Given a fixed alphabet, a word of length n is an n-tuple with entries in the alphabet. A hole is a character outside the alphabet that is viewed as representing any letter of the alphabet. A partial word is a string where each character is a hole or belongs to the alphabet. Two partial words having the same length are compatible if they agree at each position where neither has a hole.
A square is a word formed by concatenating two copies of a single word (no holes). A partial word W contains a square S if S is compatible with some (consecutive) subword of W. Let g(h,s) denote the maximum length of a binary partial word with h holes that contains at most s distinct squares. We prove that g(h,s)=∞ when s≥4 and when s=3 with h∈{0,1,2}; otherwise, g(h,s) is finite. Furthermore, we extend our research to cube-free binary partial words.
This is joint work with Dr. Francine Blanchet-Sadri and Robert Mercas.

Toric topology is the study of various aspects of topological spaces
with torus actions. In this talk, I will introduce quasitoric
manifolds and cohomological rigidity problems in toric topology. And
then I will give partial affirmative answers to the cohomological
rigidity problems; if the cohomology ring of a quasitoric manifold is
isomorphic to that of the projectivization of the Whitney sum of
complex line bundles over a 4-dimensional toric manifold, then the
cohomology ring determines not only the orbit space but also the
homeomorphism type of the quasitoric manifold in some cases. This talk
is based on the joint work with Suyoung Choi.

A graph $n$-braid group is the fundamental group of the configuration
space of $n$ points on a graph. It was introduced by Ghrist and Abrams
to apply topology to robotics in 1999.
This talk is an introduction to graph braid groups with some
discussions of their presentaions and the homology groups.

Let $\Sigma$ be a compact, connected, orientable surface of genus
$g\ge 1$ with boundary and $\bar{\mathbf{x}}^0=\{x_1^0,\dots,x_n^0\}$
be a distinct points in the interior of $\Sigma$.
Then the braid group $\mathbf{B}_n(\Sigma,\bar{\mathbf{x}}^0)$ is
defined as the fundamental group of configuration space.
For given automorphism $\phi$ on
$\mathbf{B}_n(\Sigma,\bar{\mathbf{x}}^0)$, we say that $\phi$ is {\em
geometric} if there exists an automorphism $f$ on
$(\Sigma,\bar{\mathbf{x}}^0)$ such that the induced map $f_*$ on
$\mathbf{B}_n(\Sigma,\bar{\mathbf{x}}^0)$ is $\phi$.
In this talk, we present the necessary and sufficient condition for
$\phi$ to be geometric.

In this talk, I will give a brief introduction to Nonparametric (NP) Bayesian statistical modeling. First, I will describe some key components of Bayesian statistical inference. Then, I will begin with a statistical modeling example for which parametric modeling may have limitations and introduce the NP Bayes methodology for more flexible modeling. Focuses will be on NP Bayes approaches involving Dirichlet process (DP). I will also discuss computation-based inference procedure focusing on Markov Chain Monte Carlo (MCMC). I will conclude with a summary and some discussions of future research directions.

The porous medium equation is a nonlinear diffusion equation modelling
gas flow through a porous medium. It has a peculiar property called
`finite propagation'; the solution is compactly supported if initial
data is. In this talk, we will derive the equation and survey its
basic properties related to finite propagation. Also concept of
`intermediate asymptotics' will be introduced.

In complex analysis of several variables, the celebrated theorem,
which is called Cartan uniqueness theorem, is the following: Let
$\Omega$ be a domain in $\mathbb{C}^n$ and $p$ be a point in $\Omega$.
If a holomorphic function $f$ from $\Omega$ to $\Omega$ satisfies that
$f(p)=p$ and $df_p=\mathrm{Id}$, then $f$ is the identity map. In this
talk, first we discuss the proof of Cartan uniqueness theorem briefly.
Then we discuss the unique theorem for CR and conformal mappings.

http://mathsci.kaist.ac.kr/~manifold/Arithmetics.html

A phylogenetic (i.e evolutionary) tree can be interpreted as a compatible split system, that is a collection of bipartitions of a finite set X such that, for all four elements of X, there are no two bipartitions in the collection that induce different splits of those four elements into two pairs. Such a split of a 4-set into two 2-sets is called a quartet, and a split system is said to display a quartet, if there is at least one split in the system that induces this quartet. In phylogenetics, it is often useful to allow more general than compatible split systems, in order to display contradicting signals in the data or to find evidence for reticulate evolution. One natural such generalization are weakly compatible split systems, where for every 4-set at most two of the three possible quartets are allowed to be displayed. The split decomposition algorithm (implemented in the Splitstree software) is a successful tool to construct weakly compatible split systems from distance data. However, weakly compatible split systems are not as well understood as compatible ones. For example, maximal compatible split systems, i.e. compatible split systems which become incompatible whenever a new split is added, correspond to binary trees and display one quartet for every 4-set. In contrast, maximal weakly compatible split systems often display less than the two quartets per 4-set that are allowed by definition. Indeed there are examples where no quartet is displayed for almost all 4-sets. This leaves the question what is the minimum cardinality of maximal weakly compatible split systems for given cardinality of X.
In my talk I will introduce weakly compatible split systems and explain their relevance for phylogenetics, and I will present upper and lower bounds for the smallest number of quartets in maximal weakly compatible split systems.

One of the most fundamental question in algebraic geometry is whether a certain algebraic variety is birational to a projective space or not. Even in the case of hypersurfaces in projective spaces, this question is far from being easy. Our toy in the talk is a certain quartic hypersurface in 7-dimensional projective space, called the 'Coble quartic'. It is the moduli space of (S-equivalent classes of) semi-stable vector bundles of rank 2 on a non-hyperelliptic curve of genus 3 with canonical determinant. We introduce the rationality problem and explain some geometry of this space.

Several years ago, nonconforming linear finite element spaces were suggested
over quadrilateral meshes. As a pressure component, it makes a simple
mini element pair to solve Stokes equations. In another application, we
will introduce a divergence-free subspace of it over square meshes. The dimension
of the proposed space is the number of squares contained in the
interior of the domain. The discrete H1 interpolation error for a continuous
divergence-free function is analyzed to be O(h) which is optimal as a linear
approximation. A basis for the proposed space is clarified for practical usage
and several numerical tests are shown.

The aim of the Schubert calculus is observing multiplications between the cohomology classes of a given manifold whose cohomology is a free module. The purposes of this seminar are to look into what the Schubert calculus is and to provide the computing algorithm by using Young tableaux. The cup product between the cohomology classes is related with the intersection of the proper submanifolds, so we can get the solution of Schubert's quiz, which is finding number of lines in 3-space which intersect four given lines. This quiz is the origin of the Schubert calculus. The key which solves this quiz is observing the Schubert calculus in flag varieties.

Microscopic properties of eigenvalues of normal random matrices change drastically in a narrow belt around the edge of the spectrum. I present an elementary method to prove Borodin and Sinclair's theorem on the scaling limit of correlation kernels for the soft-edge Ginibre ensemble. This method gives new result for the hard-edge Ginibre ensemble. After a discussion of the general properties of this scaling limit, I state a universality conjecture and provide arguments to support it. This is a joint work with Y. Ameur and N. Makarov.

임의의 수학적인 구조가 있을때, 그 대칭성들의 집합은 군을 이룹니다.

역으로, 임의로 주어진 군을 연구하는 데에는 그에 대응하는 특별한 그래프가 효과적으로 쓰일 수 있습니다.

본 강연에서는 그래프처럼 미적분이 정의되지 않는 공간에서 기하학적인 성질들을 찾아내는 방법을 소개하고, 이 접근이 어떻게 무한군의 성질들을 밝히는 데에 쓰일 수 있는지 소개합니다.

A nonnegative harmonic function defined on an open ball in
\R^d(d\ge 2)can be represented as an integral over the Euclidean
boundary of the ball. For the general domain, it is represented as an
integral over the Martin boundary. We consider the problem of
identification of Martin boundary and Euclidean boundary.

We will construct the primitive generators of the ray class
fields over imaginary quadratic fields by using the singular values of
suitable powers of Siegel functions. We investigate the algorithm for
finding all Galois conjugates of singular values obtained by Gee and
Stevenhagen and Shimura's reciprocity law. And then, by comparing the
absolute values of all Galois conjugates of given singular value, we
construct the ray class invariants of a given imaginary quadratic
field K.

Furthermore, we obtain the normal bases for class fields by using the
singular values of suitable powers of Siegel functions.

We introduce a new class of rate one half codes, called complementary information set codes. A binary linear code of length 2n and dimension n is called a complementary information set code (CIS code for short) if it has two disjoint information sets. This class of codes contains self-dual codes as a subclass. It is connected to graph correlation immune functions of use in the security of hardware implementations of  cryptographic primitives. In this talk, we give optimal or best known CIS codes of length <132. We  derive general constructions based on cyclic codes, double circulant codes, strongly regular graphs, and doubly regular tournaments. We derive a Varshamov-Gilbert bound for long CIS codes, and show that they can all be classified in small lengths up to 12 by the building up construction. This is a joint work with Claude Carlet, Philippe Gaborit, and Patrick Sole.

http://mathsci.kaist.ac.kr/~manifold/Arithmetics.html

We survey the theory of non-abelian Galois representations and its applications, as developed

out of the anabelian programme of Grothendieck from the 1980's.

The Internet has evolved into a vast heterogeneous system, comprised of independent selfish users who value the benefits they derive from the network much more than the efficiency of the network as a whole. Independent selfish user behavior results in a non-cooperative distributed network environment that increasingly undermines many important congestion control schemes (e.g., TCP and CSMA/CA) relying on voluntary participation. Considering that selfish users may modify the current standard congestion protocol and exploit knowledge of the cooperative behavior of others, we have abandoned the paradigm of assumed-cooperative users in the networks and considered networks with only selfish users who pursue only their own benefit.

In this seminar, we will focus on transmission rate control algorithms with selfish users in wireless networks. This seminar covers non-cooperative games among selfish users in random access networks, specifically networks using simple slotted ALOHA or IEEE 802.11 DCF. A transmission rate control algorithm in ALOHA networks and a generalization of the rate control algorithm will be discussed. A transmission rate control algorithm for a variant of IEEE 802.11 DCF will be proposed to consider selfish user behaviors. Existence and uniqueness of equilibrium points of the proposed algorithm will be explored. Convergence properties of the equilibrium points will be studied via Lyapunov stability theory.

Multiple hovering UAVs equipped with signal capturing sensors are used for target tracking due to economic efficiency. Comparing the received signals reveals the differences in the distances to the target from different UAVs, and the statistical analysis of the data allows for the estimation of the target location. The estimation process can be simplified by employing geometric approximations suitable for practical applications. Minimizing an objective function with an Lagrange addivity constant solves the problem of finding the maximum likelihood estimator for the target location. A sensitivity analysis of this process provides a useful suggestions as to which flight formation is more efficient than others. (Joint work with prof. Sung-Ho Kim, ADD project)

This will be a series of roughly 6-8 lectures. The first half will be on well-known and classical material on equivariant cohomology and Schubert calculus, while I will concentrate on my recent joint work with Tymoczko, Bayegan, and Dewitt in the second half of the lecture series.

We discuss initial boundary value problems in the regime of classical solutions to the Vlasov-Poisson system with large data. We also talk on the exponential time decay rate of smooth solutions of small amplitude to the Vlasov-Poisson-Fokker-Planck equations to Maxwellian.

Hessenberg varieties are a class of subvarieties of the flag variety which appear in many areas, e.g. in geometric representation theory. In order to generalize Schubert calculus to Hessenberg varieties, a first step is to construct computationally convenient module bases for the (equivariant) cohomology rings of Hessenberg varieties analogous to the famous Schubert classes which are a basis for the cohomology of flag varieties. Goresky-Kottwitz-MacPherson ("GKM") theory gives a concrete combinatorial description of the equivariant cohomology of spaces with torus action which satisfy certain conditions (usually called the GKM conditions). We propose a framework for approaching the problem of constructing module bases for Hessenberg varieties which uses GKM theory. The main conceptual challenge in this context is that conventional GKM theory requires a `sufficiently large-dimensional torus' action (to be made precise in the talk), while Hessenberg varieties generally have only a circle action. To resolve this, we define the notion of GKM-compatible subspaces of GKM spaces and give applications in some special cases of Hessenberg varieties. The talk will be intended for a wide audience, and in particular I will begin with a conceptual sketch of the main ideas in Schubert calculus and of classical GKM theory.

This is mainly joint work with Tymoczko; time permitting, I will mention joint work with Bayegan, and also with Dewitt.

This will be a series of roughly 6-8 lectures. The first half will be on well-known and classical material on equivariant cohomology and Schubert calculus, while I will concentrate on my recent joint work with Tymoczko, Bayegan, and Dewitt in the second half of the lecture series.

This will be a series of roughly 6-8 lectures. The first half will be on well-known and classical material on equivariant cohomology and Schubert calculus, while I will concentrate on my recent joint work with Tymoczko, Bayegan, and Dewitt in the second half of the lecture series.

Kashaev volume conjecture suggests that the limit of the colored Jones polynomial of a hyperbolic knot gives the volume and the Chern-Simons invariant of the knot complement. It is one of the most important problems in quantum topology because it shows many non-trivial relations between many areas including knot theory, hyperbolic geometry, quantum group, extended Bloch group, and more.

In this talk, we survey many aspects of the volume conjecture. Especially, we focus on how the extended Bloch group plays a crucial role in proving the optimistic limit version of the volume conjecture.

In signal processing/communications, an analogue (or continuous)
signal is represented by its discrete counterpart which is called
samples of the analog signal. Since it is inevitable in practice that
some of the samples are missing during transfer, not only engineers
but also mathematicians have been trying to circumvent this problem
with various kind of approaches. For band-limited signals, It is well
known that any finitely many missing samples can be recovered from the
remaining known samples when the signal is oversampled at a rate
higher than the minimum Nyquist rate. In this talk, we consider a
similar problem of recovering missing samples for signals in shift
invariant spaces.

We study an optimal portfolio and consumption choice problem of a family that combines life insurance for parents who receive deterministic labor income until the fixed time T. We consider utility functions of parents and children separately and assume that parents have an uncertain lifetime. If parents die before time T, children have no labor income and they choose the optimal consumption and portfolio with remaining wealth and life insurance benefit. The object of the family is to maximize the weighted average of utility of parents and that of children. We obtain analytic solutions for the value function and the optimal policies, and then analyze how the changes of the weight of the parents’ utility function and other factors affect the optimal policies.

Given an even subset T of the vertices of an undirected graph, a T-join is a subgraph in which the subset of vertices with odd degree is exactly T. Given edge weights, the weighted T-join problem is to find a T-join of minimum weight. With nonnegative edge weights, the problem can be reduced to finding a minimum weight perfect matching on the metric completion of the vertices in T.
Given an undirected graph with nonnegative edge weights but no specific T, the Max-Min Weighted T-join problem is to find an even cardinality vertex subset T such that the minimum weight T-join for this set is maximum. The unweighted case of the problem when all weights are either unit or infinity has been well characterized by a decomposition of the underlying graph into factor critical and matching-covered bipartite subgraphs (Frank1993). We consider the weighted version which is NP-hard even on a cycle. After showing a simple exact solution on trees, we present a 2/3-approximation algorithm for the general case. Our algorithm is based on a natural cut packing upper bound obtained using an LP relaxation and uncrossing, and relating it to the T-join problem using duality.
This is a joint work with R. Ravi.

This will be a series of roughly 6-8 lectures. The first half will be on well-known and classical material on equivariant cohomology and Schubert calculus, while I will concentrate on my recent joint work with Tymoczko, Bayegan, and Dewitt in the second half of the lecture series.

Submodular functions are discrete analogues of convex functions.

Examples include cut capacity functions, matroid rank functions,

and entropy functions. Submodular functions can be minimized in

polynomial time, which provides a fairly general framework of

efficiently solvable combinatorial optimization problems.

In contrast, the maximization problems are NP-hard and several

approximation algorithms have been developed so far.

In this talk, I will review the above results in submodular

optimization and present recent approximation algorithms for

combinatorial optimization problems described in terms of

submodular functions.

This will be a series of roughly 6-8 lectures. The first half will be on well-known and classical material on equivariant cohomology and Schubert calculus, while I will concentrate on my recent joint work with Tymoczko, Bayegan, and Dewitt in the second half of the lecture series.

In the very first lecture, I intend to give a very introductory overview lecture on Schubert calculus, concentrating on the most classical case: that of the (cohomology of the) Grassmannian of k-planes in complex n-space. This lecture will be elementary, requiring (mostly) only undergraduate material (mainly linear algebra), but I hope to give the broad overview of a beautiful and elegant subject where algebra, combinatorics, and geometry come together. It will serve as a motivation for the entire lecture series.


In the next lecture(s), I intend to give a broad overview of the theory of equivariant cohomology and, in particular, the so-called `GKM' (Goresky-Kottwitz-MacPherson) theory which gives, under suitable conditions, a combinatorial description of the equivariant cohomology in terms of the data of the equivariant 1-skeleton of a space equipped with a group action. I will also discuss some of the Morse-theoretic interpretations of aspects of GKM theory, and in particular take some time to discuss how GKM theory provides a good technology for constructing computationally convenient module bases for equivariant cohomology.

In a following lecture(s) I will review the basic geometry of flag varieties associated to compact Lie groups (or complex reductive algebraic groups), including the description of the Schubert varieties sitting inside the flag varieties. I will also briefly recall the corresponding objects in equivariant cohomology, namely the Schubert classes. I will briefly describe some of the known formulas for computing products of certain of these classes. Finally, I will explain some Bruhat-order combinatorics and the Sara Billey formula for computing localizations of Schubert classes at T-fixed points, and how this is another method for a computation `in principle' of the structure constants in the equivariant cohomology ring.


In the next lecture(s) I will describe in some detail my recent work with Julianna Tymoczko and Darius Bayegan, in which we generalize classical Schubert calculus to a certain class of subvarieties of flag varieties, namely, the Peterson varieties. We exploit the GKM theory on the ambient flag variety in order to derive a computationally convenient module basis for the S^1-equivariant cohomology ring of Peterson varieties. I will explain in detail how this is done, and also explain some consequences of our approach: namely, Tymoczko and I derive a manifestly-positive and manifestly-integral Monk formula for the structure constants, and Bayegan and I similarly derive a Giambelli formula.

In the following lecture(s) I will explain how to generalize the techniques in my work with Tymoczko-Bayegan on Peterson varieties. In particular I explain the key new concept of `GKM-compatible subspaces' of an ambient GKM space. The crucial idea is that, although a subspace X of a GKM space Y may not be itself GKM, the inclusion map of X into Y (and associated map on equivariant cohomology) may have good enough properties that we can deduce properties about the equivariant geometry of X through the GKM theory on Y. Motivated by the notion of GKM-compatibility, Tymoczko and I also introduce a new combinatorial game called `poset pinball', which in some situations allow us to explicitly build computationally convenient module bases analogous to the Schubert classes in Schubert calculus.

In the last lecture(s) I will explain concrete examples and applications of the general techniques of GKM-compatibility and poset pinball introduced above, which are contained in my joint work with Tymoczko, Bayegan, and Dewitt. For example, we give an explicit construction of an equivariant lift of the classical Springer action on the cohomology of subregular Springer varieties of type A.


For further reading:

Kleiman-Laksov, Schubert Calculus. The American Mathematical Monthly, Vol. 79, No. 10, (Dec., 1972), pp. 1061-1082.

Allen Knutson, The symplectic and algebraic geometry of Horn's problem.  http://arxiv.org/abs/math/9911088

Julianna Tymoczko. An introduction to equivariant cohomology andhomology, following Goresky, Kottwitz, and MacPherson. Snowbird lectures in algebraic geometry. AMS.

Julianna Tymoczko. Permutation actions on equivariant cohomology of flag varieties. Toric Topology conference proceedings. AMS.

Tara Holm. Act globally, compute locally: group actions, fixed points, and localization. Toric Topology conference proceedings. AMS.

Fulton, Young tableaux. Cambridge University Press.

Megumi Harada and Julianna Tymoczko, A positive Monk formula in the S^1-equivariant cohomology of type A Peterson varieties. Proc. London Math. Soc. (2011) doi: 10.1112/plms/pdq038.

Megumi Harada and Julianna Tymoczko, Poset pinball, GKM-compatible subspaces, and Hessenberg varieties. http://arxiv.org/abs/1007.2750

Darius Bayegan and Megumi Harada, A Giambelli formula for the $S^1$-equivariant cohomology of type A Peterson varieties.
http://arxiv.org/abs/1012.4053

Darius Bayegan and Megumi Harada, Poset pinball, the dimension pair algorithm, and type A regular nilpotent Hessenberg varieties.
http://arxiv.org/abs/1012.4054

Barry Dewitt and Megumi Harada, Poset pinball, highest forms, and (n-2,2) Springer varieties. http://arxiv.org/abs/1012.5265

This will be a series of roughly 6-8 lectures. The first half will be on well-known and classical material on equivariant cohomology and Schubert calculus, while I will concentrate on my recent joint work with Tymoczko, Bayegan, and Dewitt in the second half of the lecture series.

 In this seminar, I will talk about the properties and the classification of embeddings of homogeneous spaces, especially the case of affine normal embeddings of reductive groups. We might guess that as in the case of toric varieties, some specific subset of one-parameter subgroups may contribute to the classification of affine embeddings of general reductive group. To check this, we review the theory of affine normal SL(2)-embeddings, and prove that the classification cannot be solved entirely based on one-parameter subgroups. We can also observe that even though this set does not give a complete answer to the classification problem, but still contains useful information about varieties. If time permitted, I will also give examples of GL(2)-embeddings which had not previously been constructed in detail, which might be helpful in understanding the general classification of affine normal G-embeddings.

 In this talk, we establish an oscillation estimate of nonnegative harmonic functions for a large class of integro-differential
operators. Such operators are
the infinitesimal generators of pure-jump subordinate Brownian motion. As an application, we give a probabilistic proof of relative Fatou  theorem for harmonic functions
for the integro-differential operators in bounded $\kappa$-fat open set.   That is, if $u$ is a positive harmonic function in  a bounded $\kappa$-fat open set $D$ and $h$ is a positive  harmonic function in $D$ vanishing on $D^c$, then the non-tangential limit
 of  $u/h$ exists almost everywhere with respect to the Martin-representing measure of $h$.
Under the gaugeability assumption, relative Fatou theorem is true for operators obtained from the generator of pure-jump subordinate Brownian motion in bounded $\kappa$-fat open set $D$ through non-local Feynman-Kac transforms.