According to the fundamental principle of quantum mechanics, a quantum system of N particles can be described by a wave function that solves Schroedinger equation. When N is large, however, it is in general hard to derive results about the system rigorously. In this talk, some mathematically rigorous results about many-particle systems, especially Bose-Einstein condensates, will be introduced, and ideas behind the derivations of those results will be explained.

Elastography is to create a diagnostic image of the human body based on elastic stiffness variations, the motivation of which is to extend the doctors’ palpation examination. Three types of experiments have been proposed: static, dynamic, and transient where either the tissue is compressed or has a time harmonic or single pulse excitation, which lead to elliptic, Helmholtz, and wave equation, respectively.
 In each case, either ultrasound or MRI is used to determine interior displacement, which enables a high resolution images.
 In this talk, we introduce the experimental techniques, some results of mathematical analysis including uniqueness results and sensitivity results, and some modeling issues on viscoelastic media.

This talk will focus on mathematical approaches to image processing, specially using variational/PDE-based models. I will start with the review of the classical model of Total variation minimizing denoising model (Rudin-Osher-Fatemi) and Mumford-Shah functional. Various aspects of the models will be discussed and some new extensions will be presented.

We study the functional code defined on a projective algebraic variety X over a finite field. The minimum distance of this code is determined by computing the number of rational points of the intersection of X with all the hypersurfaces of a given degree. In the case where X is a non-degenerate Hermitian surface, A.B. Sørensen has formulated twenty years ago, a conjecture, which should give the exact value of the minimum distance of this code. In this talk, first of all we give a proof of Sorensen’s conjecture for quadratic surfaces and consequently the weight distribution associated to this code. Secondly we give the best upper bounds for the number of points of the quadratic section of a non-degenerate Hermitian solid and consequently the first weights, the weight distribution and an important property on the structure of the related code. Finally we generalize the results obtained for Hermitian surfaces and solids, to higher dimensional Hermitian varieties.

(Joint Topology & Discrete Math Seminar)

Inspired by the famous virtual Haken conjecture in 3--manifold theory, Gromov asked whether every one-ended word-hyperbolic group contains a surface group. One simple, but still captivating case, is when the word-hyperbolic group is given as the double of a free group with a cyclic edge group. In the first part of the talk, I will describe the polygonality of a word in a free group, and a relation between polygonality and Gromov's question. Polygonality is a combinatorial property, which is very much like solving a "topological jigsaw puzzle". In the second part, I will describe a reduction to a purely (finite) graph theoretic conjecture using the Whitehead graph. Part I is a joint word with Henry Wilton (Caltech).

We consider the problem Max-r-SAT, an extensively studied variant of the classic satisfiability problem. Given an instance of CNF (Conjunctive Normal Form) in which each clause consists of exactly r literals, we seek to find a satisfying truth assignment that maximizes the number of satisfied clauses. Even when r=2, the problem is intractable unless P=NP. Hence the next quest is how close we can get to optimality with moderate usage of compuation time/space.

We present an algorithm that decides, for every fixed r≥2 in time O(m) + 2^O(k2) whether a given set of m clauses of size r admits a truth assignment that satisfies at least ((2^r-1)m+k)/2^r clauses. Our algorithm is based on a polynomial-time preprocessing procedure that reduces a problem instance to an equivalent algebraically represented problem with O(k²) variables. Moreover, by combining another probabilistic argument with tools from graph matching theory and signed graphs, we show that an instance of Max-2-Sat either is a YES-instance or can be transformed into an equivalent instance of size at most 3k.

I will discuss two general methods for estimating multipoint Seshadri constants on surfaces, 
and review what the best known estimates are currently for finite generic subsets of the plane 
and compare them to a famous conjecture on Nagata. (This conjecture is equivalent to the 
statement that the multipoint Seshadri constant for r>9 generic points of the plane is $1/sqrt{r}$

An interesting problem is to classify finite sets of points in the 
plane according to the Hilbert functions of fat point subschemes supported at the points. For 
sets of  r≤8 points of the plane, this classification is now complete and is related to the classification of simple 8 element rank 3 matroids. The classification also allows one to 
determine the graded Betti numbers for any fat point subscheme supported at up to 6 points of
 the plane. I will discuss this work, which is joint with E. Guardo (for r≤6) and A. V. Geramita
 and J. Migliore (for 7≤r≤8). If time permits, I will mention additional recent related work joint
 with S. Cooper and Z. Teitler.

The purpose of this talk is to give a characterization of projective varieties  admitting linear 
projections from points which induce non-birational maps onto their images. 
 
As applications, we will give some results about the defining equations and their syzygies of 
smooth projective varieties, including the problem of bounding CAstelnuovo-Mumford regularity

In geometric group theory we are interested in studying finitely generated groups as geometric objects. A finitely generated group can be considered as a metric space when endowed with a `word metric'. This word metric depends on the choice of generating set but all such metrics are bilipschitz equivalent. Usually, however, finitely generated groups are studied up to `quasi-isometry'. This is a coarse version of bilipschitz equivalence that allows one to study these groups by studying proper geodesic metric spaces on which they act. I will give examples that show that these two notions are not equivalent. The proof will give a flavor of some of the various theorems and techniques used geometric group theory. 

I am going to give a survey on several basic problems of combinatorial
nature concerning random  Bernoulli matrices, including:

(1) The singularity problem: What is the probability that a random
Bernoulli matrix is singular ?

(2) The determinant problem: What is the typical value of the determinant ?

(3) The permanent problem: What is the typical value of the permanent ?

(4) The eigenvector problem: How does a typical eigenvector look like ?

If time allows, I will discuss connections to other areas of
mathematics, most importantly additive combinatorics.

Although there is a well-known characterization of Hilbert functions of reduced 0 –dimensional 
schemes in projective space(i.e., of fat point schemes where each point has multiplicity 1), 
there is no such formulation for fat point schemes where each point has some given multiplicity
 m>1. By tearing down the fat point scheme as a sequence of residuals with respect to lines, 
we obtain upper and lower bounds for the Hilbert function. Moreover, we give a simple criterion
 for when the bounds coincide, yielding a prescise calculation of the Hilbert fucntion. In this 
case, we also obtain upper and lower bounds on the graded Betti numbers for the ideal 
defining the fat point scheme. We’ll apply these results to combinatoriallly interesting examples.
 This is joint work with B. Harbourne and Z. Teitler

The study of vacuum is important in understanding the motion of gaseous stars or shallow water. Due to the degeneracy caused by vacuum, there are only a few mathematical results. We propose some interesting problems described by compressible Euler equations with vacuum and present the rigorous framework on how to study, in particular, in one space dimension.

I will discuss  a range of open problems and conjectures growing out of work on 0-
dimensional subschemes of projective spaces, including a conjecture of Nagata, the Segre-
Harbourne-Gimigliano-Hirschowitz Conjecture and a related folklore conjecture on the 
occurrence of curves of negative self-intersection on surfaces, the Ideal Generation 
Conjecture, and a question of Huneke on symbolic powers of ideals of points and related 
conjectures that grow out of my work on Huneke’s question.

We will give an introduction level survey on the works on collapsed manifolds with bounded sectional curvature and applications in Riemannian geometry by Cheeger-Gromov, Fukaya and others.

The goal is to provide a glimpse into the basic concerning wavelet expansions or wavelet representations. The attempt will be to describe some of the basics in this field via the approach of shift-invariant spaces. In view of the above, there will be essentially three (closely related) topics discussed in th lectures.

(i) Representation of elements in a Hilbert space: The analysis and synthesis operators. Riesz bases, frames, tight frames, dual systems, dual bases, the canonical dual system.

(ii) Shift-invariant spaces (one dimension, local PSI theory only): orthogonal projection, Fourier transform characterization, approximation orders, linear independence, factorization, dual systems.

(iii) wavelet systems: definition, the Haar wavelet, the sinc (Shannon) wavelet. Multiresolution analysis. Mallat's constructions. Daubechies' wavelets. Bi-orthogonal systems. Transfer operator analysis: smoothness, Riesz bases.

(iv) wavelet frames: dual Gramian analysis of SI systems. Quasi-affine systems. The characterization of wavelet frames. Tight wavelet frames from multiresolution. Extension principles. Compactly supported tight spline frames. CAP/CAMP/LCAMP representations.

The goal is to provide a glimpse into the basic concerning wavelet expansions or wavelet representations. The attempt will be to describe some of the basics in this field via the approach of shift-invariant spaces. In view of the above, there will be essentially three (closely related) topics discussed in th lectures.

(i) Representation of elements in a Hilbert space: The analysis and synthesis operators. Riesz bases, frames, tight frames, dual systems, dual bases, the canonical dual system.

(ii) Shift-invariant spaces (one dimension, local PSI theory only): orthogonal projection, Fourier transform characterization, approximation orders, linear independence, factorization, dual systems.

(iii) wavelet systems: definition, the Haar wavelet, the sinc (Shannon) wavelet. Multiresolution analysis. Mallat's constructions. Daubechies' wavelets. Bi-orthogonal systems. Transfer operator analysis: smoothness, Riesz bases.

(iv) wavelet frames: dual Gramian analysis of SI systems. Quasi-affine systems. The characterization of wavelet frames. Tight wavelet frames from multiresolution. Extension principles. Compactly supported tight spline frames. CAP/CAMP/LCAMP representations.

Ever since the pioneering work of Cox, Ross and Rubinstein, tree
models have been popular among asset pricing methods. On the other
hand, statistical estimation of parameters of tree models has not been
studied as much. In this paper, we use K Means Clustering method to
estimate the parameters of multinomial trees. By the weak convergence
property of multinomial
trees to continuous-time models, we show that this method can be in
turn used to estimate parameters in continuous time models,
illustrated by an example of jump-diffusion model.

Game theory is the study of multi-person decision-making. Game theory relies on the assumption of rationality in the analysis of inter-personal interactions. The rationality assumption allows the agents in the game theory to make inferences about the opponents’ choices and thus allows the game theorist to make predictions about the equilibrium play of the game. While the rationality assumption proves convenient and powerful in the analysis, its requirement is, in some contexts, overly restrictive and, in other contexts, not reasonable. The lecture introduces key concepts of rationality and uses them to explain a few well-known examples of game theory. We discuss the difficulties the rationality assumption causes in the explanation of real world phenomena using game theory.

In this talk I will describe the recent results on Betti tables of graded modules.
The Betti table describe numerical data related to a minimal free resolutions of a module.
The basic idea goes back to Hilbert who first proved existence of finite free resolutions.
Recently Boij and Soderberg made striking conjectures about the general shapes of Betti tables.
It allows to say (up to an integer multiple) which Betti tables actually exist.
These conjectures were subsequently proved by Eisenbud and Schreyer.

I will define all the basic notions concerning resolutions and Betti tables, no knowledge
of these questions will be assumed.

We introduce almost reverse lexicographic ideals in a polynomial ring over a field of arbitrary characteristic.Then we give a criterion for a given sequence of nonnegative integers to be the Hilbert function of an almost reverse lexicographic ideal in the polynomial ring. 

First, we will review syzygies and some related notions of a scheme $X$ in the projective space $\mathbb{P}^N$. We also consider how do the linear syzygies of $X$ behave under inner projection, i.e. the projection taking its center inside the original scheme. This is a natural 'projection' analogue to 'Restricting linear syzygies' due to D. Eisenbud, M. Green, K. Hulek, and S. Popescu which tells us some infomation about linear syzygies in case of taking linear sections. Finally, we will end this talk by considering some applications of it.

http://mathsci.kaist.ac.kr/~manifold/Arithmetics.html

The goal is to provide a glimpse into the basic concerning wavelet expansions or wavelet representations. The attempt will be to describe some of the basics in this field via the approach of shift-invariant spaces. In view of the above, there will be essentially three (closely related) topics discussed in th lectures.

(i) Representation of elements in a Hilbert space: The analysis and synthesis operators. Riesz bases, frames, tight frames, dual systems, dual bases, the canonical dual system.

(ii) Shift-invariant spaces (one dimension, local PSI theory only): orthogonal projection, Fourier transform characterization, approximation orders, linear independence, factorization, dual systems.

(iii) wavelet systems: definition, the Haar wavelet, the sinc (Shannon) wavelet. Multiresolution analysis. Mallat's constructions. Daubechies' wavelets. Bi-orthogonal systems. Transfer operator analysis: smoothness, Riesz bases.

(iv) wavelet frames: dual Gramian analysis of SI systems. Quasi-affine systems. The characterization of wavelet frames. Tight wavelet frames from multiresolution. Extension principles. Compactly supported tight spline frames. CAP/CAMP/LCAMP representations.

Uniformly hyperbolic systems are nowadays fairly well understood,
both from the topological and the ergodic point of view. Outside the hyperbolic
domain, two main phenomena occur: homoclinic tangencies and cycles involving
saddles with different indices. Homoclinic classes and chain compoments are the
natural candidates to replace hyperbolic basic sets in non-hyperbolic theory. Several
recent papers explore their ”hyperbolic-like” properties, many of which hold only
for generic dynamical systems. In this talk, we study how a C1-robust dynamic
property (i.e. a property that holds for a system and all C1 nearby ones) on the
underlying manifold would influence the behavior of the tangent map on the tangent
bundle.

We present a coefficient formula which provides some information about the polynomial map  when only incomplete information about a polynomial  is given. It is an integrative generalization and sharpening of several known results and has many applications, among these are:

1. The fact that polynomials  in just one variable have at most deg(P) roots.
2. Alon and Tarsi’s Combinatorial Nullstellensatz.
3. Chevalley and Warning’s Theorem about the number of simultaneous zeros of systems of polynomials over finite fields.
4. Ryser’s Permanent Formula.
5. Alon’s Permanent Lemma.
6. Alon and Tarsi’s Theorem about orientations and colorings of graphs.
7. Scheim’s formula for the number of edge n-colorings of planar n-regular graphs.
8. Alon, Friedland and Kalai’s Theorem about regular subgraphs.
9. Alon and Füredi’s Theorem about cube covers.
10. Cauchy and Davenport’s Theorem from additive number theory.
11. Erdős, Ginzburg and Ziv’s Theorem from additive number theory.

The goal is to provide a glimpse into the basic concerning wavelet expansions or wavelet representations. The attempt will be to describe some of the basics in this field via the approach of shift-invariant spaces. In view of the above, there will be essentially three (closely related) topics discussed in th lectures.

(i) Representation of elements in a Hilbert space: The analysis and synthesis operators. Riesz bases, frames, tight frames, dual systems, dual bases, the canonical dual system.

(ii) Shift-invariant spaces (one dimension, local PSI theory only): orthogonal projection, Fourier transform characterization, approximation orders, linear independence, factorization, dual systems.

(iii) wavelet systems: definition, the Haar wavelet, the sinc (Shannon) wavelet. Multiresolution analysis. Mallat's constructions. Daubechies' wavelets. Bi-orthogonal systems. Transfer operator analysis: smoothness, Riesz bases.

(iv) wavelet frames: dual Gramian analysis of SI systems. Quasi-affine systems. The characterization of wavelet frames. Tight wavelet frames from multiresolution. Extension principles. Compactly supported tight spline frames. CAP/CAMP/LCAMP representations.

Interval exchange maps generalize rotations and are characterized by

combinatorial and metric data.

The analysis of first return times on an interval (renormalisation)

leads to a remarkable extension

of the classical continued fraction algorithm (Rauzy, Veech, Zorich).

For almost all interval exchange maps T_0, with combinatorics of

genus g at least equal 2, we construct affine interval exchange maps

T which are semi-conjugate to T_0 and have a wandering interval.

This is a joint work with Pierre Moussa and Jean-Christophe Yoccoz

 The space of smooth rational cubic curves in projective space $\mathbb{P}^r$ ($r\geq 3$) is a smooth quasi-projective variety, which gives us an open subset of the corresponding Hilbert scheme, the moduli space of stable maps, or the moduli space of stable sheaves. By taking its closure, we obtain three compactifications $\mathbf{H}$, $\mathbf{M}$, and $\mathbf{S}$ respectively. In this talk, we compare these compactifications. First, we prove that $\mathbf{H}$ is the blow-up of $\mathbf{S}$ along a smooth subvariety parameterizing planar stable sheaves. Next we prove that $\mathbf{S}$ is obtained from $\mathbf{M}$ by three blow-ups followed by three blow-downs and the centers are described explicitly. Using this, we calculate the cohomology of $\mathbf{S}$.

For a smooth manifold M, there is a special function on M which has some nice properties ( called a Morse function.) In this seminar, we talk about  basic backgrounds of Morse theory and give some application in classical mechanics and symplectic geometry.

The goal is to provide a glimpse into the basic concerning wavelet expansions or wavelet representations. The attempt will be to describe some of the basics in this field via the approach of shift-invariant spaces. In view of the above, there will be essentially three (closely related) topics discussed in th lectures.

(i) Representation of elements in a Hilbert space: The analysis and synthesis operators. Riesz bases, frames, tight frames, dual systems, dual bases, the canonical dual system.

(ii) Shift-invariant spaces (one dimension, local PSI theory only): orthogonal projection, Fourier transform characterization, approximation orders, linear independence, factorization, dual systems.

(iii) wavelet systems: definition, the Haar wavelet, the sinc (Shannon) wavelet. Multiresolution analysis. Mallat's constructions. Daubechies' wavelets. Bi-orthogonal systems. Transfer operator analysis: smoothness, Riesz bases.

(iv) wavelet frames: dual Gramian analysis of SI systems. Quasi-affine systems. The characterization of wavelet frames. Tight wavelet frames from multiresolution. Extension principles. Compactly supported tight spline frames. CAP/CAMP/LCAMP representations.

The goal is to provide a glimpse into the basic concerning wavelet expansions or wavelet representations. The attempt will be to describe some of the basics in this field via the approach of shift-invariant spaces. In view of the above, there will be essentially three (closely related) topics discussed in th lectures.

(i) Representation of elements in a Hilbert space: The analysis and synthesis operators. Riesz bases, frames, tight frames, dual systems, dual bases, the canonical dual system.

(ii) Shift-invariant spaces (one dimension, local PSI theory only): orthogonal projection, Fourier transform characterization, approximation orders, linear independence, factorization, dual systems.

(iii) wavelet systems: definition, the Haar wavelet, the sinc (Shannon) wavelet. Multiresolution analysis. Mallat's constructions. Daubechies' wavelets. Bi-orthogonal systems. Transfer operator analysis: smoothness, Riesz bases.

(iv) wavelet frames: dual Gramian analysis of SI systems. Quasi-affine systems. The characterization of wavelet frames. Tight wavelet frames from multiresolution. Extension principles. Compactly supported tight spline frames. CAP/CAMP/LCAMP representations.

Hajos’ conjecture is false, and it seems that graphs without a subdivision of a big complete graph do not behave as well as those without a minor of a big complete graph.

In fact, the graph minor theorem (a proof of Wagner’s conjecture) is not true if we replace the minor relation by the subdivision relation. I.e, For every infinite sequence G₁,G₂, … of graphs, there exist distinct integers i_i is a minor of G_j, but if we replace ”minor” by ‘’subdivision”, this is no longer true.

This is partially because we do not really know what the graphs without a subdivision of a big complete graph look like.

In this talk, we shall discuss this issue. In particular, assuming some moderate connectivity condition, we can say something, which we will present in this talk.

Topics also include coloring graphs without a subdivision of a large complete graph, and some algorithmic aspects. Some of the results are joint work with Theo Muller.

In his 1985-paper introducing J-holomorphic curves into symplectic topology, Gromov proposed, among other things, to define invariants of symplectic manifolds as bordism classes of spaces of J-holomorphic curves. Over the years, the idea got transformed, through the work of Kontsevich as well many others, into a more general construction of Gromov-Witten invariants as intersection numbers in moduli spaces of stable maps. It is not only a subject of active research, but also deeply related to various field of mathematics including symplectic geometry, algebraic geometry and string theory.
In this talk, we will introduce the definition of the definition of Gromov-Witten invariants and give examples of applications to enumerative geometry.

http://mathsci.kaist.ac.kr/~manifold/Arithmetics.html

The current financial crisis is forcing thorough overhaul of not only the
practice but the theoretical framework of modern finance. We will talk
about how and why such crisis occurred and what kind of inadvertent, albeit
supporting, role modern finance played in creating it.. We will also
discuss some possible directions modern finance may go in. Our view is a
socio-historical as well mathematical financial one.

The goal is to provide a glimpse into the basic concerning wavelet expansions or wavelet representations. The attempt will be to describe some of the basics in this field via the approach of shift-invariant spaces. In view of the above, there will be essentially three (closely related) topics discussed in th lectures.

(i) Representation of elements in a Hilbert space: The analysis and synthesis operators. Riesz bases, frames, tight frames, dual systems, dual bases, the canonical dual system.

(ii) Shift-invariant spaces (one dimension, local PSI theory only): orthogonal projection, Fourier transform characterization, approximation orders, linear independence, factorization, dual systems.

(iii) wavelet systems: definition, the Haar wavelet, the sinc (Shannon) wavelet. Multiresolution analysis. Mallat's constructions. Daubechies' wavelets. Bi-orthogonal systems. Transfer operator analysis: smoothness, Riesz bases.

(iv) wavelet frames: dual Gramian analysis of SI systems. Quasi-affine systems. The characterization of wavelet frames. Tight wavelet frames from multiresolution. Extension principles. Compactly supported tight spline frames. CAP/CAMP/LCAMP representations.

The goal is to provide a glimpse into the basic concerning wavelet expansions or wavelet representations. The attempt will be to describe some of the basics in this field via the approach of shift-invariant spaces. In view of the above, there will be essentially three (closely related) topics discussed in th lectures.

(i) Representation of elements in a Hilbert space: The analysis and synthesis operators. Riesz bases, frames, tight frames, dual systems, dual bases, the canonical dual system.

(ii) Shift-invariant spaces (one dimension, local PSI theory only): orthogonal projection, Fourier transform characterization, approximation orders, linear independence, factorization, dual systems.

(iii) wavelet systems: definition, the Haar wavelet, the sinc (Shannon) wavelet. Multiresolution analysis. Mallat's constructions. Daubechies' wavelets. Bi-orthogonal systems. Transfer operator analysis: smoothness, Riesz bases.

(iv) wavelet frames: dual Gramian analysis of SI systems. Quasi-affine systems. The characterization of wavelet frames. Tight wavelet frames from multiresolution. Extension principles. Compactly supported tight spline frames. CAP/CAMP/LCAMP representations.

A mock modular form is the holomprphic part of a harmonic weak Maass form. In particular, Ramanujan's mock theta function and the generating series of the traces of singular moduli are mock modular forms of weights 1/2 and 3/2, respectively. In this talk, we will survey some recent progress in these mock modular forms. 

This lecture series provide a brief overview of finite element methods for electromagnetic propagations in both frequency and time domain. There are currently enormous developments in the understanding of mathematical theory of Maxwell's equations relevant to its numerical treatments. But one needs still more contributions on designs and their stable, efficient and robust simulation in particular, in following problems: scattering in unbounded domain, parameter optimization, ferromagnetics, nano optical devices.

In 1910, Max Dehn introduced a new method to construct 3-manifolds, which is now called by Dehn surgery. After nearly 50 years, two different mathematicians, Raymond Lickorish and Andrew Wallace, proved independently that any compact connected 3-manifold can be obtained from the 3-sphere by Dehn surgery on a link. For a hyperbolic knot, William Thurston showed that all but finitely many Dehn surgeries give hyperbolic 3-manifolds, in the late 1970s.
In this talk, we start with definitions of knot and link. And I will explain examples and theorems relative to Dehn surgery on knots and hyperbolic Dehn surgery. SnapPea will also be introduced, which is free software for hyperbolic 3-manifolds.

We first present how to extend Ramanujan’s method in partition congruences and show a congruence relation that the coefficients of the quotient of generating series for partitions and sums of squares satisfy. Then we observe a combinatorial interpretation of the product of them and see whether we could find some arithmetic properties of its coefficients.

일반 투자자들에게 인지도가 높은 투자수단인 ELS(주가연계증권)의 예를 통해 파생상품의 설계(Structuring)업무를 소개합니다. 현재 시장에서 유행하고 있는 상품의 예를 통해 설계시 고려해야 할 점들에 대해서도 알아봅니다. 또한 OTC파생상품 운용 분야의 다양한 업무들도 소개할 예정입니다.

.

http://mathsci.kaist.ac.kr/~manifold/Arithmetics.html

우선적으로 통계학의 시작과 본질이 무엇이며, 통계학이 사회발전에 어떻게 기여하고 있는가를 8가지로 나누어 조명하기로 한다. 다음으로 21세기 지식기반 정보화 사회에서 통계학의 역할이 무엇이며, 통계학의 도전에 대하여 밝히고자 한다. 이와 관련하여 데이터 기술(DT: Data Technology)을 정의하고, DT와 IT와의 관계, 수리과학과 통계학의 관계, 미래 지향적 통계학의 발전과정으로서의 DT의 역할 등에 대하여 설명한다.  마지막으로 통계학의 대표적인 응용사례로서 소프트웨어 산업에의 응용, 품질경영에의 응용, 여론조사 등에 대하여 설명하고, 통계학의 역할과 미래를 설명한다.

The second bounded cohomology of an amenable group is zero. On the other hand, the second bounded cohomology of a free group of rank greater than 1  is infinite dimensional as a vector space over R. Also it is known that no group which contains a free group on two generators can be amenable. 

It was conjectured that the second bounded cohomology of a discrete group is zero or infinite dimensional. Though it is shown that this conjecture is not true in general,  but it holds for a  group that has no nontrivial perfect normal subgroup, in particular, for a residually solvable group. So it seems natural to ask if there is some relationship between free groups and the dimension of the second bounded cohomology.

In this talk, we prove that the second bounded cohomology of a residually solvable group G is infinite dimensional if and only if there is a finite ordinal n such that its n-th commutator subgroup G^(n) is free of rank greater than 1. 

Rank-width of a graph G, denoted by rw(G) is a graph width parameter introduced by Oum and Seymour(2006). A random graph is a graph on n vertices such that two vertices are adjacent with the probability p independently at random. This model of random graphs was introduced by Erdös and Renýi (1960).
In this talk, I will give a brief introduction on those two different objects, random graphs and graph width parameters, of graph theory. Moreover, I will talk about some ideas of our result about rank-width of random graph G(n, p). Roughly speaking, we show that many random graphs have linear rank-width. Also, we find the sharp threshold of p = p(n) with respect to having linear rank-width. This is joint work with Choongbum Lee and Sang-il Oum.