By modifying the definition of moments of ranks and cranks, we study the odd moments of ranks and cranks. In particular, we prove the inequality between the first crank moment ‾M₁(n) and the first rank moment ‾N₁(n):

We also study new counting function ospt(n) which is equal to ‾M₁(n) – ‾N₁(n). We will also discuss higher order moments of ranks and cranks.
This is a joint work with G. E. Andrews and S. H. Chan.

For a given representation of a knot group, the twisted Alexander
polynomial(TAP) is obtained by taking the Reidemeister torsion of a certain
chain complex associated to the representation. In recent years TAPs have
been successfully used for finding many topological properties of knots
such as fiberedness, knot genus, mutation, knot concordance, and more. In
this talk, we introduce a new approach for the problem of detecting
fiberedness of knots using SL(2,C)-character varieties and TAPs. We

conjecture that TAPs associated to SL(2,C)-character varieties determine if
a knot is fiebered, and give evidence for the conjecture.

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

A general goal of noncommutative geometry (in the sense of Alain Connes) is to translate the main tools of differential geometry into the Hilbert space formalism of quantum mechanics by taking advantage of the familiar duality between spaces and algebras. In this setting noncommutative spaces are only represented through noncommutative algebras that play formally the role of algebras of functions on these (ghost) noncommutative spaces. As a result, this allows us to deal with a variety of geometric problems whose noncommutative nature prevent us from using tools of classical differential geometry. In particular, the Atiyah-Singer index theorem untilmately holds in the setting of noncommutative geometry. 

The talk will be an overview of the subject with a special emphasis on quantum space-time and diffeomorphism invariant geometry. In particular, if time is permitted,  it is planned to allude to recent projects in biholomorphism invariant geometry of complex domains and contactomorphism invariant geometry of contact manifolds. 

We show that the higher-order multilinear analogue  of the fractional integral operator due to L. Grafakos has the endpoint weak type boundedness.
Furthermore, we discuss that  the method about theorems of the multilinear fractional integral operator is related to the research of multilinear (trilinear)  Hilbert transform.

The Bergman Tau function is a holomorphic function defined over Teichm"uller spaces. This satisfies modular property with repsect to the mapping class group. In this talk, we will explain an infinite product expression of the Bergman Tau function. This can be considered as a generalization of the Dedekind eta function to higher genus case. The complex valued Chern-Simons functional will be introduced for this infinite product expression. We will also explain some corollaries of this result about the eta invariant and a Polyakov type formula. 

Weakly separated set families were first studied by Leclerc and Zelevinsky in the context of quantum flag variety. Two quantum Plücker coordinates quasi-commute whenever their indexing sets are weakly separated. It was conjectured that maximal such families always have the same size. Similar question was asked by Scott when she studied quantum Grassmannian. These conjectures were independently proved by Danilov-Karzanov-Koshevoy and Oh-Postnikov-Speyer using some planar graphs and by the author using truncation. In this talk, definitions and motivations for the weakly separated set families will be explained, including Oh-Postnikov-Speyer’s point of view on the subject. The proof of the purity conjecture using truncation will be provided, and related questions will be discussed.

With availability of high frequency financial data, recently growing numbers of related mathematical methods have been developed. Among those we explain the methods based on quadratic variation process of return series with applications to measurement, forecast and pricing.

We introduce critical surfaces, which are topological index two surfaces.

In this talk, we briefly introduce the theory of degree and its application to some elliptic PDEs.
Under certain symmetry of the system concerned, most natural deformation of PDEs
fall down to ODEs.   One can use equivariant degree formula to the system in this case.
We demonstrate the detailed analysis and gives some recent results.

An intertwine of two graphs G and H is a graph that has both G and H as a minor and is minor-minimal with this property. In 1979, Lovász and Unger conjectured that for any two graphs G and H, there are only a finite number of intertwines. This now follows from the graph minors project of Robertson and Seymour, although no ‘elementary’ proof is known.
In this talk, we consider intertwining problems for matroids. Bonin proved that there are matroids M and N that have infinitely many intertwines. However, it is conjectured that if M and N are both representable over a fixed finite field, then there are only finitely many intertwines. We prove a weak version of this conjecture where we intertwine ‘connectivities’ instead of minors. No knowledge of matroid theory will be assumed.
This is joint work with Bert Gerards (CWI, Amsterdam) and Stefan van Zwam (Princeton University).

We propose a new EIT image reconstruction algorithm using multiple
boundary voltage data from a planar array of voltage-sensing electrodes. The current
injection electrodes are placed so that the induced internal currents approximately
flow in the direction parallel to the surface of the voltage-sensing probe. The proposed
algorithm uses the interrelationship between the measured voltage differences and the
computed current, which allows us to derive a PDE-based Ohms law. Based on the
derived voltage-current relation, we produce images of admittivity changes within
a local region underneath the voltage-sensing probe. We describe the new image
reconstruction algorithm and its numerical simulation results.

We introduce a new invariant called the slope invariant
for all tunnels for tunnel number one knots in the 3-sphere, arising
from a study of the disk complex of the standard genus two handlebody
in the 3-sphere. The slope invariants have been calculated for some
well known tunnels, including all tunnels for 2-bridge knots and torus
knots, and all (1, 1)―tunnels. We introduce briefly a way to calculate
them, and their several applications in other related topics.

The study of vortex rings in incompressible 3D fluids dates back to Kelvin and Helmholtz in the mid 1800's. In 1906, Da Rios and Levi-Civita gave a formal derivation of a geometric flow for filaments of infinitely small cross section and arbitrary shape. This flow is now widely called the binormal curvature flow. In the talk, I will first review and then present recent results on stability estimates for the filament flow, and their application to so-called Schrodinger maps. 

It has been known for more than 20 years that certain semilinear
parabolic equations, such as the Allen-Cahn equation, exhibit interfaces that, in certainlimits, evolve by the mean curvature flow, a parabolic geometric evolution. We prove some analogous results relating semilinear hyperbolic
equations and certain hyperbolic geometric evolution problems.

Braids are beautiful objects in low dimensional topology. They can be seen likewise as tangles in the 3-ball or as elements of the mapping class group of the punctured disc or as automorphisms of free groups.We start by recalling the construction of the HOMFLYPT invariant for tangles , the Niesen-Thurston classification of diffeomorphisms of the punctured disc and the growth rate of  automorphisms of  free groups.
We present then our machinery for constructing 1-cocycles which produce HOMFLYPT invariants for 1-parameter families of tangles. It turns out that they contain information about the geometry of braids. There will be lots of examples in the talk.

We construct easily calculable invariants for tangles which can distinguish mutants without using cabling operations. They refine the HOMFLYPT and the 2-variable Kauffman invariant and they are no longer multiplicative under the composition of tangles.

As a first geometric application we show that the invariants can detect that a given braid is not isotopic to a rotation of the disc. We conjecture that they are a complete invariant for the geometric type of braids.

Imagine that you are cooking chicken at a party. You will cut the raw chicken fillet with a sharp knife, marinate each of the pieces in a spicy sauce and then fry the pieces. The surface of each piece will be crispy and spicy. Can you cut the chicken so that all your guests get the same amount of crispy crust and the same amount of chicken?
We show that if the number of guests is a prime power, n=p^k. Then such partition is possible. We derive this from a more general statement about equipartitions of convex bodies with respect to a measure and d-1 continuous functionals on the space of convex bodies, where d is the dimension the convex body sits in.
Our proof uses optimal transport and equivariant topology.

We construct easily calculable invariants for tangles which can distinguish mutants without using cabling operations. They refine the HOMFLYPT and the 2-variable Kauffman invariant and they are no longer multiplicative under the composition of tangles.

As a first geometric application we show that the invariants can detect that a given braid is not isotopic to a rotation of the disc. We conjecture that they are a complete invariant for the geometric type of braids.

In this introductory seminar, we first introduce the concept of "Quality of Service Guarantee"
that is important in the design of communication networks. We then investigate how to implement the concept in real communcation networks and what is the role of teletraffic theory in this regard.

We construct easily calculable invariants for tangles which can distinguish mutants without using cabling operations. They refine the HOMFLYPT and the 2-variable Kauffman invariant and they are no longer multiplicative under the composition of tangles.

As a first geometric application we show that the invariants can detect that a given braid is not isotopic to a rotation of the disc. We conjecture that they are a complete invariant for the geometric type of braids.

In the 1970s, Lovász and Plummer conjectured that every cubic bridgeless graph has exponentially many perfect matchings. This was proven by Voorhoeve for bipartite graphs and by Chudnovsky and Seymour for planar graphs. In this talk I will describe our proof of the general case, which uses elements of both aforementioned partial results as well as Edmonds’ characterization of the perfect matching polytope.
(Joint work with Louis Esperet, František Kardoš, Daniel Král’, and Sergey Norin.)

Shadow systems are often used to approximate
reaction–diffusion systems when one of the diffusion rates is large.
In this talk, we focus on the global existence and blow-up phenomena
for shadow systems. Our results show that even for these fundamental
aspects, there are serious discrepancies between the dynamics of the
reaction–diffusion systems and that of their corresponding shadow systems.

Based on Korea’s experience, this talk discusses issues related in massive capital inflows in emerging market countries and hopefully draws policy lessons for Asia and Pacific countries dealing with capital inflows problem.

Based on Korea’s experience, this talk discusses issues related in massive capital inflows in emerging market countries and hopefully draws policy lessons for Asia and Pacific countries dealing with capital inflows problem.

In this talk, we will discuss on the uniqueness for the Radon transform.

Our first problem is the global uniqueness.

It is interesting that the global uniqueness does not hold without any global decay condition on the function.

Our next problem is the uniqueness in the exterior problem. In this problem, it is well-known that it is essential to assume the rapid decay condition on the function.

In this talk, however, there is another essential condition on this problem.

Throughout these problems, we claim the importance the singularities of the function at infinity.

TBA

In this talk I will talk about the inviscid limit of Bejamin-Ono-Burgers (BOB) equation.  We prove that the Cauchy problem for the BOB equation is uniformly (with respect to the viscid parameter) globally well-posed in $H^s$ ($s\geq 1$) for all. Moreover, we show that the solution converges to that of Benjamin-Ono equation in $C([0,T]:H^s)$ ($s\geq 1$) for any $T>0$ as $\ve\to 0$. Our results give a new proof without gauge tranform for the global wellposedness of BO equation in $H^1$ which was first obtained by Tao \cite{TaoBO}, and obtain the inviscid limit behavior in $H^1$.