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.

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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.

Nonlinear hydrodynamic instability theory has made notable successes
in predicting and indeed in providing control mechanisms for
transition to turbulence. The theory has had a major impact on
problems relevant to the mechanical, chemical and aeronautical
engineering. The theory concerns the solution of the 3D unsteady
Navier Stokes equations by a combination of analytical and numerical
means. Here we discuss the relevance of the theory to geophysical
flows and in particular discuss how river patterns and migrations can
be predicted mathematically. Several new nonlinear pde evolution
equations are derived and shown to reproduce several key features of
braided rivers.

In 1770, Lagrange proved that every nonnegative integer is the sum of four squares. Waring's problem is the generalization of Lagrange's theorem. More generally, we will introduce Waring's problem for polynomials and talk about the asymptotic order of a set of some polynomials.

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

We present a mathematical model of left heart governed by the partial differential equations. This heart is coupled with a lumped model of the whole circulatory system governed by the ordinary differential equations. The immersed boundary method is used to investigate the intracardiac blood flow and the cardiac valve motions of the normal circulation in humans. We investigate the intraventricular velocity field and the velocity curves over the mitral ring and across outflow tract. The pressure and flow are also measured in the left and right heart and the systemic and pulmonary arteries. The simulation results are comparable to the existing measurements.

Inverse problems are ill-posed and have virtually no solution. However, a-priori knowledge of the medium may reduce ill-posedness significantly. One such knowledge is smallness of the inclusion. I will talk about the method of small volume expansions to image small inclusions and its applications to emerging modalities of medical imaging such as MRElastography and Photo-acoustic Imaging.

The hypertoric manifold is defined by the hyperKahler analogue of symplectic toric manifolds. In usually, the toric manifold is a 2n-dim manifold with an n-dim torus action. On the other hand, the hypertoric manifold is a 4n-dim manifold with an n-dim torus action. However, we can apply the method of toric geometry or toric topology to analyze the hypertoric manifolds. In this talk, I introduce the hypertoric manifold and the method to analyze it from topological point of view, and prove that its equivariant diffeomorphism type is determined by the equivariant cohomology. 

Tropical geometry might loosely be described as algebraic geometry over the tropical semiring. It has deep connections to numerous branches of pure and applied mathematics, including algebraic geometry, combinatorics, and computational algebra. In this talk, I will explain the definition and properties of a tropical linear space, and how it is related to various areas of mathematics and computational biology.

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

Lovász and Plummer conjectured that there exists a fixed positive constant c such that every cubic n-vertex graph with no cutedge has at least 2^cn perfect matchings. Their conjecture has been verified for bipartite graphs by Voorhoeve and planar graphs by Chudnovsky and Seymour. We prove that every claw-free cubic n-vertex graph with no cutedge has more than 2^n/18 perfect matchings, thus verifying the conjecture for claw-free graphs.

This talk studies two examples of singular perturbations for particle systems. The first example is based on classical Tichinov theory for ODEs and applied to flocking. The second example uses a new non-classical averaging method and is applied to a KdV-Burgers type equation.

The similarity structure of certain convection or diffusion equations are well-known. The fundamental solutions of such problems are given explicitly and called self-similar solutions. The N-waves for the Burgers equation, the Gaussian for the heat equation and the Barenblatt solution for the porous medium equation are examples. These self-similar solutions have been played key roles in the theoretical development. However, there is no systematic approach to handle these similarity structures in a single frame. In this talk we introduce a method to derive similarity solution which is applicable to convection and diffusion equations.

It is a classical result due to Grothendieck that every vector bundles on the projective line is a direct sum of line bundles. Using this, there have been many attempts to understand vector bundles on the projective space, for example, by W.Barth and K.Hulek. In this talk, we introduce this idea in the case of smooth quadric surface. In the first half of the talk, we explain the basic notions in the algebraic geometry that will be used in the talk and recall several results on the projective space. In the second, we introduce the notion of jumping conics and prove that the set of jumping conics associated to a stable vector bundles on a smooth quadric surface forms a hypersurface in a 3-dimensional projective space. Using this, we explicitly describe the moduli spaces of stable vector bundles in two cases and see how these description can be applied to prove other classical results.