution (LSD) exists, and via local weak convergence of associated graphs, the LSD corresponds to the spectral measure associated to the root of a graph which is formed by connecting infinitely many Poisson weighted infinite trees using a backbone structure of special edges called ``cords to infinity''. One example covered by the results are matrices with i.i.d. entries having infinite second moments, but normalized to be in the Gaussian domain of attraction. In this case, the limiting graph is $mathbb{N}$ rooted at 1, so the LSD is the semi-circle law. For this special case, we also discuss a delocalization of eigenvectors result which is joint work with Jaehun Lee.
Moment Asymptotics for the (2+1)-dimensional Directed Polymer in the Critical Window Sun, Rongfeng (NUS) 2018-11-15
The partition function of the directed polymer model on Z^{2+1} has been shown to undergo a phase transition on an intermediate disorder scale. In this talk, we focus on a window around the critical point. Exploiting a renewal process representation, we identify the asymptotics for the second and third moments of the partition function. As a corollary, we show that, viewed as a random field, the family of partition functions admits non-trivial diffusive scaling limits, and each limit point has the same covariance structure with logarithmic divergence near the diagonal. Similar results are obtained for the stochastic heat equation on R^2, extending earlier results by Bertini and Cancrini (98). Based on joint work with F. Caravenna and N. Zygouras.
On the isoperimetric ratio over scalar-flat conformal classes Jin, Tianling (HKUST) 2018-11-15
Let (M,g) be a smooth compact Riemannian manifold of dimension n with smooth boundary. Suppose that (M,g) admits a scalar-flat conformal metric. We prove that the supremum of the isoperimetric ratio over the scalar-flat conformal class is strictly larger than the best constant of the isoperimetric inequality on Euclidean space, and consequently is achieved, if either (i) n>=12 and the boundary has a nonumbilic point; or (ii) n>=10, the boundary is umbilic and the Weyl tensor does not vanish at some boundary point. A crucial ingredient in the proof is the expansion of solutions to the conformal Laplacian equation with blowing up Dirichlet boundary conditions.
Weak detection of signal in the spiked Wigner model Lee, Jioon (KAIST) 2018-11-15
We consider the problem of detecting the presence of the signal in a rank-one signal-plus-noise data matrix. In case the signal-to-noise ratio is under a certain threshold, we propose a hypothesis testing based on the linear spectral statistics of the data matrix. The proposed test does not depend on the distribution of the signal or the noise, and the error of the test matches that of the likelihood ratio test, which minimizes the sum of the Type-I and Type-II errors. As an intermediate step, we establish a central limit theorem for the linear spectral statistics of rank-one spiked Wigner matrices for a general spike. This is a joint work with Hye Won Chung.
해외석학 특별강연 시리즈
Remarks on the Riemann Hypothesis Charles M. Newman (NYU Courant) 2018-11-14
One fairly standard version of the Riemann Hypothesis (RH) is that a specific probability density on the real line has a moment generating function (Laplace transform) that, as an analytic function on the complex plane, has all its zeros pure imaginary. We'll review a series of results that span the period from the 1920's to 2018 concerning a perturbed version of the RH. In that perturbed version, due to Polya, the log of the probability density is modified by a quadratic term.
This gives rise to an implicitly defined real constant known as the de Bruijn-Newman Constant, Lambda. The conjecture and now theorem (Newman 1976, Rodgers and Tao 2018) that Lambda is greater than or equal to zero is complementary to the RH which is equivalent to Lambda less than or equal to zero; The conjecture/theorem is a version of the dictum that the RH, if true, is only barely so. Until very recently, the best upper bound, was a 2009 result of Ki, Kim and Lee that Lambda is strictly less than 1/2.
Faster Exact and Approximate Algorithms for k-Cut 이의웅 (NYU) 2018-11-13
In the k-cut problem, we are given an edge-weighted graph G and an integer k, and have to remove a set of edges with minimum total weight so that G has at least k connected components. This problem has been studied various algorithmic perspectives including randomized algorithms, fixed-parameter tractable algorithms, and approximation algorithms. Their proofs of performance guarantees often reveal elegant structures for cuts in graphs.
It has still remained an open problem to (a) improve the runtime of exact algorithms, and (b) to get better approximation algorithms. In this talk, I will give an overview on recent progresses on both exact and approximation algorithms. Our algorithms are inspired by structural similarities between k-cut and the k-clique problem.
Modular forms and higher Chow cycles Ramesh Sreekantan (Indian Statistical Institute, Bangalore) 2018-11-08
Gross and Zagier made a conjecture on the algebraicity of values of certain `higher' Greens functions at special points. Mellit proved a few cases by linking it to the existence of certain higher Chow cycles. Viazovska proved a few cases by linking it to Borcherds lifts of modular forms. We formulate a conjecture linking modular forms and higher Chow cycles which relates the two approaches and also describe a construction of higher Chow cycles which allows us to prove special cases of the Gross-Zagier conjecture as well as provide evidence for our conjecture.
On the rational Turán exponents conjecture 강동엽 (KAIST) 2018-11-05
The extremal number ex(n,F) of a graph F is the maximum number of edges in an n-vertex graph not containing F as a subgraph. A real number r∈[1,2] is realisable if there exists a graph F with ex(n , F) = Θ(nr). Several decades ago, Erdős and Simonovits conjectured that every rational number in [1,2] is realisable. Despite decades of effort, the only known realisable numbers are 1,7/5,2, and the numbers of the form 1+(1/m), 2-(1/m), 2-(2/m) for integers m≥1. In particular, it is not even known whether the set of all realisable numbers contains a single limit point other than two numbers 1 and 2.
We discuss some recent progress on the conjecture of Erdős and Simonovits. First, we show that 2-(a/b) is realisable for any integers a,b≥1 with b>a and b≡±1 (mod a). This includes all previously known ones, and gives infinitely many limit points 2-(1/m) in the set of all realisable numbers as a consequence.
Secondly, we propose a conjecture on subdivisions of bipartite graphs. Apart from being interesting on its own, we show that, somewhat surprisingly, this subdivision conjecture in fact implies that every rational number between 1 and 2 is realisable.
This is joint work with Jaehoon Kim and Hong Liu.
SIR Models: From Micro to Macro and Back Again Grzegorz A. Rempala (Ohio State University) 2018-11-01
Recently, there has been considerable interest in both inference and predictions for compartmental epidemic models on multiple physical scales. For instance, one could be interested in analyzing response of immune system to infection within a single host or in describing infectuous interactions in a population of hosts. Both viral invasions and global pandemics are often described by similar mathematical constructs known as SIR models. In this talk I will review some basic concepts related to such models across scales and present a simple unifying framework that allows to conceptually connect both deterministic (e.g., population level) and stochastic (e.g., molecular level) SIR models with the help of tools of statistical theory of survival analysis.
Periodic orbits in symplectic geometry 강정수 (서울대학교) 2018-10-25
Symplectic geometry has one of its origins in Hamiltonian dynamics. In the late 60s Arnold made a fundamental conjecture about the minimal number of periodic orbits of Hamiltonian vector fields. This is a far-reaching generalization of Poincaré's last geometric theorem and completely changed the field of symplectic geometry. In the last 30 years symplectic geometry has been tremendously developed due to the theory of holomorphic curves by Gromov and Floer homology theory by Floer. I will give a gentle introduction to the field of symplectic geometry and explain how modern methods give rise to existence results for periodic orbits and discover rigidity phenomena in symplectic geometry.
Representation theory and geometry I Geordie Williamson (University of Sydney) 2018-10-23
One of the most fundamental questions in representation theory asks for a description of the simple representations. I will give an introduction to this problem with an emphasis on the representation theory of Lie groups and algebras.
A recurring theme is the appearance of geometric techniques in seemingly algebraic problems. I will also emphasise the important role played by invariant forms and signature.
Representation theory and geometry II Geordie Williamson (University of Sydney) 2018-10-23
2018년도 Nobel Prize 수상분야 해설 강연
2018년도 물리 Nobel Prize 수상분야 해설 강연 안재욱 (KAIST) 2018-10-11
2018년도 화학 Nobel Prize 수상분야 해설 강연 박희성 (KAIST) 2018-10-11
2018년도 생리의학 Nobel Prize 수상분야 해설 강연 강석조 (KAIST) 2018-10-11
Quotients of affine spaces by finite groups: from Noether's problem to Colliot-Thélène conjecture Danny Neftin (Technion) 2018-10-11
Given an action of a finite group G on an affine space A^n, the quotient variety A^n/G have been a subject of interest for over a century, partially motivated by Noether's problem concerning its rationality. Weaker properties of this variety were extensively studied, relating it to the Inverse Galois problem, parametrization of Galois extensions and essential dimension. One of its weakest possible properties, (very) weak approximation, is the subject of a more recent conjecture by Colliot-Th'el`ene. We shall discuss the above rationality properties, the conjecture, and the (recent) progress towards it.
Spatially balanced sampling Yves Tillé (Institute of Statistics, University of Neuchatel) 2018-10-04
Spatial sampling is particularly important for environmental statistics. A simple reasoning developed in Grafström and Tillé (2013) shows that under a self-correlated linear model, it is more efficient to select a well-spread sample in space. If we select two neighbouring units in a sample, we will tend to collect partially redundant information. Grafström and Lundström (2013) discuss at length the concept of spreading, also known as spatial balancing and its implication on estimation. The Generalized Random Tesselation Sampling GRTS design has been proposed by Stevens Jr. and Olsen (1999, 2004, 2003) to select spread samples. The pivotal method has been proposed by Deville and Tillé (2000). Grafström et al. (2012) proposed to use the pivotal method for spatial sampling. This method, called the local pivotal method, consists, at each step, in comparing two neighboring units. If the probability of one of these two units is increased, the probability of the other is decreased, which induces a repulsion between the units. The natural extension of this idea is to confront a group of units. The local pivot method was generalized by Grafström and Tillé (2013) to provide samples that are both well-balanced in space and balanced on the totals of the auxiliary variables. This method is called the local cube method. We also propose a new method that enables us to select spreader samples that all existing methods and allows the construction of periodic sampling plans when these plans exist.
Mekler's Construction on NTP1 Theory 안진후 (연세대학교) 2018-10-01
Any structure whose language is finite has a model of graph theory which is bi-interpretable with it. From this idea, Mekler further developed a way of interpreting a model into a group. This Mekler's construction preserves various model-theoretic properties such as stability, simplicity, and NTP2, thus helps us find new group examples in model theory. In this talk, I will introduce to you what Mekler's construction is and briefly show that this preserves NTP1.
Optimal cluster recovery in the labeled stochastic block model 윤세영 (KAIST) 2018-09-27
We consider the problem of community detection or clustering in the labeled Stochastic Block Model (LSBM) with a finite number K of clusters of sizes linearly growing with the global population of items n. Every pair of items is labeled independently at random, and label ℓ appears with probability p(i,j,ℓ) between two items in clusters indexed by i and j, respectively. The objective is to reconstruct the clusters from the observation of these random labels. Clustering under the SBM and their extensions has attracted much attention recently. Most existing work aimed at characterizing the set of parameters such that it is possible to infer clusters either positively correlated with the true clusters, or with a vanishing proportion of misclassified items, or exactly matching the true clusters. We find the set of parameters such that there exists a clustering algorithm with at most smisclassified items in average under the general LSBM and for any s=o(n), which solves one open problem raised in Abbe et al 2015. We further develop an algorithm, based on simple spectral methods, that achieves this fundamental performance limit within O(n polylog(n)) computations and without the a-priori knowledge of the model parameters.
Information Center for Basic Sciences KAIST
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