# Seminars & Colloquia

구글 Calendar나 iPhone 등에서 구독하면 세미나 시작 전에 알림을 받을 수 있습니다.

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자연과학동(E6-1) Room 3433
Discrete Math
Jaehoon Kim (Birmingham University)
Spanning trees in a randomly perturbed graphs

We extend this result to trees with unbounded maximum degree. More precisely, for a given n

^{ε}≤ Δ≤ cn/log n and α>0, we determined the precise number (up to a constant factor) of random edges that we need to add to an arbitrary n-vertex graph G with minimum degree αn in order to guarantee with high probability a copy of any fixed T with maximum degree at most Δ. This is joint work with Felix Joos.

Let X be a smooth complete intersection of two quadrics in P^5. We study the moduli space of Ulrich bundles on X. Using the fact that the derived category of X contains the derived category of certain hyperelliptic curve C, we associate Ulrich bundles on X with semistable vector bundles on C, and use the moduli space of semistable vector bundles on C to describe the moduli space of Ulrich bundles on X. This is a joint work with Yeongrak Kim and Kyoung-Seog Lee.

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E6-1, ROOM 3433
Discrete Math
Hong Liu (University of Warwick, Warwick, UK)
On the maximum number of integer colourings with forbidden monochromatic sums

Let f(n,r) denote the maximum number of colourings of A⊆{1,…,n} with r colours such that each colour class is sum-free. Here, a sum is a subset {x,y,z} such that x+y=z. We show that f(n,2) = 2^(⌈n/2⌉), and describe the extremal subsets. Further, using linear optimisation, we asymptotically determine the logarithm of f(n,r) for r≤5.

Identification of differences between multiple groups in molecular and cellular phenotypes measured by high-throughput sequencing assays is frequently encountered in genomics applications. For example, common problems include identifying genetic variants associated with gene expression using RNA-seq data and detecting differences in chromatin accessibility across tissues/conditions using DNase-seq or ATAC-seq data. These high-throughput sequencing data provide high-resolution measurements on how traits vary along the whole genome in each sample. However, typical analyses fail to exploit the full potential of these high-resolution measurements, instead aggregating the data at coarser resolutions, such as genes, or windows of fixed length. In this talk, I will present two multi-scale methods that more fully exploit the high-resolution data. In the first part of my talk, I will introduce a wavelet-based approach and demonstrate that the proposed wavelet-based approach has more power than simpler window-based approaches in identification of genetic variants associated with chromatin accessibility. I will also illustrate how the estimated shape of the genotype effect can help in understanding the potential mechanisms underlying the identified associations. The second part will discuss potential limitations of the wavelet based approach in analyses of data sets with small sample sizes or low sequencing depths. To address these issues, I will present another approach that models the count nature of the sequencing data directly using multi-scale models for inhomogeneous Poisson processes, and demonstrate that the proposed models have substantially more power than the wavelet-based approach in analyses of data sets with small sample sizes or low sequencing depths. While we developed these methods with specific applications to sequencing data in mind, these methods have natural applications for analysis of many functional phenotypes.

Analyses of molecular phenotypes, such as gene expression, transcription factor binding, chromatin accessibility, and translation, is an important part of understanding the molecular basis of gene regulation and eventually organismal-level phenotypes, such as human disease susceptibility. The development of cheap high-throughput sequencing (HTS) technologies with experiment protocols has increased the use of HTS data as measurements of the molecular phenotypes (e.g., RNA-seq, ChIP-seq, and ATAC-seq). The HTS data provide high-resolution measurements across the whole genome that represent how the molecular phenotypes vary along the genome. We develop multiple statistical methods that better exploit the high-resolution information in the data and apply them to different biological questions in genomics. In this talk, I will briefly introduce two projects: 1) wavelet-based methods for identification of genetic variants associated with chromatin accessibility, and 2) mixture of hidden Markov models for inference of translated coding sequences.

제8회 CMC 정오의 수학산책

일시: 12월 1일(금) 12:00 - 13:15

장소: KAIST 자연과학동 E6-1 3435호

강연자: 김재광 교수 (KAIST)

제목: 빅데이터시대의 통계학

내용: 빅데이터 시대를 맞이하여 빅데이터를 이용하여 사회 과학을 연구하고자 할 때 어떠한 통계학적 이슈들이 있는지 그리고 그러한 문제점들을 해결하고자 할 때 어떤 점들을 주의해야 하는지에 대한 전반적인 내용들을 다루었다. 특히, 빅데이터에서 발생하기 쉬운 선택 편향과 정보 편향에 대한 통계학적 점검과 이것들을 어떻게 해결할 수 있을지에 대한 내용도 다루었다.

참가: https://goo.gl/forms/lJdtJG2HGToWdYMO2 를 통해 사전등록

Computational electromagnetics (CEM) that develops efficient algorithms for solving Maxwell's equations has been a popular subfield of applied mathematics since the invention of computers. Computational nanophotonics applies the CEM techniques to studying nanophotonics, a field of optics that concerns the interaction of light with nanometer-scale objects. However, nanophotonics poses new challenges to CEM, because physical conditions of interest in nanophotonics are very different from those in micro- and radio-wave engineering, which have been the main application areas of CEM. In this talk, I will discuss three exemplary nanophotonic contexts whose unique physical conditions give rise to unexpected computational challenges: plasmonics, dynamic modulation, and nanophotonic laser design. I will demonstrate that it is possible to overcome such challenges by developing new computational techniques.

During the past 10 years, most of acoustic metamaterial research has been done within a theoretical frame in which the medium is at rest. However, such acoustic metamaterials cannot preserve their unique properties or functions in the presence of ow. For example, the well-known acoustic cloak for a cylindrical object fails even at low subsonic flow. In a previous study, the wave operator couldn't take into account the effect of non-uniformity of the flow around the metamaterial as well as the density inhomogeneity due to the compressibility of fluid. Therefore, in this study, we propose a theoretical framework to consider the effect of non-uniform mean flow on acoustic metamaterials aiming at understanding the underlying physics and designing a new type of acoustic metamaterial.

In this talk, we study a rumor spreading model. We employ a steady state analysis to obtain the

2017년 제7회 정오의 수학산책

일시: 11월 17일(금) 12:00 - 13:15

장소: KAIST 수리과학과 E6-1 3435호

강연자: 양현미 교수 (서울대학교)

제목: 4th Industrial Revolution and the New Talents We Need

내용: TBA

등록: https://docs.google.com/forms/d/1l1px6mJC7YtDeZspNpn0f0MmRlfJm-j_JnkCCl--xQk/edit?ts=59fbeb7e

* 11월 24일(금) 채동호 교수님 강연은 연사분 사정으로 취소되었습니다.

I will introduce the dual of the formal affine Demazure algebra, which is the algebraic replacement of T-equivariant oriented cohomology of complete flag variety. I will also mention the proof of the generalized Borel Isomorphism. In the second half of this talk, I will define the two Hecke actions on the dual of the formal affine Demazure algebra. Then I will define the push-pull operators of the oriented cohomology and define perfect pairings on the equivariant cohomology of complete and partial flag varieties. I will also mention hyperbolic cohomology and its relation with Kazhdan-Lusztig basis.

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Room 3433, Bldg. E6-1
Discrete Math
Noel Jonathan (University of Warwick, UK)
The Best Way to Spread a Rumour in a High-Dimensional Grid

Bootstrap percolation is a simple process which is used as a model of propagation phenomena in real world networks including, for example, the spread of a rumour in a social network, the dynamics of ferromagnetism and information processing in neural networks. Given a graph G and an integer r, the r-neighbour bootstrap process begins with a set of “infected” vertices and, in each step, a “healthy” vertex becomes infected if it has at least r infected neighbours. A central problem in the area is to determine the size of the smallest initial infection which will spread to every vertex of the graph. In this talk, I will present a trick for obtaining lower bounds on this quantity by transforming the problem into an infection problem on the edges of the graph and applying some basic facts from linear algebra. In particular, I will outline a proof of a conjecture of Balogh and Bollobás (2006) on the smallest infection which spreads to every vertex of a high-dimensional square lattice and mention some potential applications to analysing the behaviour of a random infection in this setting. This talk is based on joint work with Natasha Morrison.

I will introduce the concept of oriented cohomology in the sense of Levine and Morel, and the work of Kostant and Kumar on algebraic construction of singular cohomology and Grothendieck group of flag varieties. Then I will introduce the formal group algebra of Calmes-Petrov-Zainoulline, which is the algebraic replacement of T-equivariant oriented cohomology of a point. I will introduce the definition of formal affine Demazure algebra and sketch the proof of the structure theorem.

본 강연에서는 산업계에서 실제 적용되고 있는 산업수학의 예제들을 소개하고 계산 유체역학 분야의 특허와 관련하여 제안할 수 있는 아이디어를 논하고자 한다. 주된 논의점은 어떻게 산업수학이 산업계가 대면하고 있는 실질적인 문제들에 의미있는 해법을 제시할 수 있는지에 있다. 이를 위해서, 첫 번째로 수치 편미방 분야의 예제를 설명한다. 수치 편미분 방정식에서 가장 기본적으로 사용되는 방법론을 설명하고 계산 유체 역학의 실무적 관점에서 이해될 수 있는 근본적 문제들을 토론해본다. 더불어 현실적인 측면을 반영한 알고리즘을 생각해 보고, 산업수학자에게 의미있는 문제들을 제시하고자 한다. 두 번째 예제에서는 최적화 기법을 이용해서 간단한 2차원 스케치로부터 3차원 모델을 구성하는 모형을 논의한다. 실제로 만화 제작에 쓰이는 이 알고리즘으로부터 응용될 수 있는 문제들도 다룰 예정이다. 셋째, 영상처리분야에서의 Euler’s elastica를 효율적으로 계산하는 방법론을 간단히 설명하고, 이를 바탕으로 실무에 적용 가능한 연구 방향을 논의해 본다.

Hyperbolic dynamical systems are nowadays fairly well understood from the topological and ergodic point of view. In this talk, we discuss some recent and ongoing works on the dynamics beyond hyperbolicity. In the ﬁrst part, we will provide a characterization of robustly shadowable chain transitive sets for C1-vector ﬁelds on compact smooth manifolds. In the second part, we extend the concepts of topological stability and pseudo-orbit tracing property from homeomorphisms to Borel measures, and prove that every expansive measure with the pseudo-orbit tracing property is topologically stable. This represents a measurable version of the stability theorem by Peter Walters. The ﬁrst part is joint work with M. Reza, and the second part is joint work with C.A. Morales.

The geometry of the moduli space of sheaves on $\mathbb{P}^2$ has been studied in various viewpoints, for instance curve counting, the strange duality conjecture, and birational geometry via Bridgeland stability. For small degree cases, it was possible to classify all rational contractions and compute the cohomology ring of the moduli space.

In the absence of the entropy condition, we construct an $L^\infty$ solution to the Cauchy problem of a scalar conservation law in one space dimension that exhibits microscopic oscillation in the interior of its support when the initial function is non-constant, continuous and compactly supported. As a result, such a solution is nowhere continuous in the interior of the support. Our method of proof is to convert the main equation into a suitable partial differential inclusion and to rely on the convex integration method of M\"uller and \v{S}ver\'ak. In doing so, we find an appropriate subsolution by solving certain ordinary differential equations and make use of it to tailor an in-approximation scheme that reflects persistence of oscillations.

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자연과학동(E6-1) Room 3435
KAIST CMC noon lectures
Byeong-Kweon Oh (Seoul National University)
Representations of quadratic forms

2017년 제6회 정오의 수학산책

일시: 11월 3일(금) 12:00 - 13:15

장소: KAIST 수리과학과 E6-1 3435호

강연자: 오병권 교수 (서울대학교)

제목: Representations of quadratic forms (이차형식의 표현)

내용: 이 강연에서는 가우스(K. F. Gauss)이래 발전을 거듭하여 온 정수 계수의 이차형식에 의한 정수 표현에 대하여 살펴본다. 특히, 주어진 이차형식의 차원이 4이상인 경우, 표현되는 정수를 모두 구하는 방법을 알아본다. 또한 이차형식의 차원이 3인 경우 표현되는 정수를 모두 구하기 위하여 시도되고 있는 다양한 방법을 소개한다.

등록: https://goo.gl/forms/hFwMk8xiDJVBPgD03

In a constantly changing world, animals must account for fluctuations and changes in their environment when making decisions. They must make use of recent information, and appropriately discount older, irrelevant information. But to do so they need to learn the rate at which the environment changes. Recent experimental studies show that humans and other animals can indeed do so. However it it is unclear what underlying computations they use to achieve this. Developing normative models of evidence accumulation is a first step in quantifying such decision-making processes. While optimal, these algorithms are computationally intensive. To address this problem we developed an approximation of the normative inference process, and show how this approximate computation can be implemented in neural circuits. In the second part of the talk I will discuss evidence accumulation on networks where private information can be shared between neighboring nodes.