# 세미나 및 콜로퀴엄

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E6-1, ROOM 3433
Discrete Math
Xavier Goaoc (Université Paris-Est, Marne-la-Vallée, France)
Shatter functions of (geometric) hypergraphs

In combinatorial and computational geometry, the complexity of a system of sets is often studied via its shatter function. I will introduce these functions, and discuss how their asymptotic growth rate is governed from a single of its values, in the spirit of the classical notion of “Vapnik-Chernonenkis dimension” of hypergraphs. In particular, I will describe a probabilistic construction that refutes a conjecture of Bondy and Hajnal. This is joint work with Boris Bukh (https://arxiv.org/abs/1701.06632). The talk will start from first principles.

The main aim of the lectures is to show some recent applications of

positive characteristic version of non-abelian Hodge theory to study

of algebraic varieties and their invariants in positive and zero characteristics.

I will also review classical Simpson's correspondence and relate

it to recent developments.

In these two talks, I will introduce geometric problems related to mass in general relativity, such as a series of geometric inequalities, and

conjectures regarding a notion of quasi-local mass proposed by Bartnik.

The geometric inequalities we consider include the angular momentum-mass

inequality for axially symmetric initial data for the Einstein equations.

Note that the special cases treating maximal data have been proved by Dain

et al. Here I will explain how to reduce the general (non-maximal) case to

the known maximal case, and then discuss the solvability of the system of

Elliptic PDEs arose in the process, for near maximal case.

The second part of the talk will mainly provide an introduction to the

static/stationary metric extension conjectures, related to Bartnik

quasi-local mass. I will briefly discuss some known results for the static

metric extension conjecture by Anderson, Anderson/Khuri, Miao et al., and

show a local existence theorem for the solutions of axially symmetric,

stationary vacuum Einstein equations.

We show how to compute for n-vertex planar graphs in roughly O(n^(11/6)) expected time the diameter and the sum of the pairwise distances. These are the first algorithms for these problems using time O(n^c) for some constant c<2, even when restricted to undirected, unweighted planar graphs.

The main aim of the lectures is to show some recent applications of

positive characteristic version of non-abelian Hodge theory to study

of algebraic varieties and their invariants in positive and zero characteristics.

I will also review classical Simpson's correspondence and relate

it to recent developments.

The main aim of the lectures is to show some recent applications of

positive characteristic version of non-abelian Hodge theory to study

of algebraic varieties and their invariants in positive and zero characteristics.

I will also review classical Simpson's correspondence and relate

it to recent developments.

빅데이터(Big-Data), 단어는 많이 들어보았는데 과연 빅데이터의 본질은 무엇일까? Volume, Velocity, Variety의 3V로 정의되는 빅데이터는 과연 우리의 미래를 예측할 수 있을까? 이번 강연에서는 우리가 살고 있는 복잡한 사회를 이해하는 도구로서의 빅데이터와 네트워크 과학에 관한 설명과 함께, 그 성공적 응용사례로 구글의 검색엔진을 이용한 선거의 예측, N-gram 프로젝트를 통한 과거의 검색엔진 개발, 알파고의 AI 알고리즘, 대규모 특허분석을 통한 기술예측, 인문학 및 예술에서의 빅데이터 분석사례 등을 소개하고자 합니다. 또한 성공의 이면에 숨겨있던 빅데이터의 한계점과 어두운 그림자까지 빅데이터의 모든 것을 함께 고민하고 이를 통해 우리의 미래의 가능성을 살짝 엿보고자 합니다.

The spatially varying coefficient process model is a nonstationary approach to explaining spatial heterogeneity by allowing coefficients to vary across space. To accommodate geographically hierarchical data, we develop a methodology for generalizing this model. We consider two-level hierarchical structures and allow for the coefficients of both low-level and high-level units to vary over space. We assume that the spatially varying low-level coefficients follow the multivariate Gaussian process, and the spatially varying high-level coefficients follow the multivariate simultaneous autoregressive model that we develop by extending the standard simultaneous autoregressive model to incorporate multivariate data. We apply the proposed model to transaction data of houses sold in 2014 in a part of the city of Los Angeles. The results show that the proposed model predicts housing prices and fits the data effectively.

2017년 제5회 정오의 수학산책

일시: 10월 13일(금) 12:00 - 13:15

장소: KAIST 수리과학과 E6-1 3435호

강연자: 임선희 교수 (서울대)

제목: On the work of Mirzakhani : from counting geodesics to the classification of measures

(미르자카니의 결과들 : 측지선의 개수부터 측도의 분류까지)

내용: TBA

등록: 아래 링크를 통해 사전등록 바랍니다.

https://goo.gl/forms/DbAazUTgVwyc7vQp1

Right-angled Artin groups (RAAGs) are defined from finite simplicial graphs.

It is a fundamental question whether or not, given two RAAGs, there is an embedding from one group to the other.

Extension graphs are useful in solving this problem.

In this talk, I will briefly review RAAGs and extension graphs,

and show some results on solving the embeddability problem in RAAGs by using extension graphs.

In these two talks, I will introduce geometric problems related to mass in general relativity, such as a series of geometric inequalities, and

conjectures regarding a notion of quasi-local mass proposed by Bartnik.

The geometric inequalities we consider include the angular momentum-mass

inequality for axially symmetric initial data for the Einstein equations.

Note that the special cases treating maximal data have been proved by Dain

et al. Here I will explain how to reduce the general (non-maximal) case to

the known maximal case, and then discuss the solvability of the system of

Elliptic PDEs arose in the process, for near maximal case.

The second part of the talk will mainly provide an introduction to the

static/stationary metric extension conjectures, related to Bartnik

quasi-local mass. I will briefly discuss some known results for the static

metric extension conjecture by Anderson, Anderson/Khuri, Miao et al., and

show a local existence theorem for the solutions of axially symmetric,

stationary vacuum Einstein equations.

It was shown by Schaffer that for a dense class of compactly supported smooth initial data,the number of shock curves is finite for large time. It was not known if there is a smooth data for which the number of shock curves is unbounded. In this talk, using the structure of entropy solutions, one can construct a smooth initial data for which the number of shocks curves is infinite.

For a motivation of stochastic integrals, we start with stochastic (random) versions of deterministic systems and then we discuss the Ito integral as a non-anticipating (adapted) stochastic integral.

Based on the quantum decomposition of Brownian motion, we study the Skorohod integral and Stratonovich integral as anticipating (non-adapted) stochastic integrals.

Note that the Skorohod integral is the adjoint action of the Malliavin derivative.

I discuss recent progress in the classification of simply

connected Godeaux surfaces. There are two parts. The first

describes remarkable recent work of Isabel Stenger

(Kaiserslautern). She constructs an 8-dimensional rationally

parametrised moduli family of "general" surfaces, for which

the bicanonical pencil has no hyperelliptic fibres. Her

family contains a 7-dimensional subfamily with a single

hyperelliptic fibre and a 6-dimensional subfamily with two

hyperelliptic fibres. The second part describes my different

attack on the same problem, on which there has been some

progress over the last 25 years.