학과 세미나 및 콜로퀴엄
Abstract:
Sequences of regressions in joint or single responses
are a flexible subclass of graphical Markov models
that include directed acyclic graphs and two types
of fully undirected graph as special cases.
They provide a framework for assessing pathways of dependence,
as they develop over time. Such pathways may often be
modelled locally and it may become simpler to understand
them when a graph captures corresponding conditional
independences. It is to be shown when and why this
happens and how the recent results, on sequences
regressions and regression graphs, can be exploited
in analysing and understanding pathways of
dependences and in planning follow-up studies.
The key is to define models that are complex enough
to capture important types of structure, especially
to permit changes arising simultaneously in several
observable features when there is an intervention, but
that are simple enough to permit estimation and
interpretation even for large numbers of variables.
Among the tools for tracing pathways of dependence
and deriving consequences of a given structure are
matrix operators for closing paths in regression
graphs and for deriving consequences of a given
graph when some of the variables are ignored
and others are conditioned on.
are a flexible subclass of graphical Markov models
that include directed acyclic graphs and two types
of fully undirected graph as special cases.
They provide a framework for assessing pathways of dependence,
as they develop over time. Such pathways may often be
modelled locally and it may become simpler to understand
them when a graph captures corresponding conditional
independences. It is to be shown when and why this
happens and how the recent results, on sequences
regressions and regression graphs, can be exploited
in analysing and understanding pathways of
dependences and in planning follow-up studies.
The key is to define models that are complex enough
to capture important types of structure, especially
to permit changes arising simultaneously in several
observable features when there is an intervention, but
that are simple enough to permit estimation and
interpretation even for large numbers of variables.
Among the tools for tracing pathways of dependence
and deriving consequences of a given structure are
matrix operators for closing paths in regression
graphs and for deriving consequences of a given
graph when some of the variables are ignored
and others are conditioned on.
Lecture 3 (2013년 10월31일 14:30~15:30):
Definitions, properties and results for regression graphs
Lecture 4 (2013년 10월31일 15:45~16:45):Constructing graphs from data and tracing pathways of
development
development
Small cancellation theory is one of representative geometric techniques in group theory initiated from Dehn in 1911. In order to understand what small cancellation theory is and how it can be applied, we will first consider an easy question concerning the fundamental group of an orientable closed surface of genus 2. In contrast with a topological proof to the question using universal covering space introduced by Gyuntae Kim, a group theoretic proof using small cancellation theory will be introduced. Also some recent results obtained jointly with Makoto Sakuma by applying small cancellation theory to 2-bridge link groups and even Heckoid groups will be briefly discussed.
E6-1, ROOM 1409
Discrete Math
유미수 (KIAS)
Schur expansion of the integral form of Macdonald polynomials
E6-1, ROOM 1409
Discrete Math
In this talk, we consider the combinatorial formula for the Schur coefficients of the integral form of the Macdonald polynomials. As an attempt to prove Haglund’s conjecture that ⟨Jλ[X;q,qk]/(1-q)n,sμ(X)⟩∈ℕ[q], we have found explicit combinatorial formulas for the Schur coefficients in one row case, two column case and certain hook shape cases. Egge-Loehr-Warrington constructed a combinatorial way of getting Schur expansion of symmetric functions when the expansion of the function in terms of Gessel’s fundamental quasi symmetric functions is given. We apply this method to the combinatorial formula for the integral form Macdonlad polynomials of Haglund-Haiman-Loehr in quasi symmetric functions to get the Schur coefficients and prove the Haglund’s conjecture in more general cases.
Lecture 2: Knot groups.
The 'group' of knot in 3-space is the fundamental group of its complement. I will discuss how to calculate a presentation for the group of a knot and some properties of knot groups. In particular I'll prove that knot groups are locally indicable, and therefore left-orderable.
Abstract:
Sequences of regressions in joint or single responses
are a flexible subclass of graphical Markov models
that include directed acyclic graphs and two types
of fully undirected graph as special cases.
They provide a framework for assessing pathways of dependence,
as they develop over time. Such pathways may often be
modelled locally and it may become simpler to understand
them when a graph captures corresponding conditional
independences. It is to be shown when and why this
happens and how the recent results, on sequences
regressions and regression graphs, can be exploited
in analysing and understanding pathways of
dependences and in planning follow-up studies.
The key is to define models that are complex enough
to capture important types of structure, especially
to permit changes arising simultaneously in several
observable features when there is an intervention, but
that are simple enough to permit estimation and
interpretation even for large numbers of variables.
Among the tools for tracing pathways of dependence
and deriving consequences of a given structure are
matrix operators for closing paths in regression
graphs and for deriving consequences of a given
graph when some of the variables are ignored
are a flexible subclass of graphical Markov models
that include directed acyclic graphs and two types
of fully undirected graph as special cases.
They provide a framework for assessing pathways of dependence,
as they develop over time. Such pathways may often be
modelled locally and it may become simpler to understand
them when a graph captures corresponding conditional
independences. It is to be shown when and why this
happens and how the recent results, on sequences
regressions and regression graphs, can be exploited
in analysing and understanding pathways of
dependences and in planning follow-up studies.
The key is to define models that are complex enough
to capture important types of structure, especially
to permit changes arising simultaneously in several
observable features when there is an intervention, but
that are simple enough to permit estimation and
interpretation even for large numbers of variables.
Among the tools for tracing pathways of dependence
and deriving consequences of a given structure are
matrix operators for closing paths in regression
graphs and for deriving consequences of a given
graph when some of the variables are ignored
Lecture 1 (2013년 10월29일 14:30~15:30) :
Illustrating different types of structures expressible
as regression graphs by using symmetric binary variables
Lecture 2 (2013년 10월29일 15:45~16:45):
Illustrating different types of structures expressible
as regression graphs by using symmetric binary variables
Lecture 2 (2013년 10월29일 15:45~16:45):
More complex motivating examples from observational
and intervention studies
and intervention studies
We discuss an optimal regularity theory for the gradients of weak solutions to nonlinear elliptic and parabolic equations in divergence form in the setting of various function spaces including Lebesgue spaces, generalized Lebesgue spaces, weighted Lebesgue spaces, Morrey spaces, Orlicz spaces and Lorentz spaces.
Lecture 1: Introduction to orderable groups.
This will be an introduction to the theory of ordered groups and the related idea of local indicability, and why these properties of groups are interesting. New proofs of some of the basic theorems will be given using ideas from topology.
Not only does the study of positive definite matrices remain a flourishing area of mathematical investigation, but positive definite matrices have become fundamental computational objects in many areas of engineering, statistics, quantum information, and applied mathematics. A variety of metric-based computational algorithms for positive definite matrices have arisen for approximations, interpolation, filtering, estimation, and averaging, the last being the concern of this talk. A natural and attractive candidate of averaging procedures is the least squares mean (or Karcher mean, center of mass) for the Riemannian trace metric. The Strong Law of Large Number, established by T. Sturm on Hadamard spaces, plays a crucial role for the monotonicity conjecture and gives rise to a problem of finding deterministic approximation to the Karcher mean, namely the no dice conjecture. We provide an affirmative answer to the no dice conjecture on the general setting of Hadamard (or CAT(0), NPC) spaces.
E6-1, ROOM 1409
Discrete Math
박보람 (국가수리과학연구소)
Counterexamples to the List Square Coloring Conjecture
E6-1, ROOM 1409
Discrete Math
The square G2 of a graph G is the graph defined on V(G) such that two vertices u and v are adjacent in G2 if the distance between u and v in G is at most 2. Let χ(H) and χl(H) be the chromatic number and the list chromatic number of H, respectively. A graph H is called chromatic-choosable if χl(H) = χ(H). It is an interesting problem to find graphs that are chromatic-choosable. Kostochka and Woodall conjectured that χl(G2) = χ(G2) for every graph G, which is called List Square Coloring Conjecture. In this paper, we give infinitely many counterexamples to the conjecture. Moreover, we show that the value χl(G2) − χ(G2) can be arbitrary large.
In previous talks at KAIST I derived the balance laws for isometric embedding. These lectures are now available on the web (nicely typed). Now I will emphasize PDE issues for getting existence theorems, especially the use of embedding three dimensional manifolds in six dimensional Euclidean space.
In previous talks at KAIST I derived the balance laws for isometric embedding. These lectures are now available on the web (nicely typed). Now I will emphasize PDE issues for getting existence theorems, especially the use of embedding three dimensional manifolds in six dimensional Euclidean space.
In previous talks at KAIST I derived the balance laws for isometric embedding. These lectures are now available on the web (nicely typed). Now I will emphasize PDE issues for getting existence theorems, especially the use of embedding three dimensional manifolds in six dimensional Euclidean space.
A split of a graph is a partition (A,B) of the vertex set V having subsets A' of A and B' of B such that |A|,|B| > 1 and a vertex a in A is adjacent to a vertex b in B if and only if a is in A' and b is in B'. A graph is prime (with respect to the split decomposition) if it has no split.
We prove that for each n, there exists N such that every prime graph on at least N vertices contains a vertex-minor isomorphic to either a cycle of length n or a graph consisting of two disjoint cliques of size n joined by a matching.
In this talk, we plan to describe a main tool, which is called a blocking sequence in a prime graph, and we will describe two big steps of the proof. And we will pose some open problems behind this result.
This is a joint work with Sang-il Oum.
In gas dynamics, Y. Sone (U of Kyoto) has used the term "ghost" effect for problems of transpirational flow of a gas. The reason for the word "ghost" is that the gas moves with constant pressure. In every day Newtonian mechanics that is analogous to saying a body initiates motion in the absence of force violating $F=ma$. So one might say a "ghost" has moved the gas. Of course there is no "ghost" only a need for better mathematical models and PDEs. This not hard to do but unlike the Kyoto group I will use PDEs instead of the Boltzmann equation of Kinetic theory.
In gas dynamics, Y. Sone (U of Kyoto) has used the term "ghost" effect for problems of transpirational flow of a gas. The reason for the word "ghost" is that the gas moves with constant pressure. In every day Newtonian mechanics that is analogous to saying a body initiates motion in the absence of force violating $F=ma$. So one might say a "ghost" has moved the gas. Of course there is no "ghost" only a need for better mathematical models and PDEs. This not hard to do but unlike the Kyoto group I will use PDEs instead of the Boltzmann equation of Kinetic theory.
E6-1 #1409
Discrete Math
김장수 (KIAS)
Combinatorics of continued fractions and its application to Jacobi’s triple product identity
E6-1 #1409
Discrete Math
In this talk we will see a combinatorial way to expand certain continued fractions using lattice paths called Motzkin paths. This method allows us to prove a formula for q-secant numbers. I will explain that this approach can be used to find a finite version of Jacobi’s triple product identity. This talk is based on my paper with Matthieu Josuat-Vergès: “Touchard-Riordan formulas, T-fractions, and Jacobi’s triple product identity”, The Ramanujan Journal, Volume 30, Issue 3, pp 341-378.
Log del Pezzo pairs frequently appear motivated by developments in the minimal model program. However, they are not very well understood from the viewpoint of classification. We define suitable smooth models of del Pezzo pairs, and we show that they are the same. This result with the comprehensive study of anticanonical morphisms of ruled surfaces by Sakai and Badescu reduces the problem of classification of log del Pezzo pairs to that of del Pezzo surfaces. In particular, we completely classify non-rational weak log canonical del Pezzo pairs, and we characterize when they have finitely generated Cox rings. This is a joint work with DongSeon Hwang. At the end of the talk, we will discuss a higher dimensional generalization of these results. This is a joint work in progress with DongSeon Hwang and Sung Rak Choi.
In this talk we will consider some aspects of abstract harmonic analysis on compact groups and its quantum extension, namely operator amenabilities of the L^1 algebras. Basic concepts on compact groups and quantum groups will be briefly reviewed focusing on the examples of SU(2) and its q-deformations.
For a smooth projective curve over a number field the motivic finiteness conjectures predict that SK_1 should be a torsion group. In this survey talk we motivate this conjecture, and explain some of the difficulties
that arise in an attempt to compute this group.
All possible (minimal good) resolution graphs of complex surface singularities that admit a rational homology disk smoothing (that is, a smoothing whose Milnor number is $0$) are classified by Stipsicz--Szab'o--Wahl~[J. Topol. 2008], except the well-known class of linear graphs, into three classes: (i) graphs with only one star, called ``star-shaped graphs'' (here, a emph{star} is a vertex with valency $ge 3$); (ii) non-star-shaped graphs with all stars with valencies $le 3$; or (iii) non-star-shaped graphs with all stars with valencies $le 3$ but only one star with valency $4$. Thanks to the work of Bhupal and Stipsicz~[Amer. J. Math. 2011], one has a complete list of star-shaped graphs having a rational homology disk smoothing. In contrast we prove that any non-star-shaped graphs with all stars with valencies $le 3$ cannot be the resolution graphs of complex surface singularities with a rational homology disk smoothing. Our result supports the conjecture of Wahl~[Geom. Topol. 2011] that the only complex surface singularities admitting a rational homology disk smoothing are the known weighted homogeneous examples, whose resolution graphs are linear or star-shaped. This is a joint work with Heesang Park and Andr'as I. Stipsicz.