본 강연은 KMRS(KIAST Math Research Station)에서 제공하는 집중 강연으로 기하학적 입장에서 유도되어지는 편미분 방정식을 소개하는 것을 목적으로 하고 있다. 이제까지 주로 연구 되어진 다양한 편미분 방정식은 주로 물리학적인 문제들에서 유도 되어진 것이다. 본 강연을 통해서 기존의 편미분 방정식을 보다 기하학적인 관점에서 이해할 뿐 아니라 기하학적인 문제를 편미분 방정식의 형태로 소개 하고자 한다.  

This lecture series is provided by KMRS and aiming to introduce PDEs which are derived from geometry view point. So far PDEs are mostly derived from physical view point. In these lectures PDEs will be understood in geometry view point and PDEs related to geometry will be introduced.

In 1975, Y. Morita conjectured that if an abelian variety defi ned over a number fi eld has the Mumford-Tate group which does not have any non-trivial Q-rational unipotent element, then it has potential good reduction everywhere. In this talk, we explain a proof of this conjecture. The main ingredients of proof include some newly established cases of the conjecture due to Vasiu, a generalization of a criterion of Paugam on good reduction of abelian varieties, and the local-global principle of isotropy for Mumford-Tate groups of abelian varieties.

A local factor of a Dedekind zeta function can be btained as a partition function of an appropriate C*-dynamical system. For a Dedekind zeta function itself such a dynamical system is related to the Galois group of the given number field and is called a BC system. We will ee how to extend this idea in the case of Hecke L-functions.

Using Todd operator one can express Euler-McLaurin formula in simple form. For recent decades, along with development of toric geometry and theory of polytopes, Euler-McLaurin formula has been generalized to a category of lattice cones and polytopes by Brion-Vergne, Karshon-Sternberg-Weitsman, Garoufalidis-Pommersheim and several others. In particular, Garoufalidis-Pommersheim expressed special values of zeta function associated to Todd operator associated to a certain cone decomposition. We extend the category of cones by Grothendieck group construction of ordinary cones. This new category contains 'virtually decomposed cones' considering cones with negative weight and the appropriate form of Todd operator construction generalizes the Euler-Mclaurin formula on it. We apply this generalization to obtain an alternating sum expression of special values of (partial) zeta functions at nonpositive integers associated (virtual) cone decomposition. This expression enables us to read polynomial behavior of special values of zeta function at nonpositive integers in some family of real quadratic fields. It is joint work with Byungheup Jun.

In this talk, I will first survey some important examples of simply connected closed positively curved manifolds as well as known nonnegatively curved manifolds and their constructions. After that, I plan to focus on exotic positively curved metrics on P_2 homeomorphic to a unit 3-sphere bundle over a 4-sphere, which have been recently constructed by Grove, Verdiani, and Ziller. Finally, I will consider some interesting problems related to positively curved manifolds and manifolds with almost positive sectional curvatures.

본 강연은 KMRS(KIAST Math Research Station)에서 제공하는 집중 강연으로 기하학적 입장에서 유도되어지는 편미분 방정식을 소개하는 것을 목적으로 하고 있다. 이제까지 주로 연구 되어진 다양한 편미분 방정식은 주로 물리학적인 문제들에서 유도 되어진 것이다. 본 강연을 통해서 기존의 편미분 방정식을 보다 기하학적인 관점에서 이해할 뿐 아니라 기하학적인 문제를 편미분 방정식의 형태로 소개 하고자 한다.  

This lecture series is provided by KMRS and aiming to introduce PDEs which are derived from geometry view point. So far PDEs are mostly derived from physical view point. In these lectures PDEs will be understood in geometry view point and PDEs related to geometry will be introduced.

In view of scheme language, we start from the beginning of the theory of elliptic curves and geometric modular forms, and further cover the topics about Jacobians and Galois representation as well as modularity problems.

Wavelets are useful for many applications including signal/image processing. Tensor product has been a predominant method in constructing multivariate wavelets. In this talk, I will first provide a brief overview of the wavelet analysis and the use of tensor product in constructing multivariate wavelets. Then I will introduce a new alternative to tensor product, to which we refer as Coset Sum. We will discuss the similarity and difference between the two methods. We will also see that some of known limitations of tensor product can be overcome by Coset Sum, albeit in a limited sense.

Since the inception of Donaldson theory and Seiberg-Witten theory in late 20 century, the invariants induced from these gauge theories, in particular, Seiberg-Witten invariants have become so powerful tools in study of smooth and symplectic 4-manifolds. Nevertheless, it has not been much known that one can distinguish smooth and symplectic 4-manifolds to some extent using Seiberg-Witten invariants. In this talk I'd like to show that there exist an infinite family of non-simply connected, non-diffeomorphic, but homeomorphic, 4-manifolds with the same Seiberg-Witten invariants. The main techniques used in the construction are a knot surgery technique and a covering method.

A path cover of a graph is a set of disjoint paths such that every vertex in the graph appears in one of the paths.
We prove an upper bound for the minimum size of a path cover in a connected
4-regular graph with n vertices, confirming a conjecture by Graffiti.pc.
We also determine the minimum number of vertices in a connected k-regular graph that is not Hamiltonian, and we solve the analogous problem for Hamiltonian paths.

This is a partly joint work with Gexin Yu and Rui Xu.

http://dl.dropbox.com/u/12483374/Lectures/poster.pdf

http://dl.dropbox.com/u/12483374/Lectures/poster%20seminar.pdf

The heat equation describes time variation of temperature. Poisson's equation ignores the time derivative in the heat equation so that it might describe the steady state temperature. In fact, if we consider the whole Euclidean space with dimension greater than or equal to three, solutions of the heat equation converge to a solution of Poisson's equation as time passes by. However, if the dimension is one or two, solutions of the heat equation diverge in general. In this talk, a sufficient condition to guarantee the convergence will be given. Also, if time permits, we will consider related questions for nonlinear problems. This talk is based on ongoing project with Jieun Choi and Yong Jung Kim.

A mass linear function is an affine function on a simple convex polytope whose value on the center of mass depends linearly on the positions of the supporting hyperplanes. A Delzant polytope is a simple, rational, and smooth convex polytope. In this talk, I will introduce the relationship between Delzant polytopes and symplectic toric manifolds, and then geometric implications of mass linear functions. This talk is based on the work by D. McDuff and S. Tolman, Polytopes with mass linear functions.

우리나라의 역사적 기록을 토대로 우리의 선조들이 정치, 경제, 복지, 과학분야에서 어떤 업적을 이루었는지를 재조명하는 시간을 갖는다. 아래 내용은 수학과 관련된 내용의 일부이다.

洪大容(1731-1783)

수학서 <주해수용(籌解需用)>의 내용

(1) 구체의 체적이 62,208척이다. 이 구체의 지름을 구하라.

正弦=sinA 餘弦=cosA 正切=tanA 餘切=cotA

正割=secA 餘割=cosecA 正矢=1-cosA 餘矢=1-sinA

 

正弦 30도=sin30도=0.5

正弦 25도 42분 51초=sin25.4251。=0.4338883739118

正弦 45도=sin45=0.7070167811865

(2) 甲地와 乙地는 동일한 子午眞線에 있다. 甲地는 北極出地 37도에 있고, 乙地는 36도 30분에 있다. 甲地에서 乙地로 직선으로 가는데 고뢰(鼓擂)가 12번 울리고, 종뇨(鍾鬧)가 125번 울렸다. 이 때 지구 1度의 里數와 지구의 지름, 지구의 둘레를 구하라.

The existence of topologically slice knots that are of infinite

Nonsmooth optimization problems are generally considered to be more difficult than smooth problems. Among those, optimization problem with sparsity, which has wide applicability in machine learning, satistics, and image processing, are usually structured. Hence many efficient optimization methods have been developed to solve such problems. In this talk, we introduce several optimization problems with sparsity arising in applications and optimization methods for solve them.

Basic concepts, examples and applications of quantum
(=non-commutative) probability will be presented.
Preliminaries would be a mild familiarity of random variables and
bounded linear operators on Hilbert spaces (or just matrices).

We consider a well-known combinatorial search problem. Suppose that there are n identical looking coins and some of them are counterfeit. The weights of all authentic coins are the same and known a priori. The weights of counterfeit coins vary but different from the weight of an authentic coin. Without loss of generality, we may assume the weight of authentic coins is 0. The problem is to find all counterfeit coins by weighing sets of coins (queries) on a spring scale. Finding the optimal number of queries is difficult even when there are only 2 counterfeit coins.
We introduce a polynomial time randomized algorithm to find all counterfeit coins when the number of them is known to be at most m≥2 and the weight w(c) of each counterfeit coin c satisfies α≤|w(c)|≤β for fixed constants α, β>0. The query complexity of the algorithm is O((m log n)/log m), which is optimal up to a constant factor. The algorithm uses, in part, random walks.
We will also discuss the problem of finding edges of a hidden weighted graph using a certain type of queries.

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I will start with a pivot-minor containment problem in graphs. A graph H is a pivot-minor of a graph G if H is obtained from G by a sequence of pivoting edges and vertex deletions. In recent, we have a question that any incidence graph of a tree does not have binary tree of depth at least 5 as a pivot-minor. This comes true and I gives two proofs it. First, I prove it by using the fact that an adjacency matrix of a tree is nonsingular if and only if it has a perfect matching. Second, I will discuss how this problem is related to a fundamental graph of a binary matroid. Then we can convert original problem into a graph minor containment problem and we can solve it.

We discuss the structure of braid groups on complexes that
is embedded in a surface using configuration spaces of complexes. We
show that the discrete configuration space of some cube complex that
is homeomorphic to a given complex is an Eilenberg-MacLane space of a
braid group on the complex.

Heron's formula relates the square of the area of a triangle to the 4-dimensional volume of a hyper-rectangle. As such, it should lend itself to a 4-dimensional proof. In this talk, I show how to use a scissors congruence proof of the Pythagorean Theorem to create a scissors congruence proof of Heron's formula. The talk will be an excursion into some interesting aspects of 4-dimensional hyper-solids. 

The generalized Dirichlet Problem in a plane region is established by Perron for the regions whose boundary has positive logarithmic capacity. Let be a region whose boundary has positive logarithlmic capacity. For a bounded continuous function on, there exists a unique harmonic function (the Perron function) on whose boundary function is n.e. (i.e., outside a set of capacity zero) on The solution is related with the Green's function on but the explicit form of the Green's function is known only for special class of regions, like disks, half plane, annulus and their conformally equivanent regions by means of conformal maps. In this talk, the Greens functions will be given by a boundary preserving Nevanlinna class function from the unit disk onto. In this way the geometric property of is shown to be related with the function theoretic property of the analytic function. I wish to make this talk accessible to the first year graduate students by starting with the Dirichlet problem on the unit disk with explicit Green's function and Possion integral for the solution.

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강연 30분전 세미나실 앞에서 다과가 있습니다.

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In this talk, I will introduce the idea of an n-dimensional foam which generalizes trivalent graphs, and the usual notion of a surface foam. Such foams can be knotted in (n+2)-dimensional space. Local pictures for the crossing points are obtained in all dimensions. There are different crossing types that are easy to parametrize. Also local crossings have signs associated to them. In all dimensions it is possible to examine quandle colorings and group-flows on n-foams. As a result, group-families of quandles, and cocycles that are associated to these can be used to distinguish different knotted foams. The subject of this talk is being developed in conjunction with Masahico Saito.

In this talk, I will give a brief introduction to Nonparametric (NP) Bayesian statistical modeling. First, I will describe some key components of Bayesian statistical inference. Then, I will begin with a statistical modeling example for which parametric modeling may have limitations and introduce the NP Bayes methodology for more flexible modeling. Focuses will be on NP Bayes approaches involving Dirichlet process (DP). I will also discuss computation-based inference procedure focusing on Markov Chain Monte Carlo (MCMC). I will conclude with a summary and some discussions of future research directions.

In this talk, we are interested in the stability and dynamic bifurcation for the two dimensional Swift-Hohenberg equation with an odd periodic condition. It is shown that an attractor bifurcates from the trivial solution as the control parameter crosses the critical value. The bifurcated attractor consists of finite number of singular points and their connecting orbits. Using the center manifold theory, we verify the nondegeneracy and the stability of the singular points.

We consider an optimal financial planning problem of an
economic agent with labor income when the agent has limited
opportunities to borrow against future labor income. The economic
agent determines his/her inter-temporal consumption, portfolio, and
contribution on annuity contract to maximize his/her utility of
lifetime consumption. We transform the agent’s inter-temporal problem
into a dual problem to derive the optimal policies. It can be shown
that constraints on the borrowing opportunities are necessary to
remove the arbitrage opportunities.

Recently, imaging techniques in science, engineering and medicine have evolved to expand our ability to visualize internal information of an object such as the human body.  In particular, there has been marked progress in electromagnetic property imaging techniques where cross-sectional image reconstructions of conductivity, permittivity and susceptibility distributions inside the human body are pursued. They will widen applications of imaging methods in medicine, biotechnology, non-destructive testing, monitoring of industrial process and others. 

This lecture focuses on mathematical modeling and analysis on electromagnetic tissue property imaging. The imaging problems can be formulated as inverse problems that are intrinsically nonlinear, and finding solutions with practical significance and value requires deep understanding of underlying physical phenomenon (Maxwell's equations) with data acquisition systems as well as implementation details of image reconstruction algorithms. We will explain strategies dealing with these complicated structures using a simple linear algebra.

Garside theory was initiated from the work of F. A. Garside on the word and conjugacy 

For centuries, researchers have attempted to grapple with the basic question of what risk is and how to measure risk. Especially, financial markets are becoming increasingly sophisticated in pricing, isolating, repackaging, and transferring risks thanks to tools such as derivatives and securitization. Therefore, financial risk management is vital to the survival of financial institutions and the stability of the financial system. At this point, financial risk management highly depends on a quantitative assessment of risk involved in a financial position. In this talk, we will discuss various issues related on financial risk analysis and management and discover the importance of advanced mathematics in those issues.

The minimal model program (MMP) refers to a series of theorems and
conjectures which arise naturally when one attempts to classify
projective varieties in terms of their pluricanonical line bundles.
The theory of multiplier ideal sheaves has played a central role in
the recent development of MMP.

A multiplier ideal sheaf is determined by a singular hermitian metric
of a line bundle. In fact, a singular hermitian metric contains more
information than its multiplier ideal sheaf. We will give an overview
of these fundamental notions and their applications in the context of
MMP.

On the other hand, in the subclass of algebraic multiplier ideal
sheaves, it is known that not every integrally closed ideal is an
algebraic multiplier ideal. We extend this statement to the full class
of analytic multiplier ideal sheaves, answering a question asked by
Lazarsfeld.

The cycle counting rook numbers, hit numbers, and q-rook numberes and q-hit numbers have been studied by many people, and Briggs and Remmel introduced the theory of p-rook and p-hit numbers which is a rook theory model of the weath product of the cyclic group C_p and the symmetric group S_n.
We extend the cycle-counting q-rook numberes and q-hit numbers to the Briggs-Remmel model. In such a settinig, we define multivariable version of the cycle-counting q-rook numbers and cycle-counting q-hit numbers where we keep track of cycles of pernutation and partial permutation of C_pwearth product with S_n according to the signs of the cycles.
This work is a joint work with Jim Haglund at University of Pennsylvania and Jeff Remmel at UCSD.

We consider the time evolution of hypersurfaces immersed in Euclidean space with the speed of the square root of the scalar curvature times a positive conformal factor.  This is an example of the geometric flow deforming the immersions which are similar to the mean curvature flow and the Gauss curvature flow.  The main ingredient for the convergence and the existence is the pinching estimate modifying that by B. Andrews.  In dimension two, a monotone quantity is obtained from the divergence structure for the Gauss curvature.  This is a joint work with Lami Kim and Kiahm Lee.

For a group G with a finite presentation and a subgroup H of G, the Reidemeister-Schreier method enables us to find a presentation of H. By applying the Reidemeister-Schreier method to right-angled Artin groups, Bell(2011) obtained a result concerning a kind of subgroups of a given right-angled Artin group. 

Following Bell's idea, we apply the Reidemeister-Schreier method to right-angled Coxeter groups.