학과 세미나 및 콜로퀴엄
The main aim of this talk is to design efficient and novel numerical algorithms for highly oscillatory dynamical systems with multiple time scales. Classical numerical methods for such problems need temporal resolution to resolve the finest scale and become very inefficient when the longer time intervals are of interest. In order to accelerate computations and improve the long time accuracy of numerical schemes, we take advantage of various multiscale structures established from a separation of time scales. The framework of the heterogeneous multiscale method (HMM) will be considered as a general strategy both for the design and for the analysis of multiscale methods.
E6-1, ROOM 1409
Discrete Math
Boris Aronov (Department of Computer Science and Engineering, Po)
The complexity of unions of shapes
E6-1, ROOM 1409
Discrete Math
Over the years, the following class of problems has been studied quite a lot: Given a class of simply-shaped objects in the plane (disks, unit disks, squares, axis-aligned squares, isosceles triangles, shapes definable with a small number of polynomial equations and inequalities), how complicated can be the union of N shapes from the class? There are several different ways in which one can measure this (combinatorial) complexity. Two popular measures are the number of connected components of the complement, and the number of places where two object boundaries intersect on the boundary of the union (so-called “vertices” of the union).
It is easy to see that, if each object is “simple,” the union of N objects cannot be larger than O(N^2) and a matching construction is easy. Are there classes of objects for which this quantity is near-linear in N? (Yes, there are: disks, axis-aligned squares, and more.) The quest for such classes, over the years, motivation for the problem, generalizations to higher dimensions, and other puzzles will constitute the content of this talk.
If I ever get to it, the latest and most amazing result in this area is joint work with Mark de Berg, Esther Ezra, and Micha Sharir. It is quite technical and I will not be able to say much about this during the talk, but if anyone is interested, I can provide lots of technical details on request. An overview of the subject will be mostly based on a survey of Agarwal, Pach, and Sharir.
In this talk, we will discuss about the following question posed by S. Tolman.
Question : If a smooth compact sympletic manifold (M, ω) admits a Hamiltonian torus action with only isolated fixed points, then is the sequence of even Betti numbers of M unimodal?
R. Stanley proved that the above question is “yes” when our torus has a full dimension, i.e. (M, ω) is a toric variety. One of the main purpose of this talk is to show that the answer is also “yes” if dimM ≤ 8.
Secondly, we will see that we can deal with the above question using graph theory. Let’s assume that T be a compact torus of dimension k and (M, ω) be a 2n-dimensional closed Hamiltonian T -manifold with some nice condition.
Then a 1-skeleton of the corresponding moment polytope, which is called a GKM graph, is an n-valent graph embedded in R^k satisfying several nice properties. In this case, we will see that our question is equivalent to the followings.
Question (2nd version) : If we have a GKM graph Γ, let ξ∈R^k be any vector such that ξ is not orthogonal to any edge of Γ. Then ξ gives an orientation overrightarrow{e} for each edge e in Γ such that <overrightarrow{e},ξ> 0. Let b_r(Γ, ξ) be the number of vectices with index r. Then is the sequence b_0, b_1,... unimodal?
자연과학동 Room 1409
Discrete Math
김은정 (CNRS, LAMSADE, Paris, France)
On subexponential and FPT-time inapproximability
자연과학동 Room 1409
Discrete Math
Fixed-parameter algorithms, approximation algorithms and moderately exponential algorithms are three major approaches to algorithms design. While each of them being very active in its own, there is an increasing attention to the connection between these different frameworks. In particular, whether Independent Set would be better approximable once allowed with subexponential-time or FPT-time is a central question. Recently, several independent results appeared regarding this question, implying negative answer toward the conjecture. They state that, for every 0<r<1, there is no r-approximation which runs in better than certain subexponential-function time. We outline the results in these papers and overview the important concepts and techniques used to obtain such results.
The research on flips is a crucial part to study minimal model program (MMP). To understand flips, it is worthwhile to get some numerical invariants of flips to understand it better. The minimal log discrepancy (mld) is one of the important invariant to give a geometrical information about flips, and it appear naturally in global contexts. There have been several conjectures on mld related with the termination of flips. I will introduce basic concepts of mld and talk about some relations between mld and other invariants with some examples of flips.
In this talk I will discuss the spreading properties of solutions of a prey-predator type reaction-diffusion system. This system belongs to the class of reaction-diffusion systems for which the comparison principle does not hold. For such class of systems, little has been know about the spreading properties of the solutions. Here, by a spreading property, we mean the way the solution propagates when starting from compactly supported initial data. We show that propagation of both the prey and the predator occur with a definite spreading speed. Furthermore, quite intriguingly, the spreading speed of the prey and that of the predator are different in some situations. This is joint work with Arnaud Ducrot and Thomas Giletti.
The classical p-typical Witt vectors were contrived by Teichm?ller and Witt to build unramified extensions of the field of p-adic numbers from their residue fields in a functorial way. Dress and Siebeneicher introduced a fascinating generalization of them called "Witt-Burnside rings" in a group-theoretical way. In this talk, we will briefly review the basic theory of Witt vectors and Witt-Burnside rings. Recent developments in this area, in particular, some open problems concerned with Witt vector construction will be also dealt with.
The rational homology groups of the matching complexes are closely related to the syzygies of the Veronese embeddings. In this talk, I will give a proof of the shellability of certain skeleta of matching complexes, thus proving that the coordinate rings of the Veronese varieties satisfy property $N_{2d-2}$. Using duality and explicit computation of homology groups of matching complexes, we will deduce the Ottaviani-Paoletti conjecture for fourth Veronese embeddings.
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 세미나
Peter Schenzel (Martin Luther University)
ON THE VISUALIZATION OF BLOWING UPS OF THE PLANE IN POINTS
자연과학동(E6) Room 2411
ASARC 세미나
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.