- Age-at-death, time-until-death random variables
- Some parametric survival models
- Tabulation of basic mortality functions

Hamiltonian cycle in a graph G is a cycle, which contains every vertex of the graph G. The problem of existence of a hamiltonian cycle in a graph is a well known NP-complete problem. While some theoretical necessary and sufficient conditions are known, to date, but no practical characterization of hamiltonian graphs has been found. There are several ways how to generalize the notion of a hamiltonian cycle.

For any integer r>1, an r-trestle is a 2-connected graph F with maximum degree ∆(F)≤ r. We say that a graph G has an r-trestle if G contains a spanning subgraph which is an r-trestle. This concept can be viewed as an interesting variation on the notion of Hamilton cycle. Another such variation is a concept of k-walks, where a k-walk in a graph G is a closed spanning walk visiting each vertex at most k times, where k is a positive integer.

We present several results and problems concerning mainly with k-walks and r-trestles and relations between them.

- Cash Flow Modeling: Concepts and Applications
- Interest Rates: Definitions and relationships
- Valuation Principles

In an attempt to transfer the loss rate risks in motor insurance to the capital market, we use the tranche technique to hedge the motor insurance risks. Though this application is new, this transaction is based on a concept similar to CDOs. The pricing methods of the tranches are illustrated, and possible suggestions to improve the pricing method and the design of these new securities follow.

It is a task of central importance in understanding an ambient closed
3-manifold to find an essential surface - that is, a
$\pi_1$-injectively immersed closed surface of non-positive Euler
characteristic. Extending this question, one naturally asks what
conditions guarantee or prohibit the existence of essential surfaces
in non-positively curved cubical complexes. In this talk, I will
survey results on this latter question specialized in the case of
right-angled Artin groups. This talk will be self-contained and all
the necessary definitions will be given.

- Actuary의 정의와 그 역할
- Actuary 시험제도 및 처우
- 우수 Actuary들을 양성할 수 있는 국내 대학에서의 교육 프로그램 개발 방안

A total coloring is a combination of a vertex coloring and an edge coloring of a graph: every vertex and every edge is assigned a color and any two adjacent/incident objects must receive distinct colors. One of the main open problems in the area of graph colorings is the Total Coloring Conjecture of Behzad and Vizing from the 1960’s asserting that every graph has a total coloring with at most D+2 colors where D is its maximum degree.

Fractional colorings are linear relaxation of ordinary colorings. In the setting of fractional total colorings, the Total Coloring Conjecture was proven by Kilakos and Reed. In the talk, we will present a proof of the following recent conjecture of Reed:

For every real ε>0 and integer D, there exists g such that every graph with maximum degree D and girth at least g has total fractional chromatic number at most D+1+ε.

For D=3 and D=4,6,8,10,…, we prove the conjecture in a stronger
form: there exists g such that every graph with maximum degree D and girth at least g has total fractional chromatic number equal to D+1.

Joint work with Tomás Kaiser, František Kardoš, Andrew King and Jean-Sebastien Sereni.

We present recent developments of syzygy theory and applications of Koszul cohomology to 
the geometry of complex projective varieties. The lectures will begin with a review of definitions
 and basic properties, examples, and will end with a discussion on the curve case. Voisin's 
description of Koszul cohomology in terms of Hilbert schemes will also be discussed during 
the lectures.
The main source is the joint book with J. Nagel with the same title, published by the AMS 
(Univ. Lect. Series 52)

We present recent developments of syzygy theory and applications of Koszul cohomology to
the geometry of complex projective varieties. The lectures will begin with a review of definitions and basic properties, examples, and will end with a discussion on the curve case. Voisin's
description of Koszul cohomology in terms of Hilbert schemes will also be discussed during the lectures.
The main source is the joint book with J. Nagel with the same title, published by the AMS
(Univ. Lect. Series 52)

The Catalan number $\frac{1}{n+1}{2n \choose n}$ is perhaps the most frequently occurred number in combinatorics. Richard Stanley has collected more than 170 combinatorial objects counted by the Catalan number. Noncrossing partition, which has received great attention recently, is one of these, so called, Catalan objects. Noncrossing partitions are generalized to each
finite Coxeter group. In this talk, we will interpret noncrossing partitions of type B in terms of noncrossing partitions of type A. As applications, we can prove interesting properties of noncrossing partitions of type B.

We study algebraic analogues of the graph Removal Lemma. Vaguely speaking, the graph Removal Lemma says that if a given graph does not contain too many subgraphs of a given kind, then all the subgraphs of this kind can be destroyed by removing few edges. In 2005, Green conjectured the following analogue of it for systems of equations over integers:

For every k x m integral matrix A with rank k and every ε>0, there exists δ>0 such that the following holds for every N and every subset S of {1,…N}: if the number of solutions of A x = 0 with x ∈ S^m is at most δ N^{m-k}, then it is possible to destroy all solutions x ∈ S^m of A x = 0 by removing at most ε N elements from the set S.

We prove this conjecture by establishing its variant for not necessarily homogenous systems of equations over finite fields. The core of our proof is a reduction of the statement to the colored version of hypergraph Removal Lemma for (k+1)-uniform hypergraphs. Independently of us, Shapira obtained the same result using a reduction to the colored version of hypergraph Removal Lemma for O(m²)-uniform hypergraphs. The talk will be self-contained and no previous knowledge of the area related to the graph Removal Lemma will be assumed.

Joint work with Oriol Serra and Lluis Vena.

subprime crisis, LTCM, KIKO, 다이아몬드 펀드 등 사례를 중심으로
금융공학과 파생상품에 대한 이해를 높이는 내용입니다.

 In this talk, we propose an orientation-matching functional minimization for image denoising and image inpainting. Following the contemporary two-step TV-Stokes algorithm, a regularized tangential vector field with zero divergence condition is first obtained. Then a novel approach to reconstruct the image is proposed. Instead of finding an image that fits the regularized normal direction from the first step, we propose to minimize an orientation matching cost measuring the alignment between the image gradient and the regularized normal direction. This functional yields a new nonlinear partial differential equation for reconstructing denoised and inpainted images. The equation has an adaptive diffusivity depending on the orientation of the regularized normal vector field, providing reconstructed images which have sharp edges and smooth regions. The additive operator splitting scheme is used for discretizing Euler-Lagrange equations. We present the results of various numerical experiments that illustrate the improvements obtained with the new functional.

The implied volatility from Black and Scholes (1973) model has been empirically tested for the forecasting performance of future volatility and commonly shown to be biased. Based on the belief that the implied volatility from option prices is the best estimate of future volatility, this study tries to find out a better model, which can derive the implied volatility from option prices, to overcome the forecasting bias from Black and Scholes (1973) model. Heston (1993)’s model which improves on the problems of Black and Scholes (1973) model the most for pricing and hedging options is one candidate, and VIX which is the expected risk neutral value of realized volatility under the discrete version is the other. This study conducts a comparative analysis on the implied volatility from Black and Scholes (1973) model, that from Heston (1993)’s model, and VIX for the forecasting performance of future volatility. From the empirical analysis on KOSPI200 option market, it is found that Heston (1993)’s implied volatility eliminates the bias mostly which Black and Scholes (1973) implied volatility has. VIX, on the other hand, does not show any improvement for the forecasting performance.

Price sensitivities(Greeks) are mathematically defined as the derivatives of a derivative security's price with respect to various model parameters. The traditional way to compute a Greek is through its finite difference approximation. However, the finite difference method has a slow convergence rate when dealing with discontinuous payoffs. To overcome this poor convergence rate, a new theoretical tool recently has been introduced in the literature. This new tool uses the so-called Malliavin calculus in order to devise efficient Monte-Carlo methods for finance. In this lecture we discuss the basic ideas of Malliavin calculus and discuss how it may be used in Mathematical Finance.

Price sensitivities(Greeks) are mathematically defined as the derivatives of a derivative security's price with respect to various model parameters. The traditional way to compute a Greek is through its finite difference approximation. However, the finite difference method has a slow convergence rate when dealing with discontinuous payoffs. To overcome this poor convergence rate, a new theoretical tool recently has been introduced in the literature. This new tool uses the so-called Malliavin calculus in order to devise efficient Monte-Carlo methods for finance. In this lecture we discuss the basic ideas of Malliavin calculus and discuss how it may be used in Mathematical Finance.

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 $\mathbf{S}$.