TBA

The Koszul complex is perhaps the most important complex in
commutative algebra. Its most striking features are its algebra
structure, its grade-sensitivity and its relations with syzygies.
Koszul cycles and homology themselves have an algebra structure. In a
regular local ring or in a polynomial ring the Koszul complex of the
maximal (homogeneous) ideal gives a resolution of the residue field.
In a non-regular ring the residue field has an infinite resolution
(which has also an algebra structure) "containing" the Koszul complex.
Commutative Koszul algebras are graded algebras whose residue field
has a free resolution which has a property in common with the Koszul
complex: it is linear. Here linear means that the matrices describing
the maps in the resolution have entries of degree 1. Koszul algebras
have several important properties. For certain homological aspects
they behave like polynomial rings, for other like complete
intersections. The main goal of the talks will be discuss these
features of Koszul algebras, their syzygies and bounds on the
regularity of Koszul homology in general. I will also present several
combinatorial/algebraic methods for proving that an algebra is Koszul
with some application to Veronese varieties and their projections.

I will present a new weighted voting classification ensemble method that uses two weight vectors: a weight vector of classifiers and a weight vector of instances.  The instance weight vector assigns higher weights to observations that are hard to classify.  The weight vector of classifiers puts larger weights to classifiers that perform better on hard-to classify instances. One weight vector is designed to be calculated in conjunction with the other through an iterative procedure. We proved that the iterated weight vectors converge to the optimal weights which can be directly calculated from the performance matrix of classifiers in an ensemble. The final prediction of the ensemble is obtained by the voting using the optimal weight vector of classifiers.  To compare the performance between a simple majority voting and the proposed weighted voting, we applied both of the voting methods to bootstrap aggregation and investigated the performance on 28 data sets. The result shows that the proposed weighted voting performs significantly better than the simple majority voting in general.

Abstract: 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 I'd like to review briefly some classical invariants of 4-manifolds and Seiberg-Witten theory with applications. And then, I'll survey various constructions of smooth 4-manifolds such as a logarithmic transform, fiber-sum and a knot surgery. Especially I'll review a knot surgery in some details which turned out to be a powerful tool in the study of smooth 4-manifolds.

Raviart-Thomas finite elements are very useful for problems
posed in H(div) since they are H(div)-conforming. We introduce two
domain decomposition methods for solving vector field problems posed
in H(div) discretized by Raviart-Thomas finite elements. A two-level
overlapping Schwarz method is developed. The coarse part of the
preconditioner is based on energy-minimizing extensions and the local
parts consist of traditional solvers on overlapping subdomains. We
also consider a balancing domain decomposition by constraints (BDDC)
method. The BDDC preconditioner consists of a coarse part involving
primal constraints across the interface between subdomains and local
parts related to the Schur complements corresponding to the local
subdomain problems.

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.