Department Seminars & Colloquia
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I will give the introductory exposition of MMP and the abundance conjecture. In this talk, I will touch on the extension and injectivity theorem and give one approach to prove the abundance conjecture. And I will talk about importance of semi-log canonical singularities of pairs.
I will give the introductory exposition of MMP and the abundance conjecture. In this talk, I will touch on the extension and injectivity theorem and give one approach to prove the abundance conjecture. And I will talk about importance of semi-log canonical singularities of pairs.
We revisit the optimal investment and consumption model of Davis and Norman (1990) and Shreve and Soner (1994), following a shadow-price approach similar to that of Kallsen and Muhle-Karbe (2010). Making use of the completeness of the model without transaction costs, we reformulate and reduce the HJB equation for this singular stochastic control problem to a non-standard free-boundary problem for a first-order ODE with an integral constraint. Having shown that the free boundary problem has a smooth solution, we use it to construct the solution of the original optimal investment/consumption problem in a self-contained manner and without any recourse to the dynamic programming principle. Furthermore, we provide an explicit characterization of model parameters for which the value function is finite. This is a joint work withMihai Sirbu and Gordan Zitkovic.
://kmrs.kaist.ac.kr/activities/registration/?ee=48
Lecture 3: 7:00-9:00pm, Nov. 27, 2013
As an example to use the theory we treated in Lecture2, we consider Merton's portfolio optimization problem with transaction costs, follow the paper by Shreve & Soner (1994).
Lecture 2: 7:00-9:00pm, Nov. 26, 2013
We review the definition of the viscosity solution of HJB equation. Then by using DPP, we see that the value function of the control problem is a viscosity solution of HJB equation.
Lecture 1: 7:00-9:00pm, Nov. 25, 2013
We consider a stochastic control problem and formally derive Hamilton–Jacobi–Bellman(HJB) equation from Dynamic Programming Principle(DPP). And we see the standard verification argument in case the HJB equation admits a classical solution. As an example, we consider Merton's portfolio optimization problem.
The main subject of this lecture is global dynamics and behavior of
solutions for nonlinear dispersive wave equations, such as the
nonlinear Schrodinger equation and the nonlinear Klein-Gordon
equation. Starting from basic materials in the analysis of partial
differential equations, the specific goal is to introduce the recent
results in joint work with Wilhelm Schlag, which give classification
and prediction for the global dynamics including various types of
behavior: scattering, soliton, blowup, and transition among them. The
lecture will consist of the following sections:
1. Overview
2. The Cauchy problem and blowup
3. Variational method and the ground state
4. Space-time estimate and the scattering theory
5. Classification of the global dynamics
Nonlinear wave equations have solutions with various types of behavior, such as dispersive waves, solitary waves (solitons), and blow-up in finite time. Heuristically, they can be distinguished by which is stronger on each solution, the dispersive effect or the nonlinear one. Rigorous analysis of the dynamics has been well developed in small neighborhoods around special solutions, typically the trivial one and some solitons, where all solutions exhibit the same behavior. However, rather little is known about the dynamics away from such neighborhoods: if and how different types of solutions can coexist or some solutions can change their behavior along time, etc. Numerical studies suggested that in some cases the two sets of solutions in stable regimes (dispersive waves and stable blow-up) are separated by a hypersurface of the third set of solutions which are unstable. Similar phenomena are well known for nonlinear diffusion equations, but they can be easily understood by the comparison principle, which does not apply to wave equations. In the joint work started with Wilhelm Schlag, we have rigorously obtained such a trichotomy in some simple settings such as the nonlinear Schrodinger and Klein-Gordon equations with unstable ground states, under some energy constraint. I will explain how we can construct the threshold hypersurface, describe the dynamics off and on the hypersurface, capture the stable transition between dispersion and blow-up, and thereby predict global behavior of solutions from the initial data. I will also discuss about open questions.
The main subject of this lecture is global dynamics and behavior of
solutions for nonlinear dispersive wave equations, such as the
nonlinear Schrodinger equation and the nonlinear Klein-Gordon
equation. Starting from basic materials in the analysis of partial
differential equations, the specific goal is to introduce the recent
results in joint work with Wilhelm Schlag, which give classification
and prediction for the global dynamics including various types of
behavior: scattering, soliton, blowup, and transition among them. The
lecture will consist of the following sections:
1. Overview
2. The Cauchy problem and blowup
3. Variational method and the ground state
4. Space-time estimate and the scattering theory
5. Classification of the global dynamics
E6-1, ROOM 1409
Discrete Math
Hermanshu Kaul (Illinois Institute of Technology)
Finding Large induced subgraphs and allocation of resources under dependeny
E6-1, ROOM 1409
Discrete Math
Given a graph, we are interested in studying the problem of finding an induced subgraph of a fixed order with largest number of edges. More generally, let G = (V, E) be an undirected graph, with a weight (budget) function on the vertices, w: V → ℤ+, and a benefit function on vertices and edges b: E ∪ V → ℤ. The benefit of a subgraph H =(VH,EH) is b(H) = ∑ v∈VH b(v) + ∑ e∈EH b(e) while its weight is w(H) = ∑ v∈VH w(v). What can be said about the maximum benefit of an induced subgraph with the restriction that its weight is less than W?
This problem is closely related to the Quadratic Knapsack Problem, the Densest Subgraph Problem, and classical problems in Extremal Graph Theory. We will discuss these connections, give applications in resource allocation, and present new results on approximation algorithms using methods from convex optimization and probability. This is joint work with Kapoor.
The main subject of this lecture is global dynamics and behavior of
solutions for nonlinear dispersive wave equations, such as the
nonlinear Schrodinger equation and the nonlinear Klein-Gordon
equation. Starting from basic materials in the analysis of partial
differential equations, the specific goal is to introduce the recent
results in joint work with Wilhelm Schlag, which give classification
and prediction for the global dynamics including various types of
behavior: scattering, soliton, blowup, and transition among them. The
lecture will consist of the following sections:
1. Overview
2. The Cauchy problem and blowup
3. Variational method and the ground state
4. Space-time estimate and the scattering theory
5. Classification of the global dynamics
The main subject of this lecture is global dynamics and behavior of
solutions for nonlinear dispersive wave equations, such as the
nonlinear Schrodinger equation and the nonlinear Klein-Gordon
equation. Starting from basic materials in the analysis of partial
differential equations, the specific goal is to introduce the recent
results in joint work with Wilhelm Schlag, which give classification
and prediction for the global dynamics including various types of
behavior: scattering, soliton, blowup, and transition among them. The
lecture will consist of the following sections:
1. Overview
2. The Cauchy problem and blowup
3. Variational method and the ground state
4. Space-time estimate and the scattering theory
5. Classification of the global dynamics
Part 5. Extended Born-Infeld model.
It is well-known that all the magnetostatic solutions with finite energy for Born-Infeld model in R^3 are trivial, and it was conjectured that incorporation of perturbation into Born-Infeld functional could provide non-trivial solutions. In this part we examine the extended Born-Infeld equations for the magnetostatic case in bounded domains. Under various boundary conditions the existence of non-trivial solutions is proved for small boundary data. The main feature of the extended Born-Infeld functionals is their degree one growth in the curl of the vector fields, which causes lack of weak compactness in the natural admissible spaces. To overcome this difficulty we introduce modified functionals and estimate their minimizers or critical points.
Part 6. A quasilinear degenerate systems with operator curl
In this part we study a quasilinear degenerate system with the operator curl. The leading order term of the associated energy functional is q-power of curl. It is interesting that the lower order terms play an important role in the existence of solutions. When the lower order part of the functional is convex we obtain weak solutions by minimizing the functional in some suitable spaces. When the lower order part is concave we look for critical points of the truncated functional and obtain weak solutions of a nonlinear eigenvalue problem. The interior regularity of the weak solutions is also examined.
References:
[8] J. Chen and X.B. Pan, Functionals with operator curl in an extended magnetostatic Born-Infeld model, SIAM J. Math. Anal., 45 (4) (2013), 2253-2284.
[9] J. Chen and X.B. Pan, An extended magnetostatic Born-Infeld model with a concave lower order term, J. Math. Phys., to appear.
A small cover is a topological analogue of real toric varieties, and is an important object in toric topology. It is noted that the formula of the ℤ2-cohomology ring of small cover is well-known. However, the integral cohomology ring of small covers has not been known well.
In this talk, we discuss about the Betti numbers and its torsion of the small covers associated to some nestohedra including graph associahedra. Interestingly, the Betti numbers can be computed by purely combinatorial method (in terms of graphs and hypergraphs). To our surprise, for specific families of graphs, these numbers are deeply related to well-known combinatorial sequences such as the Catalan numbers and Euler zigzag numbers.
In the first talk we intend to present a few techniques which will be useful for the proof of the main result. We will discuss an analytic proof of the Y. Miyaoka generic semi-positivity result, as well as a few basic facts concerning the Zariski decomposition and the finite generation of the canonical ring.
Part 3. Linear systems involving curl.
We begin with the first order system: the div-curl system. Solvability and regularity is discussed. Then we consider a second order linear system involving curl. We main consider the Dirichlet boundary condition. Estimates in Sobolev norms and in Holder norms are obtained. The results in this part will be useful in our study on quailinear systems.
Part 4. A quasilinear system for Meissner states of superconductors.
Type II superconductors in an increasing applied magnetic field undergo phase transitions from the Meissner state to the mixed state as the applied field increases to the superheating field. In this part we examine a simplified model for Meissner states and derive Holder estimates for weak solutions. Then we show that, if the penetration length is small, the solutions concentrate in a boundary layer at the surface of the domain. In particular, if the sample is subjected to a homogeneous magnetic field, the current is maximal at the surface of the sample where the applied field is tangential to the surface.
References:
[4] P. Bates and X.B. Pan, Nucleation of instability in Meissner state of 3-dimensional superconductors, Comm. Math. Phys., 276 (3) (2007), 571-610. Erratum, 283 (2008), 861.
[5] X.B. Pan, On a quasilinear system involving the operator Curl, Calc. Var. PDE, 36 (3) (2009), 317-342.
[6] X.B. Pan, Asymptotics of solutions of a quasilinear system involving curl, J. Math. Phys., 52 (2) (2011), article no. 023517.
[7] G. Lieberman and X.B. Pan, On a quasilinear system arising in the theory of superconductivity, Proc. Royal Soc. Edinburgh, 141 A (2) (2011), 397-407.
자연과학동(E6) Room 2411
ASARC Seminar
Peter Schenzel (Martin Luther University)
ON THE VISUALIZATION OF BLOWING UPS OF THE PLANE IN POINTS
자연과학동(E6) Room 2411
ASARC Seminar
http://mathsci.kaist.ac.kr/asarc/etc/abstract-Peter Schenzel.pdf
References
[Brodmann(1995)] Markus Brodmann. Computer-pictures of blowing-ups. (Computerbilder von Aufbla-
sungen.). Elem. Math., 50(4):149{163, 1995.
[Fischer(1986)] Gerd Fischer, editor. Mathematische Modelle. Vieweg-Verlag, 1986.
[Hironaka(1964)] Heisuke Hironaka. Resolution of Singularities of an Algebraic Variety Over a Field of
Characteristic Zero: I & II. The Annals of Mathematics, 79:109{203, 205{326, 1964. ISSN 0003486X.
URL http://www.jstor.org/stable/1970486.
[Stussak(2007)] Christian Stussak. Echtzeit-Raytracing algebraischer Flachen auf der GPU. Diplo-
marbeit, Institut fur Informatik, Martin-Luther-Universitat Halle-Wittenberg, 2007. URL
http://realsurf.informatik.uni-halle.de.
Martin-Luther-Universitat Halle-Wittenberg, Institut fur Informatik, D | 06 099
Halle (Saale), Germany
E-mail address:
Part 1. Partial differential systems involving curl.
In this introductory part we briefly discuss some elementary properties of the operators curl and curl^2. There are many partial differential systems (PDS) involving the operator curl in various areas of science. A few examples will be presented: div-curl system, Ginzburg-Landau system for superconductivity, quasilinear system for Meissner states, Landau-de Gennes system for liquid crystals, extended Born-Infeld model in magnetostatic case.
Part 2. Spaces of vector fields defined by curl and divergence.
In this basic part we introduce several spaces of vector fields, including H(U,curl) and H(U,div). Elementary properties of these spaces and the trace theorems are presented, and the relation between these spaces and the Sobolev spaces is discussed. The mapping properties of the operator curl is studied and the kernel and image of curl are described, and decomposition of vector fields is also discussed.
References:
[1] R. Dautray and J. L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, vol. 3: Spectral Theory and Applications, Springer-Verlag, New York, 1990.
[2] V. Girault and P. -A. Raviart, Finite Element Methods for the Navier-Stokes Equations, Theory and Algorithms, Springer, Berlin, 1986.
[3] M. Cessenat, Mathematical Methods in Electromagnetism-Linear Theory and Applications, World Scientific Publishing Co., Inc., River Edge, NJ, 1996.
We aim to review the efforts for systematic and organized global collaborations in mathematics starting from late 19th century. We also summarize the activities in Korean math research community that are sometimes sporadic but are becoming increasingly organized and systematic.
Given a simple polygon P, a point t in P, and a set of positive weights, we want to place the weights on the boundary of P in such a way that their barycenter comes to t. We show that there is always such a placement if the weights are balanced, i.e., no weight is larger than half of the total weight, and the placement can be found efficiently. We also study three-dimensional versions of the problem.
Joint work with Luis Barba, Jean Lou De Carufel, Rudolf Fleischer, Akitoshi Kawamura, Matias Korman, Yuan Tang, Takeshi Tokuyama, Sander Verdonschot, and Tianhao Wang.
Lecture 3: Fibred knots and the Alexander polynomial.
Some knot groups have orderings which are invariant under multiplication on both sides, while others do not. I will define fibred knots, monodromy and the Alexander polynomial and discuss the role this polynomial has in the question of whether the group of a fibred knot has a 2-sided invariant ordering.
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
Meesue Yoo (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
Boram Park (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.