II. RIGHT-ANGLED ARTIN SUBGROUPS OF MAPPING CLASS GROUPS

In this lecture, we will discuss the primary result of [3], which roughly says that if we take any collection of mapping classes, say {f1,...,fk} and replace them by sufficiently high powers {f1^N,...,fk^N}, they generate a right-angled Artin subgroup of the mapping class group of the expected type. Unless otherwise noted, all examples and statements can be found with proof (or appropriate reference) in [3].

[1] Sang-hyun Kim and Thomas Koberda. Actions of right-angled Artin groups on quasi–trees. In preparation.

[2] Sang-hyun Kim and Thomas Koberda. Embedability of right-angled Artin groups. Preprint.

[3] Thomas Koberda. Right-angled Artin groups and a generalized isomorphism problem for finitely generated subgroups of mapping class groups. To appear in Geom. Funct. Anal.

I will give a quick conceptual introduction to motivic homotopy theory of Morel and Voevodsky, in such a way that those who know what derived functors in homological algebra can understand this subject expressed in terms of "homotopical algebra" of Quillen. In motivic homotopy theory, one hopes to do some homotopy theory using algebraic varieties as one does the usual homotopy theory for topological spaces. We use "motivic weak-equivalences" in this subject.

Next, I will explain some "descent theorems" in motivic homotopy theory. These theorems will allow us to handle the motivic weak-equivalences a bit better. 

I will sketch some applications of these machines.

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

In 1935 Erdős and Szekeres showed that every “sufficiently large” set of points in general position in the plane contains a “large” subset which is in convex position. Since then many mathematicians have tried to determine good bounds for “sufficiently large” in terms of “large”, as well as given numerous generalizations and refinements. In this talk I will survey this famous problem and extend it to a natural object which we call generalized wiring diagram. This unifies several proposed generalizations, and as a result we will settle several conjectures in this area.

This is joint work with Michael Dobbins and Alfredo Hubard.

A way to study the geometry of a homogeneous variety under a semi-simple algebraic group is to investigate its Chow group of algebraic cycles modulo the rational equivalence relation. In general, the problem of determining the Chow group of a projective homogeneous variety reduces to computing the torsion. In this talk, we discuss the latter problem including the cases of Severi-Brauer varieties and Spin-flags. 

아인슈타인의 브라운 운동에 관한 연구는 균일한 조건에서의 확산이 mean free path과 collision time interval time에 의해 주어진다는 것을 보였다. 

A graph G is called H-saturated if it does not contain any copy of H, but for any edge e in the complement of G the graph G+e contains some H. The minimum size of an n-vertex H-saturated graph is denoted by sat(n,H). We prove sat(n,C_k) = n + n/k + O((n/k²) + k²) holds for all n≥k≥3, where C_k is a cycle with length k.
Joint work with Zoltan Füredi.

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

When a prey population is infected, we study a predator-prey system with a ratio-dependent functional response under no-flux boundary condition. First, all nonnegative equilibria are  investigated, and then conditions which gives a stability at these equilibria are  found. Especially, disease-free and biological control states are discussed in view of  biological interpretations. Lastly,  the existence of nonconstant positive steady-states is studied. The methods employed  are  a comparison principle for a parabolic problem and Leray-Schauder Theorem. 

TBA

This is a prequel to Agol's lecture series on Virtual Haken Conjecture. We survey basic facts on cube complexes and discuss how those facts are related to the study of subgroups of right-angled Artin groups. The following notions will be defined while doing so: hyperbolic group, relatively hyperbolic group, cube complex, non-positive curvature, right-angled Artin groups, Salvetti complex, local isometry and pi-1-injectivity, hyperplane, special cube complex and subgroup separability.

There are no prerequsites for this talk, except for algebra and topology at undergraduate level.

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

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.

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.

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.