We show that any supersingular K3 surface of Artin invariant 1 in characteristic p=11, or >13 has an automorphism the entropy of which is the natural logarithm of a Salem number of the maximal degree 22, in particular nonliftable ones to characteristic zero. This is a joint work with Professors H. Esnault and K. Oguiso.

In this talk, Bayesian semiparametric methods for function estimation and model selection problems are presented. This talk is designed to provide graduate students and researchers with an introduction to Bayesian semiparametric inference. The orientation is methodological rather than theoretical, but such asymptotic theory as is necessary for a proper understanding and validating specific Bayesian methods will be also covered in detail. The materials will include three aspects of Bayesian inference, (i) Fundamentals (ii) Asymptotics and (iii) Advanced models, focusing on nonparametric and semiparametric methods. For function estimation, the Bayesian semiparametric models using Gaussian process priors are discussed. For model selection, Bayes factors are explained for dealing with lack of fit in regression and goodness of fit in density estimation problems.

  Classification of topological types of Calabi-Yau threefolds is still an open problem. In this talk, we explain how to construct some new Calabi-Yau threefolds via McKay correspondence. Such construction is based on classification of automorphism groups of smooth quintic threefolds.

  The F-conjecture is a long-standing conjecture on the structure of the nef cone of the moduli space of (pointed) curves. In this talk, I will explain how one can translate the $S_n$-invariant version of the conjecture into a feasibility problem in polyhedral geometry, which is purely computational. The main ingredients are the graphical algebra in classical invariant theory, embeddings into toric varieties, and the tropical compactification. Also I will propose a stronger statement which might be true for the genus zero case in general, and give some evidences. This is joint work with David Swinarski.

  In this talk we will introduce the varieties of minimal rational tangents (VMRT), and show how the projective geometry of VMRT plays a key role in understanding the geometry of projective manifolds.

  The F-conjecture is a long-standing conjecture on the structure of the nef cone of the moduli space of (pointed) curves. In this talk, I will explain how one can translate the $S_n$-invariant version of the conjecture into a feasibility problem in polyhedral geometry, which is purely computational. The main ingredients are the graphical algebra in classical invariant theory, embeddings into toric varieties, and the tropical compactification. Also I will propose a stronger statement which might be true for the genus zero case in general, and give some evidences. This is joint work with David Swinarski.

In this talk, aimed at a general mathematical audience, I will revisit the classical problem of simplifying polynomials in one variable by Tschirnhaus transformations. Surprisingly, many of the old questions are still open. I will restate them in geometric terms and discuss recent progress in this area.

Let $overline{M_{g, n}}$ be the moduli space of
stable curves of genus $g$ with $n$ marked points.
It is a classical problem in algebraic geometry
to determine which of these spaces are rational over $mathbb C$. 
In this talk, based on joint work with Matthieu Florence,
I will address the rationality problem for twisted forms
of $overline{M_{g, n}}$.  Twisted forms of $overline{M_{g, n}}$
are of interest because they shed light on the arithmetic
geometry of $overline{M_{g, n}}$,
and because they are coarse moduli spaces for natural moduli problems
in their own right. 

A classical result of Yu.~I.~Manin and P.~Swinnerton-Dyer
asserts that every form of $overline{M_{0, 5}}$ is rational.
(Recall that the $F$-forms $overline{M_{0, 5}}$ are precisely
the del Pezzo surfaces of degree $5$ defined over $F$.) 
Mathieu Florence and I recently showed that this theorem
continues to hold for forms of  $overline{M_{0, n}}$ if
$n geq 5$ is odd but fails if $n$ is even.  We also have 
similar results for forms of $overline{M_{g, n}}$, 
where $g leq 5$ (and $n$ is small). Our approach is based on reducing
the rationality problem for twisted forms of $overline{M_{g, n}}$
to certain special cases of the Noether problem. In the talk I plan
to survey some background material on $overline{M_{g, n}}$ and 
twisted forms, and explain how the Noether problem 
arises in this context. 

일시: 5월 20일(금) 12:00 ~ 13:15

장소: 자연과학동 E6-1 3435호

주제: 기하학자의 꿈 - 공간의 분류와 대칭성

강연자: 김영훈 교수 (서울대)

Abstract:  유클리드, 데카르트, 리만, 클라인, 그로덴딕을 통해 기하학의 역사를 조망해보고 공간의 분류와 대칭성의 탐구라는 기하학자의 꿈이 어떤 것인지 얘기하고자 한다. 구체적인 예로서 곡선의 경우에 대해 생각해 본다.

참가 링크: http://goo.gl/forms/mqr7RHvXl7

In 2009, US National Research Council of the National Academies published a report, called “A New Biology for the 21st Century”. One of the major emphases in the New Biology is the integration between biology and mathematics. As explosion of biological measurements takes place in biology due to rapid technology development in recent years, the challenge lies in how to connect and make sense of the massive experimental data collected in various forms at different spatial and temporal scales. Mathematical modeling is becoming an increasingly important tool that enables better understanding of the complex data in biology. In this talk, I will use research carried out in my group as examples to illustrate how mathematics can help discover new knowledge in biology as well as how biological models introduce new challenges and opportunities in mathematics.

In a family of $S_{d+1}$-fields ($d=2,3,4$), we obtain the conjectured upper and 

lower bounds of the residues of Dedekind zeta functions except for a density zero set. 

For $S_5$-fields, we need to assume the strong Artin conjecture. 

We also show that there exists an infinite family of number fields with the upper and lower bounds, resp.

This is a joint work with Henry Kim

The classic Plateau's problem asks if given a smooth simple closed curve in space, does there exist a smooth orientable (embedded) surface-with-boundary spanning that curve, and having least area among all surfaces spanning the given curve? While the answer is in the affirmative, to solve Plateau's problem requires developing the theory of currents; these are generalized orientable surfaces-with-boundary. In this talk we study Plateau's problem for $C^{1,alpha}$ tangentially immersed boundaries.

 Harmonic classes for a simplicial complex are the kernel of its combinatorial Laplacians. They contain more information than homology classes for various data analytic purposes. 

In this talk, we will discuss harmonic classes as a new method for analyzing simplicial complexes associated with a given data set. As an application, we will present an intriguing connection between harmonic classes and effective conductance for a network.

  2016년 3월, 세계를 떠들썩하게 만든 세기의 대결이 펼쳐졌다. 구글 딥마인드(Google DeepMind)가 선보인 인공지능 알파고(AlphaGo)와 바둑기사 이세돌 9단의 5번기였다. "구글 딥마인드 챌린지 매치”의 뚜껑을 열어보니 결과는 알파고의 4대 1 승리, 이세돌 9단의 완패로 대결의 막을 내리게 된다.

 본 강연에서는, 알파고의 기보를 통해 알파고가 세기의 대결에서 보여준 여러 능력들을 ‘바둑이론’을 통해 점검하고, 알파고를 통해 구글이 궁극적으로 성취하고 도달하고자 하는 지향점을 검토하며, 구글 인공지능 프로젝트에 바둑적 관점이 제공할 수 있는 시사점은 무엇인지 말하고자 한다.

A new algorithm will be presented for the path tsp, with an improved analysis and ratio. After the starting idea of deleting some edges of Christofides’ trees, we do parity correction and eventual reconnection, taking the salesman to travel through a linear program determining the conditional probabilities for some of his choices; through matroid partition of a set of different matroids for a better choice of his initial spanning trees; and through some other adventures and misadventures.
The proofs proceed by global and intuitively justified steps, where the trees do not hide the forests.

One more pleasant piece of news is that we get closer to the conjectured approximation ratio of 3/2, and a hopefully last misadventure before finishing up this problem is that we still have to add 1/34 to this ratio, and also for the integrality gap. (The previous result was 8/5 with slight improvements.)

This is joint work with Anke van Zuylen.

현재 의료와 헬스케어 산업은 변혁의 시기를 지나고 있다. 의료와 헬스케어 분야를 근본적으로 변화시키고 있는 이 변혁의 시발점은 기하급수적인 디지털 기술의 발전이다. '디지털 헬스케어’ 라고 불리는 이 새로운 분야는 기존의 헬스케어 및 의료 기술이 디지털 기술과 융합되면서 태동되었다. 스마트폰, 웨어러블 디바이스, 사물인터넷, 인공지능, 3D 프린터, 클라우드 컴퓨팅, 증강 현실 등의 디지털 기술 혁신들은 헬스케어에 접목되어 기존의 건강 관리와 질병 진단 및 치료의 판도를 뒤집어 놓고 있다. 디지털 헬스케어 분야에서 어떠한 변화가 일어나고 있으며, 이 변화를 우리는 어떠한 시각으로 바라봐야 할지, 또한 우리가 맞이할 새로운 숙제와 이슈들은 무엇인지를 포괄적으로 살펴본다. 

Combinatorial optimization has recently discovered new usages of probability theory, information theory, algebra, semidefinit programming, etc. This allows addressing the problems arising in new application areas such as the management of very large networks, which require new tools. A new layer of results make use of several classical methods at the same time, in new ways, combined with newly developed arguments.

After a brief panorama of this evolution, I would like to show the new place of the best-known, classical combinatorial optimization tools in this jungle: matroids, matchings, elementary probabilities, polyhedra, linear programming.

More concretely, I try to demonstrate on the example of the Travelling Salesman Problem, how strong meta-methods may predict possibilities, and then be replaced by better suited elementary methods. The pillars of combinatorial optimization such as matroid intersection, matchings, T-joins, graph connectivity, used in parallel with elements of freshmen's probabilities, and linear programming, appropriately merged with newly developed ideas tailored for the problems, may not only replace difficult generic methods, but essentially improve the results. I would like to show how this has happened with various versions of the TSP problem in the past years (see results of Gharan, Saberi, Singh, Mömke and Svensson, and several recent results of Anke van Zuylen, Jens Vygen and the speaker), essentially improving the approximation ratios of algorithms.

The green sea turtle Chelonia midas travels for thousands of miles from the coast of Brazil to a small island in the Atlantic Ocean, Ascension Island. There the turtles lay their eggs into the warm sand on the beach. It is a classic scientific challenge to understand the navigational skills of the turtles and several orienteering mechanisms are discussed, such as geomagnetic information, chemotaxis, atlantic flow patterns etc.
In this talk I will present a mathematical model for the homing of sea turtles and discuss how it can be used to identify the navigational mechanisms of sea turtles. (joint work with K.J. Painter)

We will present an approach to the Bloch-Beilinson filtration via Voevodsky’s triangulated category of mixed motives DM.

We will present an approach to the Bloch-Beilinson filtration via Voevodsky’s triangulated category of mixed motives DM.

작년 8월 시작된 중국의 외환과 금융의 불안은 글로벌경제에 위험요인으로 부각되고 있다. 미 연준은 통화정책의 정상화 일정을 연기하고 ECB를 비롯한 유럽의 중앙은행과 일본 중앙은행은 마이너스 금리정책을 시행하고 있다.

현재 중국이 겪는 어려움에는 구조적인 문제가 내재되어 있다. 고성장시대의 마감과 기업부문을 중심으로 늘어난 부채가 거시경제의 불균형을 가져오는 가운데 금융시장과 외환시장은 대외여건에 취약한 모습을 보인다.

중장기적으로는 저성장 하에서 부채에 의존한 과잉투자의 문제가 내재되고 있으나 단기적으로는 글로벌경제와 통합하는 과정에서 외환 및 통화정책수행 시 통화당국이 안게 되는 불가능한 삼위일체(Impossible Trinity)에 근거한다.

중국의 통화국제화는 통상 先 개혁 後 국제화를 추진했던 나라들과 다른 경로를 취하고 있다. 즉 호주, 일본, 싱가폴의 선례를 따르면 금융시장발전, 중앙은행 독립성 강화, 규제 선진화, 자본 개방을 단행하고 싱가폴을 제외하면 변동환율제도로 이행한 후 정부가 통화국제화를 추진하거나 용인하였다. 한편 중국의 경우 위안화 국제화를 추진하는 과정에서 상당한 정도의 금융 개방이 단행되었으나 그에 걸맞은 수준의 금융개혁은 여전히 숙제로 남아있다.

한편 중국의 외환 불안이 일어난 시점부터 원화환율과 엔화환율은 기존의 패턴과 다른 모습을 보이고 있는데 이와 같이 변화된 환율의 패턴은 투자자들이 위험자산인 원화 자산을 매도하고 안전자산으로서 엔화 자산에 대한 수요를 늘리게 되는 안전자산선호(Flight to quality)현상이 일어나고 있기 때문이다. 이 패턴은 중국의 외환 불안이 수면 아래로 잠복할 때 다시 안전자산선호는 위험자산선호로 그 모드가 바뀌게 되어 원화환율은 절상압력을, 반대로 엔화환율은 절하 압력을 받게 되어 반대로 작동할 것으로 기대된다.

중국당국이 당면한 도전은 자본유출압력을 슬기롭게 극복하는 것이나 현실은 녹녹하지 않다. 한때 4조달러에 육박했던 외환보유액은 2년도 채 안되 8,000억불 가까이 줄어들었다. 혹자는 대외 자산이 대외부채를 1조7천억불 가까이 초과하고 여전히 3조2천억불이 넘는 보유외환으로 왜 시장 신뢰를 담보하기 어려운 지 의문을 가질 수 있다. 그것은 비록 경제 전체로서 미스 매치는 없으나 금융회사와 기업과 같이 특정 부문에 잠재적 위험이 집중되었기 때문이다. 더욱이 국제기준에서 볼 때 열악한 기업부문의 재무상태는 잠재 부실의 문제를 안고 있다.

향후 차이나 리스크는 대외적으로는 미 연준의 통화정책 방향에, 대내적으로는 중국경제의 성장 경로에 영향을 받을 것으로 보인다. 경제주체들이 왜곡된 동기부여를 가지지 않도록 하는 한편 시장 신뢰를 조성하는 중국당국의 높은 관리 역량이 필요한 시점이다.

작년 8월 시작된 중국의 외환과 금융의 불안은 글로벌경제에 위험요인으로 부각되고 있다. 미 연준은 통화정책의 정상화 일정을 연기하고 ECB를 비롯한 유럽의 중앙은행과 일본 중앙은행은 마이너스 금리정책을 시행하고 있다.

현재 중국이 겪는 어려움에는 구조적인 문제가 내재되어 있다. 고성장시대의 마감과 기업부문을 중심으로 늘어난 부채가 거시경제의 불균형을 가져오는 가운데 금융시장과 외환시장은 대외여건에 취약한 모습을 보인다.

중장기적으로는 저성장 하에서 부채에 의존한 과잉투자의 문제가 내재되고 있으나 단기적으로는 글로벌경제와 통합하는 과정에서 외환 및 통화정책수행 시 통화당국이 안게 되는 불가능한 삼위일체(Impossible Trinity)에 근거한다.

중국의 통화국제화는 통상 先 개혁 後 국제화를 추진했던 나라들과 다른 경로를 취하고 있다. 즉 호주, 일본, 싱가폴의 선례를 따르면 금융시장발전, 중앙은행 독립성 강화, 규제 선진화, 자본 개방을 단행하고 싱가폴을 제외하면 변동환율제도로 이행한 후 정부가 통화국제화를 추진하거나 용인하였다. 한편 중국의 경우 위안화 국제화를 추진하는 과정에서 상당한 정도의 금융 개방이 단행되었으나 그에 걸맞은 수준의 금융개혁은 여전히 숙제로 남아있다.

한편 중국의 외환 불안이 일어난 시점부터 원화환율과 엔화환율은 기존의 패턴과 다른 모습을 보이고 있는데 이와 같이 변화된 환율의 패턴은 투자자들이 위험자산인 원화 자산을 매도하고 안전자산으로서 엔화 자산에 대한 수요를 늘리게 되는 안전자산선호(Flight to quality)현상이 일어나고 있기 때문이다. 이 패턴은 중국의 외환 불안이 수면 아래로 잠복할 때 다시 안전자산선호는 위험자산선호로 그 모드가 바뀌게 되어 원화환율은 절상압력을, 반대로 엔화환율은 절하 압력을 받게 되어 반대로 작동할 것으로 기대된다.

중국당국이 당면한 도전은 자본유출압력을 슬기롭게 극복하는 것이나 현실은 녹녹하지 않다. 한때 4조달러에 육박했던 외환보유액은 2년도 채 안되 8,000억불 가까이 줄어들었다. 혹자는 대외 자산이 대외부채를 1조7천억불 가까이 초과하고 여전히 3조2천억불이 넘는 보유외환으로 왜 시장 신뢰를 담보하기 어려운 지 의문을 가질 수 있다. 그것은 비록 경제 전체로서 미스 매치는 없으나 금융회사와 기업과 같이 특정 부문에 잠재적 위험이 집중되었기 때문이다. 더욱이 국제기준에서 볼 때 열악한 기업부문의 재무상태는 잠재 부실의 문제를 안고 있다.

향후 차이나 리스크는 대외적으로는 미 연준의 통화정책 방향에, 대내적으로는 중국경제의 성장 경로에 영향을 받을 것으로 보인다. 경제주체들이 왜곡된 동기부여를 가지지 않도록 하는 한편 시장 신뢰를 조성하는 중국당국의 높은 관리 역량이 필요한 시점이다.

We give a very brief summary of the recent developments in the theory of 3-dimensional manifolds, and discuss how it is connected to Thurston's universal circle program. At the end, we suggest future directions to complete Thurston's picture.

Roughly speaking, Shimura varieties are a certain generalisation of modular curves and Siegel modular varieties (moduli spaces of principally polarised abelian varieties), and have been one of the central objects in number theory. Especially important for number theoretic applications is p-adic analytic study of Shimura varieties, which has features analogous to complex analytic geometry and scheme theory.

Recently, there have been many exciting new developments in the study of p-adic geometry and cohomology of Shimura varieties. Underlying many of these developments is the new technique to study “infinite-level” Shimura varieties as p-adic analytic spaces, as well as new developments in p-adic Hodge theory (both of which are built upon P. Scholze’s theory of perfectoid spaces).

For the majority of the talk, we use the examples of modular tower and moduli of principally polarised abelian varieties to illustrate the geometric results on Shimura varieties proved in the recent paper by Caraiani and Scholze, which “decomposes" certain Shimura varieties in a way that gives a meaningful decomposition of the cohomology. We will conclude by describing my work in progress to generalise this result for unramified Hodge-type Shimura varieties. We will not necessarily strive to give a precise definition or construction of each main object, and may settle for giving some simple examples (even at the risk of over-simplifying).

Thurston showed that sometimes one can construct a pseudo-Anosov surface diffeomorphism from a given one-dimensional dynamical system. We discuss examples of this construction and see how general this could be made.

Many important question in the theory of surfaces and in algebraic geometry have been solved thanks to explicit constructions of algebraic surfaces as abelian coverings branched over special configurations of lines. After recalling  the classical configurations (Pappus, Desargues, Fano, Hesse) I shall describe simple equations for such surfaces, as  the Fermat, and Hirzebruch-Kummer coverings. As the configuration of lines becomes special some interesting geometry shows up, as in the case of the six lines of a complete quadrangle, relted to the Del Pezzo surface of degree 5 and its icosahedral symmetry. After mentioning many important such  examples and applications, by  several authors, I shall concentrate on a  recent simple series of such surfaces, studied in my joint work with Ingrid Bauer and Michael Dettweiler, discussing  new results and quite general open questions.

The bounded negativity conjecture predicts that on a smooth complex surface X, there is a bound b such that for every (reduced) curve C on X, the self-intersection of C satisfies C^2 >b. It was stated one century ago by the Italian geometers but it is still quite open. Recently people introduced new tools -called Harbourne constants- in order to study that conjecture. In these lectures, we will explain these tools and give an overview of the present knowledge on that conjecture.

Sparsity and compressive sensing have had a tremendous impact in science, technology, medicine, imaging, machine learning and now, in solving multiscale problems in applied partial differential equations, developing sparse bases for Elliptic eigenspaces and connections with viscosity solutions to Hamilton-Jacobi equations. ℓ1 and related optimization solvers are a key tool in this area. The special nature of this functional allows for very fast solvers: ℓ1 actually forgives and forgets errors in Bregman iterative methods.
I will describe simple, fast algorithms and new applications ranging from image processing, machine learning to sparse dynamics for PDE.

The bounded negativity conjecture predicts that on a smooth complex surface X, there is a bound b such that for every (reduced) curve C on X, the self-intersection of C satisfies C^2 >b. It was stated one century ago by the Italian geometers but it is still quite open. Recently people introduced new tools -called Harbourne constants- in order to study that conjecture. In these lectures, we will explain these tools and give an overview of the present knowledge on that conjecture.

The bounded negativity conjecture predicts that on a smooth complex surface X, there is a bound b such that for every (reduced) curve C on X, the self-intersection of C satisfies C^2 >b. It was stated one century ago by the Italian geometers but it is still quite open. Recently people introduced new tools -called Harbourne constants- in order to study that conjecture. In these lectures, we will explain these tools and give an overview of the present knowledge on that conjecture.

Anisotropic diffusion describes random walk with different diffusivities in different directions. The fully anisotropic formulation, sometimes called myopic random walk, is based on active random walk of individuals and it has a Fokker-Planck like of diffusion term. In this talk I give intuitive reasons for anisotropic diffusion and I present scaling limits of kinetic equations which, quite naturally, lead to the fully anisotropic formulation. I show how this framework can be used for the modelling of sea turtle navigation, wolf movement, and brain tumor spread.

Linear programming has played a key role in the study of algorithms for combinatorial optimization problems. In the field of approximation algorithms, this is well illustrated by the uncapacitated facility location problem. A variety of algorithmic methodologies, such as LP-rounding and primal-dual method, have been applied to and evolved from algorithms for this problem. Unfortunately, this collection of powerful algorithmic techniques had not yet been applicable to the more general capacitated facility location problem. In fact, all of the known algorithms with good performance guarantees were based on a single technique, local search, and no linear programming relaxation was known to efficiently approximate the problem.

현재와 같은 저금리 환경에서 장기 투자기관들은 높은 수익률을 얻을 수 있는 구조화 채권에 투자하고 있다. 본 세미나에서는 거래 참여자들(구조화 채권 매수자, 구조화 채권의 발행자, 구조화 스왑 거래자)의 실질적 동기와 Hedge 거래 방식을 관찰하여 구조화 채권 운용 원리를 분석하고자 한다. 또한 구조화 채권에서 내재된 임의 상환 옵션의 행사 원리와 구조화 채권 발행 수익을 인식하는 방식의 관계에 대해 알아본다.