Lecture 4: Some results on commensurability of knot complements

A conjecture of Reid and Walsh asserts that there are at most 3 hyperbolic knot complements in any commensurability class. Here we discuss this conjecture, and give results under certain circumstances. The problem naturally divides itself into two cases, the case of hidden symmetries and the case of no hidden symmetries, and we discuss both. The new results presented here are joint with M. Boileau, S. Boyer, and R. Cebanu.

 

Lecture 3: Commensurability

Commensurability is an equivalence relation on manifolds and orbifolds which is a refinement of geometrization. Here we will describe the current study of commensurability of hyperbolic manifolds, focusing on commensurability of knot complements. We will describe hyperbolic knot complements and their symmetry groups, and discuss the commensurator group and the orbifold commensurator quotient of a hyperbolic non-arithmetic knot complement.

The aim of the intensive lectures is to introduce fundamental mathematical
and statistical tools, and inversion and optimal design methods to
address emerging modalities in medical imaging, nondestructive testing,
and environmental inverse problems. Many mathematical and computational
challenging problems arise in emerging imaging techniques and
they often lead to the investigation of fundamental problems in various
branches of mathematics. The intensive lectures describe state of-the-art
in asymptotic imaging, stochastic modelling, and analysis of wave propagation
phenomena. They throw a bridge across these different aspects of
mathematical imaging. The intensive lectures provide deep understanding
of the different scales in the physical problem and an accurate modelling
of the uncertainty and noise sources in order to derive the best imaging
functional in the sense that it achieves the optimal trade-off between
signal-to-noise ratio and resolution. The intensive lectures also lead the
participants to appreciate the practical implementations and performance
evaluations of the described imaging methods.

Lecture 7: Electromagnetic invisibility, cloaking enhancement, metamaterials.

The aim of the intensive lectures is to introduce fundamental mathematical
and statistical tools, and inversion and optimal design methods to
address emerging modalities in medical imaging, nondestructive testing,
and environmental inverse problems. Many mathematical and computational
challenging problems arise in emerging imaging techniques and
they often lead to the investigation of fundamental problems in various
branches of mathematics. The intensive lectures describe state of-the-art
in asymptotic imaging, stochastic modelling, and analysis of wave propagation
phenomena. They throw a bridge across these different aspects of
mathematical imaging. The intensive lectures provide deep understanding
of the different scales in the physical problem and an accurate modelling
of the uncertainty and noise sources in order to derive the best imaging
functional in the sense that it achieves the optimal trade-off between
signal-to-noise ratio and resolution. The intensive lectures also lead the
participants to appreciate the practical implementations and performance
evaluations of the described imaging methods.

Lecture 5: Properties of the polarization tensors, Resolution enhancement,
optimal control algorithms, dictionary matching algorithms, tracking algorithms.

The aim of the intensive lectures is to introduce fundamental mathematical
and statistical tools, and inversion and optimal design methods to
address emerging modalities in medical imaging, nondestructive testing,
and environmental inverse problems. Many mathematical and computational
challenging problems arise in emerging imaging techniques and
they often lead to the investigation of fundamental problems in various
branches of mathematics. The intensive lectures describe state of-the-art
in asymptotic imaging, stochastic modelling, and analysis of wave propagation
phenomena. They throw a bridge across these different aspects of
mathematical imaging. The intensive lectures provide deep understanding
of the different scales in the physical problem and an accurate modelling
of the uncertainty and noise sources in order to derive the best imaging
functional in the sense that it achieves the optimal trade-off between
signal-to-noise ratio and resolution. The intensive lectures also lead the
participants to appreciate the practical implementations and performance
evaluations of the described imaging methods.

Lecture 6: Imaging extended targets, resolution and stability analysis, shape
derivative.

TBA

The aim of the intensive lectures is to introduce fundamental mathematical
and statistical tools, and inversion and optimal design methods to
address emerging modalities in medical imaging, nondestructive testing,
and environmental inverse problems. Many mathematical and computational
challenging problems arise in emerging imaging techniques and
they often lead to the investigation of fundamental problems in various
branches of mathematics. The intensive lectures describe state of-the-art
in asymptotic imaging, stochastic modelling, and analysis of wave propagation
phenomena. They throw a bridge across these different aspects of
mathematical imaging. The intensive lectures provide deep understanding
of the different scales in the physical problem and an accurate modelling
of the uncertainty and noise sources in order to derive the best imaging
functional in the sense that it achieves the optimal trade-off between
signal-to-noise ratio and resolution. The intensive lectures also lead the
participants to appreciate the practical implementations and performance
evaluations of the described imaging methods.

Lecture 3: Statistical hypothesis testing. Detection test and localization of a
point target.
Optimal (weighted subspace) migration techniques for a point target.

The aim of the intensive lectures is to introduce fundamental mathematical
and statistical tools, and inversion and optimal design methods to
address emerging modalities in medical imaging, nondestructive testing,
and environmental inverse problems. Many mathematical and computational
challenging problems arise in emerging imaging techniques and
they often lead to the investigation of fundamental problems in various
branches of mathematics. The intensive lectures describe state of-the-art
in asymptotic imaging, stochastic modelling, and analysis of wave propagation
phenomena. They throw a bridge across these different aspects of
mathematical imaging. The intensive lectures provide deep understanding
of the different scales in the physical problem and an accurate modelling
of the uncertainty and noise sources in order to derive the best imaging
functional in the sense that it achieves the optimal trade-off between
signal-to-noise ratio and resolution. The intensive lectures also lead the
participants to appreciate the practical implementations and performance
evaluations of the described imaging methods.

Lecture 4: Small-volume asymptotic expansions. Least-square imaging, reversetime
imaging, Kirchhoff imaging, weighted subspace imaging, topological
derivative based imaging.
Basic resolution theory in homogeneous media.

2012/08/21, 23, 28, 30: 4 - 5 pm.


Lecture 1:
2-dimensional orbifolds

In this lecture we will define and describe orbifolds and set
notation.  In particular, we will discuss orbifold Euler
characteristic, orbifold covers, good orbifolds, bad orbifolds, and
the orbifold fundamental group.  Explicit examples of spherical,
Euclidean and hyperbolic 2-orbifolds will be given.  We will also
prove that there is a smallest closed hyperbolic 2-orbifold.


Lecture 2: 3-dimensional orbifolds

Here we will explore 3-dimensional orbifolds, restricting mainly to
good orbifolds.  Although we will give explicit examples of many
different types of 3-dimensional orbifolds, the focus will be on
hyperbolic 3-orbifolds.  To this end, we will discuss hyperbolic
isometries and the geometry of hyperbolic orbifolds and hyperbolic
orbifolds.  We will discuss how useful orbifolds are to the study of
3-manifolds, and give a statement of geometrization.

Lecture 3: Commensurability

Commensurability is an equivalence relation on manifolds and orbifolds
which is a refinement of geometrization. Here we will describe the
current study of commensurability of hyperbolic manifolds, focusing on
commensurability of knot complements. We will describe hyperbolic knot
complements and their symmetry groups, and discuss the commensurator
group and the orbifold commensurator quotient of a hyperbolic
non-arithmetic knot complement.


Lecture 4:  Some results on commensurability of knot complements

A conjecture of Reid and Walsh asserts that there are at most 3
hyperbolic knot complements in any commensurability class.  Here we
discuss this conjecture, and give results under certain circumstances.
The problem naturally divides itself into two cases, the case of
hidden symmetries and the case of no hidden symmetries, and we discuss
both.  The new results presented here are joint with M. Boileau, S.
Boyer, and R. Cebanu.

The aim of the intensive lectures is to introduce fundamental mathematical
and statistical tools, and inversion and optimal design methods to
address emerging modalities in medical imaging, nondestructive testing,
and environmental inverse problems. Many mathematical and computational
challenging problems arise in emerging imaging techniques and
they often lead to the investigation of fundamental problems in various
branches of mathematics. The intensive lectures describe state of-the-art
in asymptotic imaging, stochastic modelling, and analysis of wave propagation
phenomena. They throw a bridge across these different aspects of
mathematical imaging. The intensive lectures provide deep understanding
of the different scales in the physical problem and an accurate modelling
of the uncertainty and noise sources in order to derive the best imaging
functional in the sense that it achieves the optimal trade-off between
signal-to-noise ratio and resolution. The intensive lectures also lead the
participants to appreciate the practical implementations and performance
evaluations of the described imaging methods.

Lecture 1: Applications of wave imaging, Introduction to active array imaging,
passive array imaging, time reversal experiments. Born approximations.

The aim of the intensive lectures is to introduce fundamental mathematical
and statistical tools, and inversion and optimal design methods to
address emerging modalities in medical imaging, nondestructive testing,
and environmental inverse problems. Many mathematical and computational
challenging problems arise in emerging imaging techniques and
they often lead to the investigation of fundamental problems in various
branches of mathematics. The intensive lectures describe state of-the-art
in asymptotic imaging, stochastic modelling, and analysis of wave propagation
phenomena. They throw a bridge across these different aspects of
mathematical imaging. The intensive lectures provide deep understanding
of the different scales in the physical problem and an accurate modelling
of the uncertainty and noise sources in order to derive the best imaging
functional in the sense that it achieves the optimal trade-off between
signal-to-noise ratio and resolution. The intensive lectures also lead the
participants to appreciate the practical implementations and performance
evaluations of the described imaging methods.

Lecture 2: Introduction to integral equations techniques; Some useful identities:
reciprocity, Green’s identities, Helmholtz-Kirchhoff identity.
A quick introduction to geometric optics. Structure of the response matrix
in the presence of electronic noise.

We say that a graph property is first order expressible if it can be written as a logic sentence using the universal and existential quantifiers with variables ranging over the nodes of the graph, the usual connectives AND, OR, NOT, parentheses and the relations = and ~, where x ~ y means that x and y share an edge. For example, the property that G contains a triangle can be written as
Exists x,y,z : (x ~ y) AND (x ~ z) AND (y ~ z).

 

Starting from the sixties, first order expressible properties have been studied extensively on the most commonly studied model of random graphs, the Erdos-Renyi model. A number of very attractive and surprising results have been obtained, and by now we have a fairly full description of the behaviour of first order expressible properties on this model.
The Gilbert model of random graphs is obtained as follows. We take n points uniformly at random from the d-dimensional unit torus, and join two points by an edge if and only their distance is at most r.
In this talk I will discuss joint work with S. Haber which tells a nearly complete story on first order expressible properties of the Gilbert random graph model. In particular we settle several conjectures of McColm and of Agarwal-Spencer.
(Joint with S. Haber)

We consider mixed methods for linear elastodynamics and linear viscoelasticity problems using mixed finite elements for elasticity. We use mixed finite elements for elasticity with weak symmetry of stress, which are advantageous in implementation and computational costs, and prove a priori error estimates. Our mixed methods have robustness for nearly incompressible materials in elastodynamics and provide other benefits for some viscoelastic materials.

Variational Models and Fast Numerical Schemes in Image Profcessing and Computer Vision Lecture 5 ● Mathematical preliminaries ● Image restoration, inpainting and Deblurring ● Fast numerical schemes ● Image segmentation and geometrical PDEs

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Inquiry: Prof. Chang-Ock Lee()

Variational Models and Fast Numerical Schemes in Image Profcessing and Computer Vision Lecture 4 ● Mathematical preliminaries ● Image restoration, inpainting and Deblurring ● Fast numerical schemes ● Image segmentation and geometrical PDEs

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Inquiry: Prof. Chang-Ock Lee()

We prove that the Lam-Shimozono “down operator” on the affine Weyl group induces a derivation of the affine Fomin-Stanley subalgebra of the affine nilCoxeter algebra. We use this to verify a conjecture of Berg, Bergeron, Pon and Zabrocki describing the expansion of k-Schur functions of “near rectangles” in the affine nilCoxeter algebra. Consequently, we obtain a combinatorial interpretation of the corresponding k-Littlewood–Richardson coefficients. This can be found in arxiv:1112.4460 and arxiv:1112.4460.

Variational Models and Fast Numerical Schemes in Image Profcessing and Computer Vision Lecture 3 ● Mathematical preliminaries ● Image restoration, inpainting and Deblurring ● Fast numerical schemes ● Image segmentation and geometrical PDEs

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Inquiry: Prof. Chang-Ock Lee()

Since the inception of gauge theory - Donaldson theory and Seiberg-Witten theory - in late 20 century, a mystery of smooth 4-manifolds has been unveiled and studying 4-manifolds has been the most active and central research area in geometry and topology. One of the fundamental problems in smooth 4-manifolds is to classify simply connected smooth and symplectic 4-manifolds. Topologists and geometers working on 4-manifolds have obtained many fruitful and striking results in this direction in last 30 years. In this lecture series first I'd like to review briefly Seiberg-Witten theory. And then, I'll survey various constructions of smooth 4-manifolds such as a generalized Logarithmic transform, a nullhomologous surgery, a p/q-surgery, a Luttinger surgery and a reverse engineering technique. Especially I'll review a reverse engineering technique in some details which turned out to be a powerful tool in the construction of fake 4-manifolds with small Euler characteristic. Finally I'll show how to construct a fake CP^2#3(-CP^2) using this technique.

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Schedule:

Lecture I (Aug. 6 (Mon)): A brief review on Seiberg-Witten theory (E6-1 #1409)
Lecture II (Aug. 7 (Tue)): A generalized Logarithmic transform surgery and a nullhomologous surgery (E6-1 #2411)
Lecture III (Aug. 13 (Mon)): A p/q-surgery and a Luttinger surgery (E6-1 #2412)
Lecture IV (Aug. 14 (Tue)): A reverse engineering - Statements and properties (E6-1 #2412)
Lecture V (Aug. 20 (Mon)): A reverse engineering - Sketch of Proof (E6-1 #1409)
Lecture VI (Aug. 21 (Tue))): A reverse engineering - A fake CP^2#3(-CP^2) (E6-1 #1409)

We will give  three lectures on points in a zero dimensional complete intersection

 in a projective space. We start with Macaulay's theorem about Hibert functions of homogeneous

 algebra and then present its modern development. This is a purely algebraic approach but it has

 many geometric consequences. I will introduce many related geomtric conjecures and their methology.

We will give  three lectures on points in a zero dimensional complete intersection

in a projective space. We start with Macaulay's theorem about Hibert functions of homogeneous

 algebra and then present its modern development. This is a purely algebraic approach but it has

 many geometric consequences. I will introduce many related geomtric conjecures and their methology.

In this talk, we introduce a novel class of active contour models for image segmentation. It makes use of non-local comparisons between pairs of patches within each region to be segmented. The corresponding variational segmenta-tion problem is implemented using a level set formulation that can handle an arbitrary number of regions. The pairwise interaction of features only constrains the local homogeneity of image features, which is crucial to capture regions with smoothly spatially varying features. This segmentation method is generic and can be adapted to various segmentation problems by designing an appropriate metric between patches. We instantiate this framework using several classes of features and metrics. Piecewise smooth grayscale and color images are handled using L2 distance between image patches. We show examples of ecient segmen-
tation of natural color images. Locally oriented textures are segmented using the L2 distance between patches of Gabor coecients. We use a Wasserstein distance between local empirical distributions for locally homogenous random textures. A correlation metric between local motion signatures is able to seg-ment piecewise smooth optical ows.

In this talk, we introduce a new method to segment an image into multiple regions. A multiple region segmentation problem is unstable since the result considerably depends on the number of regions given a priori. Therefore, one of the most important tasks in solving the problem is to automatically find the number of regions. The method we propose is able to find the reasonable number of distinct regions not only for clean images but also for noisy ones. Our method is made up of two procedures. First, we develop the adaptive global maximum clustering. In this procedure, we deal with an image histogram and automatically obtain the number of significant local maxima of the histogram. This number indicates the number of different regions in the image. Second, we derive a simple and fast calculation to segment an image composed of multiple regions. Then we split an image multiple regions according to the previous procedure. In the section of numerical results, we show the efficiency of our method by comparing it with other, previous methods.

Image restoration problems, such as image denoising, are the fundamental and important steps in various image processing method, such as image segmentation and object recognition. Due to the edge preserving property of the convex total variation (TV), variational model with TV is commonly used in restoring the clean image. However, staircase artifacts are frequently observed in restored smoothed region. To remove the staircase artifacts in smoothed region, convex higher-order TV (HOTV) regularization methods are introduced. But the valuable edge information of the image is also attenuated. In this paper, we propose non-convex hybrid TV regularization method to significantly reduce staircase artifacts while well preserving the valuable edge information of the image. To efficiently find a solution of the variation model with the proposed regularizer, we use the iterative reweighted method with the augmented Lagrangian based algorithm. The proposed model shows the best performance in terms of the signal-to-noise ratio(SNR) with comparable computational complexity.

We consider the linearly constrained `1-`2 minimization and propose the accelerated Bregman method for solving this minimization problem. The proposed method is based on the extrapolation
technique, which is used in accelerated proximal gradient methods studied by Nesterov, Nemirovski, and others, and the equivalence between the Bregman method and the augmented
Lagrangian method. O( 1k2 ) convergence rate is proved for the proposed method when it is applied
to solve a more general linearly constrained nonsmooth convex minimization problem.We numerically test our proposed method on the synthetic problem from compressive sensing. Numerical results confirm that the accelerated Bregman method is faster than the original Bregman
method.

Variational Models and Fast Numerical Schemes in Image Profcessing and Computer Vision Lecture 2

● Mathematical preliminaries

● Image restoration, inpainting and Deblurring

● Fast numerical schemes

● Image segmentation and geometrical PDEs

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Inquiry: Prof. Chang-Ock Lee()

Variational Models and Fast Numerical Schemes in Image Profcessing and Computer Vision Lecture 1

● Mathematical preliminaries

● Image restoration, inpainting and Deblurring

● Fast numerical schemes

● Image segmentation and geometrical PDEs

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Inquiry: Prof. Chang-Ock Lee()

삼성전자 반도체 사업부 세미나

 

YE팀(Yield Enhancement Team)은 반도체공정에 대한 Metrology & Inspection 기술 개발 및 적용을 담당하는 부서로 Nanometer 영역에 대한 최첨단 기술개발에 도전하고자 하는 능력 있는 우수인재들과 새로운 미래를 열어가고자 합니다.

 

▶ 세미나 요약  : 반도체 제작 공정에는 수백 개의 계측, 검사 공정이 포함되어 있으며 제작 공정 난이도 증가에 따라 계측 공정의 중요성도 점차 커지고 있습니다. 계측 기술이라 함은  다양한 광학, eBeam, X-ray 등의 source로부터 얻어진 data를 가공하여 원하는 정보를 추출하는 작업이며, 이에 Image processing, Data mining, Electro-magnetic simulation 등의 다양한 수학적 기술이 필요합니다.

본 세미나에서는 이와 같은 다양한 필요 기술을 소개하고자 합니다.

 

▶ 주요기술 

  - Simulation Technique (FDTD, RCWA, Monte Carlo Simulation)

  - Image processing  (Segmentation, Inspection)

  - Data mining (Classification, Clustering, Feature extraction)

  - Optics System Design (Microscopy, Ellipsometer, Interferometer)

  - Mechanical System Design (Stage Control, System Noise Analysis)

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세미나 후 별도의 취학 상담도 진행 하오니 관심 있는 학생분들의 많은 참여 바랍니다

Recently, Kenyon and Wilson introduced a certain matrix M in order to compute
pairing probabilities of what they call the double-dimer model. They showed that the absolute value of each entry of the inverse matrix M^-1 is equal to the number of certain Dyck tilings of
a skew shape. They conjectured two formulas on the sum of the absolute values of the entries in a row or a column of M^-1. In this talk we prove the two conjectures. As a consequence
we obtain that the sum of the absolute values of all entries of M^-1 is equal to the number of
complete matchings. We also find a bijection between Dyck tilings and complete matchings.

This talk is based on the following paper: arxiv:1108.5558.

Magnetic resonance electrical impedance tomography (MREIT) aims to visualize a conductivity distribution inside the human body. When we apply the harmonic Bz algorithm to measured Bz data from animal or human subjects, there occur a few technical difficulties that are mainly related with measurement errors in Bz data especially in a local region where MR signals are very small. We investigate sources of the error and its adverse effects on the image reconstruction process. We suggest a new error propagation blocking algorithm to prevent defective data at one local region from influencing badly on conductivity images of other regions. We experimentally examine the performance of the proposed method by comparing reconstructed images with and without applying the error propagation blocking algorithm.

We present an approximate converse theorem which measures how close a given set of irreducible admissible unramified unitary generic local representations of GL(n) is to a genuine cuspidal representation. To get a formula for the measure, we introduce a quasi-Maass form on the generalized upper half plane for a given set of local representations. We also construct an annihilating operator which enables us to write down an explicit cuspidal automorphic function.

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7월 6일~8월 31일 매주금요일

In this talk, we introduce a novel gauge construction for the Yang-Mills equations on the Minkowski space $bbR^{1+3}$, utilizing the properties of the associated Yang-Mills heat flow in a crucial way. The idea of constructing a gauge using the associated geometric heat flow, which goes under the name emph{caloric} gauge, was first proposed by Tao ('04) in the context of energy-critical wave maps and further developed by Bejenaru-Ionescu-Kenig-Tataru ('11) and Smith ('11) for energy-critical Schr"odinger maps. The novel gauge for the Yang-Mills equations is seen to possess a number of desirable properties; in particular, it does not have any issues with `large' data, in contrast with the classical Coulomb gauge. As the first demonstration of the structure offered by this new gauge, we will give an alternative proof of the global existence of solutions with finite energy to the Yang-Mills equation, a result first proved by Klainerman-Machedon ('95) using local Coulomb gauges.

An old problem raised independently by Jacobson and Schönheim asks to determine the maximum s for which every graph with m edges contains a pair of edge-disjoint isomorphic subgraphs with s edges. We determine this maximum up to a constant factor and show that every m-edge graph contains a pair of edge-disjoint isomorphic subgraphs with at least c (m log m)2/3 edges for some absolute constant c, and find graphs where this estimate is off only by a multiplicative constant. Our results improve bounds of Erdős, Pach, and Pyber from 1987. Joint work with Po-Shen Loh and Benny Sudakov.