초록: 인도 수학자 라마누잔이 관심을 가졌던 여러 문제들 중 몇 가지를 소개하면서 라마누잔이 어떻게 그 문제들에 접근 했는지를 알아(상상해)보고 이를 통해서 수학 현상 및 대상을 이해 할수 있는 몇 가지 방식을 논의해 본다. 자세한 증명 없이 진행 될 것이고 partition function, hypergeometric function(series), theta function, mock theta function 순으로 시간이허용하는 내에서 얘기를 풀어나갈 것 이다.

참가 등록: http://goo.gl/forms/GefR6twzAK

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.

The classical Schubert calculus arose from classical enumerative geometric problems, and is concerned with the study of the cohomology ring of complex Grassmannians, or generally, of homogeneous varieties G/P of general Lie types. One of the most central problems in this subject is to find a manifestly positive formula of the structure constants for the cup product of Schubert cohomology classes. On the other hand, a Gelfand-Cetlin polytope is a special convex polytope, occurring in many subjects such as representation theory, toric geometry, mathematical physics, and combinatorics. It turns out to be also closely related with Schubert calculus, as was studied by Kogan and Kiritchenko-Smirnov-Timorin. In these lectures, I will talk about the relationship between both sides. In Lecture I, I will give an introduction to the classical Schubert calculus. In Lecture II, I will review some basic properties of Gelfand-Cetlin polytopes and systems, as well as some relationships between them and Schubert calculus. In Lecutre III, IV, I will talk about toric degeneration of flag varieties, together with the transversal intersection of certain Schubert varieties, based on my joint work with DongSeon Hwang, Hwayoung Lee and Jae-Hyouk Lee.

The classical Schubert calculus arose from classical enumerative geometric problems, and is concerned with the study of the cohomology ring of complex Grassmannians, or generally, of homogeneous varieties G/P of general Lie types. One of the most central problems in this subject is to find a manifestly positive formula of the structure constants for the cup product of Schubert cohomology classes. On the other hand, a Gelfand-Cetlin polytope is a special convex polytope, occurring in many subjects such as representation theory, toric geometry, mathematical physics, and combinatorics. It turns out to be also closely related with Schubert calculus, as was studied by Kogan and Kiritchenko-Smirnov-Timorin. In these lectures, I will talk about the relationship between both sides. In Lecture I, I will give an introduction to the classical Schubert calculus. In Lecture II, I will review some basic properties of Gelfand-Cetlin polytopes and systems, as well as some relationships between them and Schubert calculus. In Lecutre III, IV, I will talk about toric degeneration of flag varieties, together with the transversal intersection of certain Schubert varieties, based on my joint work with DongSeon Hwang, Hwayoung Lee and Jae-Hyouk Lee.

TBA

TBA

 In this talk, I will overview the regularity theory for p(x)-Laplace equation. The p(x)-Laplace equation is denoted by
div(|Du|^{p(x)-2}Du)=0 in Omega,
where p(x):Omega to mr satisfies 1<p_-leq p(x)leq p_+<infty. This equation is a generalization of the p-Laplace equation div(|Du|^{p-2}Du)=0, where p is a constant in (1,infty).
One can expect that the regularity of solutions to p(x)-Laplace equation depends on the one of p(x). So, I will present the conditions on p(x) in order to obtain various regularities of solutions.
In addition, I will briefly introduce Calderon-Zygmund type estimates for elliptic equations in the setting of variable exponent Lebesgue space.

In this talk, we consider nonlinear elliptic equations involving the fractional Laplacian or the pseudo-relativistic Laplacian. We shall be concerned about existence and nonexistence results, and asymptotic profile of the solutions when a parameter of the equations is close to a critical value.

The classical Schubert calculus arose from classical enumerative geometric problems, and is concerned with the study of the cohomology ring of complex Grassmannians, or generally, of homogeneous varieties G/P of general Lie types. One of the most central problems in this subject is to find a manifestly positive formula of the structure constants for the cup product of Schubert cohomology classes. On the other hand, a Gelfand-Cetlin polytope is a special convex polytope, occurring in many subjects such as representation theory, toric geometry, mathematical physics, and combinatorics. It turns out to be also closely related with Schubert calculus, as was studied by Kogan and Kiritchenko-Smirnov-Timorin. In these lectures, I will talk about the relationship between both sides. In Lecture I, I will give an introduction to the classical Schubert calculus. In Lecture II, I will review some basic properties of Gelfand-Cetlin polytopes and systems, as well as some relationships between them and Schubert calculus. In Lecutre III, IV, I will talk about toric degeneration of flag varieties, together with the transversal intersection of certain Schubert varieties, based on my joint work with DongSeon Hwang, Hwayoung Lee and Jae-Hyouk Lee.

번역기 중 가장 좋다는 구글번역기에 '비단 골이 전부가 아니다'라는 문장을 넣으면, 'Silk is not the only goal'이라고 번역한다. 이라한 오역의 근원은, 번역을 언어의 구조, 성질, 패턴 연구가 아닌 엉뚱한 통계확률로 접근하기 때문이다. 컴퓨터가 발달하고 인간의 지능을 가진 로봇을 만드려는 인간의 노력은 인간이 사용하는 자연어를 기계언어로 번역(프로그램 용어로 compile)하는 문제를 핵심 과제로 부각시켰고 그 연구에 많은 돈과 인력이 투입되고 있다.

이 발표의 목적은 한글 고유의 조사와 서술어미 변형을 구현하는(representation) 수학적 방법론 제시 및 한글 문장의 구문분석에 응용 프로그램을 시연함으로써, 한글구문분석의 올바른 방향을 제시하고자 위함이다.

issue samples from the same tumor are heterogeneous. They consist of different subclones that can be characterized by differences in DNA nucleotide sequences and copy numbers on multiple loci. Inference on tumor heterogeneity thus involves the identification of the subclonal copy number and single nucleotide mutations at a selected set of loci. We carry out such inference on the basis of a Bayesian feature allocation model. We jointly model subclonal copy numbers and the corresponding allele sequences for the same loci, using three random matrices, L, Z and w to represent subclonal copy numbers (L), the number of sub- clonal variant alleles (Z) and the cellular fractions (w) of subclones in one or more tumor samples, respectively. The unknown number of subclones implies a random number of columns. More than one subclone indicates tumor heterogeneity. Using simulation studies and a real data analysis with next-generation sequencing data, we demonstrate how posterior inference on the subclonal structure is enhanced with the joint modeling of both structure and sequencing variants on subclonal genomes. An R package is available at http://cran.r-project.org/web/packages/ BayClone2/index.html. 

Abstract: issue samples from the same tumor are heterogeneous. They consist of different subclones that can be characterized by differences in DNA nucleotide sequences and copy numbers on multiple loci. Inference on tumor heterogeneity thus involves the identification of the subclonal copy number and single nucleotide mutations at a selected set of loci. We carry out such inference on the basis of a Bayesian feature allocation model. We jointly model subclonal copy numbers and the corresponding allele sequences for the same loci, using three random matrices, L, Z and w to represent subclonal copy numbers (L), the number of sub- clonal variant alleles (Z) and the cellular fractions (w) of subclones in one or more tumor samples, respectively. The unknown number of subclones implies a random number of columns. More than one subclone indicates tumor heterogeneity. Using simulation studies and a real data analysis with next-generation sequencing data, we demonstrate how posterior inference on the subclonal structure is enhanced with the joint modeling of both structure and sequencing variants on subclonal genomes. An R package is available at http://cran.r-project.org/web/packages/ BayClone2/index.html.

Abstract: issue samples from the same tumor are heterogeneous. They consist of different subclones that can be characterized by differences in DNA nucleotide sequences and copy numbers on multiple loci. Inference on tumor heterogeneity thus involves the identification of the subclonal copy number and single nucleotide mutations at a selected set of loci. We carry out such inference on the basis of a Bayesian feature allocation model. We jointly model subclonal copy numbers and the corresponding allele sequences for the same loci, using three random matrices, L, Z and w to represent subclonal copy numbers (L), the number of sub- clonal variant alleles (Z) and the cellular fractions (w) of subclones in one or more tumor samples, respectively. The unknown number of subclones implies a random number of columns. More than one subclone indicates tumor heterogeneity. Using simulation studies and a real data analysis with next-generation sequencing data, we demonstrate how posterior inference on the subclonal structure is enhanced with the joint modeling of both structure and sequencing variants on subclonal genomes. An R package is available at http://cran.r-project.org/web/packages/ BayClone2/index.html.

Issue samples from the same tumor are heterogeneous. They consist of different subclones that can be characterized by differences in DNA nucleotide sequences and copy numbers on multiple loci. Inference on tumor heterogeneity thus involves the identification of the subclonal copy number and single nucleotide mutations at a selected set of loci

We carry out such inference on the basis of a Bayesian feature allocation model. We jointly model subclonal copy numbers and the corresponding allele sequences for the same loci, using three random matrices, L, Z and w to represent subclonal copy numbers (L), the number of sub- clonal variant alleles (Z) and the cellular fractions (w) of subclones in one or more tumor samples, respectively. The unknown number of subclones implies a random number of columns. More than one subclone indicates tumor heterogeneity.

Using simulation studies and a real data analysis with next-generation sequencing data, we demonstrate how posterior inference on the subclonal structure is enhanced with the joint modeling of both structure and sequencing variants on subclonal genomes

An R package is available at http://cran.r-project.org/web/packages/ BayClone2/index.html.

 The disk embedding problem is of fundamental importance in the study of topology of dimension four. We will discuss backgrounds on its significance and difficulty, including why dimension four is intrinsically different from other dimensions, and then present some recent advances toward the existence and non-existence of embedded disks.

Primitive (birational) automorphisms of projective manifolds, which are irreducible au-
tomorphisms of manifolds, are natural objects in birational algebraic geometry. They are
also very closely related to complex dynamics of several variables. In fact, the dynamical
degrees of birational automorphisms (a kind of renement of a more classical notion of
topological entropy of automorphisms, tting very well with birational geometry), the relative dynamical degrees (their relative version) and the product formula (a kind of Kunneth type formula), introduced by Dinh-Sibony [DS], Dinh-Nyugen [DN], provide very powerful tools also in studying primitive birational automorphisms of manifolds.
In Lecture 1, I would like to give an overview of the basic notions (entropy, dynamical
degrees, relative dynamical degrees etc) and their basic properties with concrete applications for the study of primitive (birational) automorphisms. This lecture is, in some sense, an updated version of [Og].
In Lectures II, III, I would like to prove the well-denedness and the birational invariance
of dynamical degrees, the most basic property of the dynamical degree. The original proof ([DS]) is transcendental being based on some detailed analysis of currents. Here I explain a new purely algebro-geometric proof due to Truong ([Tr]), which is based on a precise form of Chow's moving lemma ([Ro]).
In Lectures IV, V, I would like to prove the product formula, the most fundamental
and useful property of relative dynamical degrees. Here I explain again an algebraic proof
following a guideline explained in [Tr], which is a modication of original analytic proofs
([DN], [DNT]) into an algebraic one again using a precise form of Chow's moving lemma.

References

[DS] Dinh, T.-C., Sibony, N., Une borne superieure de l'entropie topologique d'une application rationnelle, Ann. of Math., 161 (2005) 1637{1644. arXiv:math/0303271.

[DN] Dinh, T.-C., Nguyen V.-A., Comparison of dynamical degrees for semi-conjugate meromorphic maps, Comment. Math. Helv. 86 (2011) 817{840. arXiv:0903.2621.

[DNT] Dinh, T.-C., Nguyen V.-A., Truong, T.-T., On the dynamical degrees of meromorphic maps preserving a bration, Commun. Contemp. Math. 14 (2012) 18pp, arXiv: 1108.4792.

[Og] Oguiso, K., Some aspects of explicit birational geometry inspired by complex dynamics, Proceedings of the International Congress of Mathematicians, Seoul 2014 (Invited Lectures) Vol.II (2015), 695{721. arXiv:1404.2982.

[Ro] Roberts, J., Chow's moving lemma, in Algebraic geometry, Oslo 1970, F. Oort (ed.), WoltersNoordhoff, Publ. Groningnen (1972), 89{96.

[Tr] Truong, T.T., (Relative) dynamical degrees of rational maps over an algebraic closed eld, arXiv:1501.01523.

Primitive (birational) automorphisms of projective manifolds, which are irreducible au-
tomorphisms of manifolds, are natural objects in birational algebraic geometry. They are
also very closely related to complex dynamics of several variables. In fact, the dynamical
degrees of birational automorphisms (a kind of renement of a more classical notion of
topological entropy of automorphisms, tting very well with birational geometry), the relative dynamical degrees (their relative version) and the product formula (a kind of Kunneth type formula), introduced by Dinh-Sibony [DS], Dinh-Nyugen [DN], provide very powerful tools also in studying primitive birational automorphisms of manifolds.
In Lecture 1, I would like to give an overview of the basic notions (entropy, dynamical
degrees, relative dynamical degrees etc) and their basic properties with concrete applications for the study of primitive (birational) automorphisms. This lecture is, in some sense, an updated version of [Og].
In Lectures II, III, I would like to prove the well-denedness and the birational invariance
of dynamical degrees, the most basic property of the dynamical degree. The original proof ([DS]) is transcendental being based on some detailed analysis of currents. Here I explain a new purely algebro-geometric proof due to Truong ([Tr]), which is based on a precise form of Chow's moving lemma ([Ro]).
In Lectures IV, V, I would like to prove the product formula, the most fundamental
and useful property of relative dynamical degrees. Here I explain again an algebraic proof
following a guideline explained in [Tr], which is a modication of original analytic proofs
([DN], [DNT]) into an algebraic one again using a precise form of Chow's moving lemma.

References

[DS] Dinh, T.-C., Sibony, N., Une borne superieure de l'entropie topologique d'une application rationnelle, Ann. of Math., 161 (2005) 1637{1644. arXiv:math/0303271.

[DN] Dinh, T.-C., Nguyen V.-A., Comparison of dynamical degrees for semi-conjugate meromorphic maps, Comment. Math. Helv. 86 (2011) 817{840. arXiv:0903.2621.

[DNT] Dinh, T.-C., Nguyen V.-A., Truong, T.-T., On the dynamical degrees of meromorphic maps preserving a bration, Commun. Contemp. Math. 14 (2012) 18pp, arXiv: 1108.4792.

[Og] Oguiso, K., Some aspects of explicit birational geometry inspired by complex dynamics, Proceedings of the International Congress of Mathematicians, Seoul 2014 (Invited Lectures) Vol.II (2015), 695{721. arXiv:1404.2982.

[Ro] Roberts, J., Chow's moving lemma, in Algebraic geometry, Oslo 1970, F. Oort (ed.), WoltersNoordhoff, Publ. Groningnen (1972), 89{96.

[Tr] Truong, T.T., (Relative) dynamical degrees of rational maps over an algebraic closed eld, arXiv:1501.01523.

Primitive (birational) automorphisms of projective manifolds, which are irreducible au-
tomorphisms of manifolds, are natural objects in birational algebraic geometry. They are
also very closely related to complex dynamics of several variables. In fact, the dynamical
degrees of birational automorphisms (a kind of renement of a more classical notion of
topological entropy of automorphisms, tting very well with birational geometry), the relative dynamical degrees (their relative version) and the product formula (a kind of Kunneth type formula), introduced by Dinh-Sibony [DS], Dinh-Nyugen [DN], provide very powerful tools also in studying primitive birational automorphisms of manifolds.
In Lecture 1, I would like to give an overview of the basic notions (entropy, dynamical
degrees, relative dynamical degrees etc) and their basic properties with concrete applications for the study of primitive (birational) automorphisms. This lecture is, in some sense, an updated version of [Og].
In Lectures II, III, I would like to prove the well-denedness and the birational invariance
of dynamical degrees, the most basic property of the dynamical degree. The original proof ([DS]) is transcendental being based on some detailed analysis of currents. Here I explain a new purely algebro-geometric proof due to Truong ([Tr]), which is based on a precise form of Chow's moving lemma ([Ro]).
In Lectures IV, V, I would like to prove the product formula, the most fundamental
and useful property of relative dynamical degrees. Here I explain again an algebraic proof
following a guideline explained in [Tr], which is a modication of original analytic proofs
([DN], [DNT]) into an algebraic one again using a precise form of Chow's moving lemma.

References

[DS] Dinh, T.-C., Sibony, N., Une borne superieure de l'entropie topologique d'une application rationnelle, Ann. of Math., 161 (2005) 1637{1644. arXiv:math/0303271.

[DN] Dinh, T.-C., Nguyen V.-A., Comparison of dynamical degrees for semi-conjugate meromorphic maps, Comment. Math. Helv. 86 (2011) 817{840. arXiv:0903.2621.

[DNT] Dinh, T.-C., Nguyen V.-A., Truong, T.-T., On the dynamical degrees of meromorphic maps preserving a bration, Commun. Contemp. Math. 14 (2012) 18pp, arXiv: 1108.4792.

[Og] Oguiso, K., Some aspects of explicit birational geometry inspired by complex dynamics, Proceedings of the International Congress of Mathematicians, Seoul 2014 (Invited Lectures) Vol.II (2015), 695{721. arXiv:1404.2982.

[Ro] Roberts, J., Chow's moving lemma, in Algebraic geometry, Oslo 1970, F. Oort (ed.), WoltersNoordhoff, Publ. Groningnen (1972), 89{96.

[Tr] Truong, T.T., (Relative) dynamical degrees of rational maps over an algebraic closed eld, arXiv:1501.01523.

변화하는 상황 속에서 인과관계를 유추하고 빠르게 학습하는 능력은 인간과 같은 고등 생명체의 고유한 특성이다. 그러나, 이러한 인간의 일반지능(general intelligence)의 뇌 과학적 원리는 밝혀져 있지 않다. 본 세미나에서는 다양한 수학적 모델을 뇌 과학 연구에 접목시켜 학습 및 추론 과정을 조절하는 인간의 인지 제어 프로세스를 이해하는 인공지능-뇌과학 융합 연구를 소개하고, 산업, 공학,정신의학 분야로의 적용 가능성을 모색하고자 한다.

Primitive (birational) automorphisms of projective manifolds, which are irreducible au-
tomorphisms of manifolds, are natural objects in birational algebraic geometry. They are
also very closely related to complex dynamics of several variables. In fact, the dynamical
degrees of birational automorphisms (a kind of renement of a more classical notion of
topological entropy of automorphisms, tting very well with birational geometry), the relative dynamical degrees (their relative version) and the product formula (a kind of Kunneth type formula), introduced by Dinh-Sibony [DS], Dinh-Nyugen [DN], provide very powerful tools also in studying primitive birational automorphisms of manifolds.
In Lecture 1, I would like to give an overview of the basic notions (entropy, dynamical
degrees, relative dynamical degrees etc) and their basic properties with concrete applications for the study of primitive (birational) automorphisms. This lecture is, in some sense, an updated version of [Og].
In Lectures II, III, I would like to prove the well-denedness and the birational invariance
of dynamical degrees, the most basic property of the dynamical degree. The original proof ([DS]) is transcendental being based on some detailed analysis of currents. Here I explain a new purely algebro-geometric proof due to Truong ([Tr]), which is based on a precise form of Chow's moving lemma ([Ro]).
In Lectures IV, V, I would like to prove the product formula, the most fundamental
and useful property of relative dynamical degrees. Here I explain again an algebraic proof
following a guideline explained in [Tr], which is a modication of original analytic proofs
([DN], [DNT]) into an algebraic one again using a precise form of Chow's moving lemma.

References

[DS] Dinh, T.-C., Sibony, N., Une borne superieure de l'entropie topologique d'une application rationnelle, Ann. of Math., 161 (2005) 1637{1644. arXiv:math/0303271.

[DN] Dinh, T.-C., Nguyen V.-A., Comparison of dynamical degrees for semi-conjugate meromorphic maps, Comment. Math. Helv. 86 (2011) 817{840. arXiv:0903.2621.

[DNT] Dinh, T.-C., Nguyen V.-A., Truong, T.-T., On the dynamical degrees of meromorphic maps preserving a bration, Commun. Contemp. Math. 14 (2012) 18pp, arXiv: 1108.4792.

[Og] Oguiso, K., Some aspects of explicit birational geometry inspired by complex dynamics, Proceedings of the International Congress of Mathematicians, Seoul 2014 (Invited Lectures) Vol.II (2015), 695{721. arXiv:1404.2982.

[Ro] Roberts, J., Chow's moving lemma, in Algebraic geometry, Oslo 1970, F. Oort (ed.), WoltersNoordhoff, Publ. Groningnen (1972), 89{96.

[Tr] Truong, T.T., (Relative) dynamical degrees of rational maps over an algebraic closed eld, arXiv:1501.01523.

The study of laminar groups was motivated by the Thurston's universal circle theory. We show that certain laminar groups act on S^1 or S^2 as a convergence group, and discuss the connection to the Cannon's conjecture.

Primitive (birational) automorphisms of projective manifolds, which are irreducible au-
tomorphisms of manifolds, are natural objects in birational algebraic geometry. They are
also very closely related to complex dynamics of several variables. In fact, the dynamical
degrees of birational automorphisms (a kind of renement of a more classical notion of
topological entropy of automorphisms, tting very well with birational geometry), the relative dynamical degrees (their relative version) and the product formula (a kind of Kunneth type formula), introduced by Dinh-Sibony [DS], Dinh-Nyugen [DN], provide very powerful tools also in studying primitive birational automorphisms of manifolds.
In Lecture 1, I would like to give an overview of the basic notions (entropy, dynamical
degrees, relative dynamical degrees etc) and their basic properties with concrete applications for the study of primitive (birational) automorphisms. This lecture is, in some sense, an updated version of [Og].
In Lectures II, III, I would like to prove the well-denedness and the birational invariance
of dynamical degrees, the most basic property of the dynamical degree. The original proof ([DS]) is transcendental being based on some detailed analysis of currents. Here I explain a new purely algebro-geometric proof due to Truong ([Tr]), which is based on a precise form of Chow's moving lemma ([Ro]).
In Lectures IV, V, I would like to prove the product formula, the most fundamental
and useful property of relative dynamical degrees. Here I explain again an algebraic proof
following a guideline explained in [Tr], which is a modication of original analytic proofs
([DN], [DNT]) into an algebraic one again using a precise form of Chow's moving lemma.

References

[DS] Dinh, T.-C., Sibony, N., Une borne superieure de l'entropie topologique d'une application rationnelle, Ann. of Math., 161 (2005) 1637{1644. arXiv:math/0303271.

[DN] Dinh, T.-C., Nguyen V.-A., Comparison of dynamical degrees for semi-conjugate meromorphic maps, Comment. Math. Helv. 86 (2011) 817{840. arXiv:0903.2621.

[DNT] Dinh, T.-C., Nguyen V.-A., Truong, T.-T., On the dynamical degrees of meromorphic maps preserving a bration, Commun. Contemp. Math. 14 (2012) 18pp, arXiv: 1108.4792.

[Og] Oguiso, K., Some aspects of explicit birational geometry inspired by complex dynamics, Proceedings of the International Congress of Mathematicians, Seoul 2014 (Invited Lectures) Vol.II (2015), 695{721. arXiv:1404.2982.

[Ro] Roberts, J., Chow's moving lemma, in Algebraic geometry, Oslo 1970, F. Oort (ed.), WoltersNoordhoff, Publ. Groningnen (1972), 89{96.

[Tr] Truong, T.T., (Relative) dynamical degrees of rational maps over an algebraic closed eld, arXiv:1501.01523.

A toric variety, which arose in the field of algebraic geometry, of dimension n is a normal algebraic variety with an algebraic action of a complex torus (ℂ*)n having a dense orbit.

For a given toric variety X, the subset consisting of points with real coordinates of X is called a real toric variety Xℝ. In particular, if Xℝ is compact and smooth, it is called a real toric manifold.

The formula for the integral cohomology ring of toric varieties (and their generalizations) have been well established. Interestingly, the formula is quite simple; according to the formula, the ring is obtained as a quotient of a polynomial ring generated by only degree 2 elements, and it has no torsion.

Nevertheless, only little is known about the topology of real toric manifolds.

The topological structures of real toric manifolds are more complicated than those of toric manifolds.

For instance, every real toric manifold is not a simply connected while every toric manifold is simply connected.

Hence, in general, it is difficult to compute topological invariants of real toric manifolds.

Only the formula of ℤ2-cohomology ring has been established by Davis-Januszkiewicz.

In this talk, we introduce the notion of real toric space as a generalization of a real toric manifold. We provide a formula of the rational cohomology ring of real toric spaces, and discuss the existence of arbitrary torsion in the integral cohomology. Furthermore, we propose several topological classification problems for real toric spaces.

A graph is (d₁, …, d_r)-colorable if its vertex set can be partitioned into r sets V₁, …, V_r where the maximum degree of the graph induced by V_i is at most d_i for each i in {1, …, r}.

Time Reversal and Cross Correlation Techniques for Inverse
Source Problems

Abdul Wahab
Department of Bio & Brain Engineering
Korea Advanced Institute of Science and Technology

Abstract. We present time reversal and cross correlation based mathematical techniques
to resolve inverse source problems, where the aim is to nd the spatial support of radiating
sources from boundary wave measurements. We rst deal with temporally localized acoustic,
elastic and electromagnetic sources and present time reversal algorithms for their resolution.
Then, we localize stationary Gaussian noise sources using cross-correlation based statistical
tools. Both spatially correlated and uncorrelated noise sources will be considered. For
correlated sources, we sketch a procedure for retrieving their correlation structure. The
eciency and robustness of the developed algorithms are substantiated through numerical
illustrations.
Joint work. This research has been jointly conducted with Prof. H. Ammari (ETH-Zurich),
Dr. E. Bretin (INSA-Lyon), Prof. J. Garnier (Paris VII), Prof. T. Hayat (QAU, Islamabad),
Dr. R. Nawaz (CIIT, Islamabad), and Dr. A. Rasheed (LUMS, Lahore).
References
[1] H. Ammari, E. Bretin, J. Garnier and A. Wahab, Time reversal in attenuating acoustic
media, in Mathematical and Statistical Methods for Imaging, Contemporary Mathematics,
vol. 548, AMS, (2011), 151{163.
[2] H. Ammari, E. Bretin, J. Garnier and A.Wahab, Time reversal algorithms in viscoeastic
media, European Journal of Applied Mathematics, 24(4):(2013), 565{600.
[3] H. Ammari, E. Bretin, J. Garnier and A.Wahab, Noise source localization in an attenuating
medium, SIAM Journal of Applied Mathematics, 72(1):(2012), 317{336.
[4] H. Ammari, E. Bretin, J. Garnier, H. Kang, and H. Lee and A.Wahab, Mathematical
Methods in Elasticity Imaging, Princeton Series in Applied Mathematics, Princeton
University Press, NJ, USA, 2015.
[5] A. Wahab, A. Rasheed, T. Hayat and R. Nawaz, Electromagnetic time reversal algorithms
and source localization in lossy dielectric media, Communications in Theoretical
Physics, 62(6):(2014), 779-789

Algebraic varieties over finite fields have their associated zeta functions. André Weil conjectured that these functions have a list of properties, including an analogue of the Riemann hypothesis, and these Weil conjectures were proved by Pierre Deligne in the 1970s. Deligne used the so-called l-adic étale cohomology theory, but it is told as a folklore that Alexander Grothendieck was not fully satisfied by this Fields Prize winning work of Deligne for not having proven the conjectures using algebraic cycles.

In this talk, I will first roughly sketch the above historical background, and then talk about how one could revisit the Weil conjectures through algebraic cycles, via 40 years' modern mathematical developments from the late 1970s to now, spanning from higher algebraic K-theory, crystalline cohomology, motivic cohomology, intersection theory, triangulated categories of motives, by Daniel Quillen, Pierre Berthelot, Spencer Bloch, Luc Illusie, Vladimir Voevodsky, Kiran Kedlaya, Hélène Esnault, etc. The main theorem is my joint work with Amalendu Krishna of the Tata Institute of Fundamental Research.