# 세미나 및 콜로퀴엄

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In this talk, we present our work of finite volume method for stochastic partial differential equations, both from the viewpoint of theoretical study and numerical simulations. In joint work with T. Funaki and D. Hilhorst [1, 2], we consider a first-order conservation law involving a Q-Brownian motion. We prove that the discrete solution converges along a subsequence in the sense of Young measures to a measure-valued entropy solution as the maximum diameter of the volume elements and the time step tend to zero. We then present the Kato's inequality and as a corollary we deduce the uniqueness of the measure-valued entropy solution as well as the uniqueness of the weak entropy solution. For the numerical simulations, we show results of stochastic Burgers equation by Monte-Carlo method [1]. And some recent simulations of phase-field model, namely Cahn-Hilliard equation and Swift-Hohenberg equation.

Using elliptic regularity results, we construct for every starting point, weak solutions to SDEs in R^d with Sobolev diffusion and locally integrable drift coefficient up to their explosion times. Subsequently, we develop non-explosion criteria which allow for linear growth, singularities of the drift coefficient inside an arbitrarily large compact set, and an interplay between the drift and the diffusion coefficient. Moreover, we show strict irreducibility of the solution, which by construction is a strong Markov process with continuous sample paths on the one-point compactification of R^d. Joint work with Haesung Lee

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E6-1, ROOM 3433
Discrete Math
권오정 (Technische Universität Berlin, Berlin, Germany)
Erdős-Pósa property of chordless cycles and its applications

A chordless cycle in a graph G is an induced subgraph of G which is a cycle of length at least four. We prove that the Erdős-Pósa property holds for chordless cycles, which resolves the major open question concerning the Erdős-Pósa property. Our proof for chordless cycles is constructive: in polynomial time, one can find either k+1 vertex-disjoint chordless cycles, or c k^2 log k vertices hitting every chordless cycle for some constant c. It immediately implies an approximation algorithm of factor O(OPT log OPT) for Chordal Vertex Deletion. We complement our main result by showing that chordless cycles of length at least ℓ for any fixed ℓ≥ 5 do not have the Erdős-Pósa property.

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E6-1, ROOM 3433
Discrete Math
이준경 (University of Oxford, Oxford, UK)
Counting tree-like graphs in locally dense graphs

We prove that a class of graphs obtained by gluing complete multipartite graphs in a tree-like way satisfies a conjecture of Kohayakawa, Nagle, Rödl, and Schacht on random-like counts for small graphs in locally dense graphs. This implies an approximate version of the conjecture for graphs with bounded tree-width. We also prove an analogous result for odd cycles instead of complete multipartite graphs.

Okounkov bodies have become a very interesting and useful tool to understand the positivity of divisors. Although the Okounkov body carries rich positivity data of a divisor, it only provides information near a single point. In this talk, we introduce a new convex body of a divisor that is effective in handling the positivity theory in a multi-point setting. We study its various properties, and observe local positivity data via this convex body.

I will explain our recent progress on the construction of exceptional vector bundles on surfaces when they admit Q-Gorestein degenerations to singularities of class T_d. This is a generalization of the result of Hacking who has studied the case d=1. We give the construction of block(=completely orthogonal exceptional collection) of length d when d>1. If the underlying spaces are del Pezzo surfaces, then our construction explains the paralleism between toric degenerations and three block collections in derived categories.

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자연과학동(E6-1) Room 3433
Discrete Math
김재훈 (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