Nonlinear hydrodynamic instability theory has made notable successes
in predicting and indeed in providing control mechanisms for
transition to turbulence. The theory has had a major impact on
problems relevant to the mechanical, chemical and aeronautical
engineering. The theory concerns the solution of the 3D unsteady
Navier Stokes equations by a combination of analytical and numerical
means. Here we discuss the relevance of the theory to geophysical
flows and in particular discuss how river patterns and migrations can
be predicted mathematically. Several new nonlinear pde evolution
equations are derived and shown to reproduce several key features of
braided rivers.

In 1770, Lagrange proved that every nonnegative integer is the sum of four squares. Waring's problem is the generalization of Lagrange's theorem. More generally, we will introduce Waring's problem for polynomials and talk about the asymptotic order of a set of some polynomials.

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

We present a mathematical model of left heart governed by the partial differential equations. This heart is coupled with a lumped model of the whole circulatory system governed by the ordinary differential equations. The immersed boundary method is used to investigate the intracardiac blood flow and the cardiac valve motions of the normal circulation in humans. We investigate the intraventricular velocity field and the velocity curves over the mitral ring and across outflow tract. The pressure and flow are also measured in the left and right heart and the systemic and pulmonary arteries. The simulation results are comparable to the existing measurements.

Inverse problems are ill-posed and have virtually no solution. However, a-priori knowledge of the medium may reduce ill-posedness significantly. One such knowledge is smallness of the inclusion. I will talk about the method of small volume expansions to image small inclusions and its applications to emerging modalities of medical imaging such as MRElastography and Photo-acoustic Imaging.

The hypertoric manifold is defined by the hyperKahler analogue of symplectic toric manifolds. In usually, the toric manifold is a 2n-dim manifold with an n-dim torus action. On the other hand, the hypertoric manifold is a 4n-dim manifold with an n-dim torus action. However, we can apply the method of toric geometry or toric topology to analyze the hypertoric manifolds. In this talk, I introduce the hypertoric manifold and the method to analyze it from topological point of view, and prove that its equivariant diffeomorphism type is determined by the equivariant cohomology. 

Tropical geometry might loosely be described as algebraic geometry over the tropical semiring. It has deep connections to numerous branches of pure and applied mathematics, including algebraic geometry, combinatorics, and computational algebra. In this talk, I will explain the definition and properties of a tropical linear space, and how it is related to various areas of mathematics and computational biology.

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

Lovász and Plummer conjectured that there exists a fixed positive constant c such that every cubic n-vertex graph with no cutedge has at least 2^cn perfect matchings. Their conjecture has been verified for bipartite graphs by Voorhoeve and planar graphs by Chudnovsky and Seymour. We prove that every claw-free cubic n-vertex graph with no cutedge has more than 2^n/18 perfect matchings, thus verifying the conjecture for claw-free graphs.

This talk studies two examples of singular perturbations for particle systems. The first example is based on classical Tichinov theory for ODEs and applied to flocking. The second example uses a new non-classical averaging method and is applied to a KdV-Burgers type equation.

The similarity structure of certain convection or diffusion equations are well-known. The fundamental solutions of such problems are given explicitly and called self-similar solutions. The N-waves for the Burgers equation, the Gaussian for the heat equation and the Barenblatt solution for the porous medium equation are examples. These self-similar solutions have been played key roles in the theoretical development. However, there is no systematic approach to handle these similarity structures in a single frame. In this talk we introduce a method to derive similarity solution which is applicable to convection and diffusion equations.

It is a classical result due to Grothendieck that every vector bundles on the projective line is a direct sum of line bundles. Using this, there have been many attempts to understand vector bundles on the projective space, for example, by W.Barth and K.Hulek. In this talk, we introduce this idea in the case of smooth quadric surface. In the first half of the talk, we explain the basic notions in the algebraic geometry that will be used in the talk and recall several results on the projective space. In the second, we introduce the notion of jumping conics and prove that the set of jumping conics associated to a stable vector bundles on a smooth quadric surface forms a hypersurface in a 3-dimensional projective space. Using this, we explicitly describe the moduli spaces of stable vector bundles in two cases and see how these description can be applied to prove other classical results.

We discuss a relationship between a class of derivations and a class of automorphisms on the noncommutative algebra of formal power series in two variables. Each class relates bijectively by exponential and logarithm maps. In this talk we define a specific class of derivations, which generates a noncommutaive Lie algebra whose defining relations are related to a classical Witt algebra. The main claim in the talk is the explicit description of the automorphisms which are corresponding to the derivations via exponential map.

(Note: Dr. Kentaro Ihara wrote a famous Compositio Math. paper with. Don Zagier on multi-zeta function, and number theorists are particularly welcomed to come.)

Packets of large amplitude internal solitary waves generated by the interaction of density-stratified flows with bottom topography have been observed in many coastal regions. Their wave amplitudes often exceed 100 m and, therefore, these waves cannot be described by classical weakly nonlinear models. A strongly nonlinear asymptotic model to describe such waves was proposed some time ago, but the model has been known to suffer from the Kelvin-Helmholtz instability and to be ill-posed. In this talk, a regularized model free from such instability will be introduced and an iterative numerical scheme to solve the regularized time-dependent model will be discussed.

We discuss structures of few financial derivatives and
explain why such financial products are built up and traded.

The graph finding problem is to find the edges of an unknown graph by
using a certain type of queries. Its extension to hypergraphs is
closely related to the problem of learning linkage in molecular
biology and artificial intelligence. In this talk, we introduce the
hypergraph finding problem and the linkage learning problem and
present our recent results for the query complexity of those problems.

Mathematical logic deals with symbols and strings, which are specialties of computers. So it is natural to use computers in practicing logic in various occasions. For instance, the verification of formal proofs, which are just the strings that follow some specific rules, is quite often tedious and time consuming--it is a job best suited for computers.

In this talk, a formal proof system called "Fitch" is introduced, and demonstrated as implemented at http://www.proofmood.com.

I will construct a moduli space of $q$ pairwise commuting nilpotents of $\mathfrak gl_d$ and give a natural compactification of it for the case $d = 3$.

Recently, a new sampling theory called compressive sampling theory was
proposed in signal processing community. According to compressive
sampling, very accurate reconstruction is possible even from very
limited data measurements which breaking Nyquist sampling limit if the
unknown signal is sparse. Furthermore, even if the signal itself is
not sparse, as long as it can be represented sparsely by appropriate
sparsifying transform, compressive sampling can be still very
effectively applied. In this talk, we introduce the basic theory for
compressive sensing, and demonstrate how this can be applied to
various bio-imaging area.

This is an introductory talk on higher Chow groups. It will be understandable for graduate students in algebra and geometry. A Chow group is used by various mathematicians in various fields. For complex geometers, Chow groups are the place where the fundamental cohomology classes originate. For number theorists, Chow groups are equal to the ideal classes groups. For some people, the group of line bundles, so called the Picard groups, is given by this. For those work in Riemann surfaces, a subgroup of a Chow group is named the Jacobian variety. I will explain how this object is related to these, and how one can see this object as an analogue of singular homology for algebraic varieties. 

In shape matching, we are given two geometric objects and we compute their distance according to some geometric similarity measure. The Fréchet distance is a natural distance function for continuous shapes such as curves and surfaces, and is defined using reparameterizations of the shapes.

Motivated by a practical application in designing safe control scheme
for automated guided vehicles or robots in industrial settings, graph
braid groups were first proposed and studied by R. Ghrist and A.
Abrams in 1999. We will quickly summarize the history of the theory
including the recent breakthrough on a conjecture by the pioneers.

How can we recognize mapping spaces from other spaces? 
We can use natural operations on them as a universal algebra and  we may use this to recognize mapping spaces up to weak equivalence. In the case of mapping spaces, n-fold loop spaces, of pointed maps from the n-sphere, we can show that any space X having such a universal algebra structure is weakly equivalent to the n-fold loop space of B(X), delooping space. I will explain its categorical frame so that it may be applicable to other problems, e.g.,moduli spaces, deformation problems etc.