Summary
In this talk inversion (=reconstruction) schemes for active thermography and time resolved diffuse optical tomography to identify unknown inclusions and their physical properties are given. The measured data are Neumann to Dirichlet map and Robin to Neuman map, respectively. By defining some indicator functions via the measured data with some inputs, the identifications are done by looking at the behavior of the indicator functions. The underlying analysis is the short time asymptotic of fundamental solution of diffusion equations with discontinuous coefficients.
11:30--12:30
Jun-Muk HWANG (KIAS)
Compactifications of $\mathbb{C}^n$
Summary
We will give an overview of the study of compactifications of ${\bf C}^n$ as projective algebraic manifolds with $b_2=1$. Our particular interest is on equivariant compactifications, an area of research initiated by Hassett-Tschinkel's work. After surveying some previous works on the subject, we will report on a recent joint work with Baohua Fu employing the method of varieties of minimal rational tangents.
Algebra Session at Main Hall
14:00--14:40
Yukinobu TODA (Univ. of Tokyo)
Stability conditions and birational geometry
Summary I will propose a conjecture which claims that the Minimal Model Program for a smooth projective variety is realized as a variation of Bridgeland moduli spaces of semistable objects in the derived category of coherent sheaves on it. I will discuss the surface case and extremal contractions for 3-folds. In the former case, the conjecture is completely solved. In the latter case, the conjecture is related to a conjectural Bogomolov-Gieseker type inequality evaluating the third characters of certain semistable objects in the derived category.
14:50--15:30
Changheon KIM (Hanyang Univ.)
Arithmetic properties of weakly harmonic Maass forms
Summary
After Zagier proved that the traces of singular moduli of the classical modular invariant $j$ at CM points are Fourier coefficients of a harmonic weak Maass form of weight 3/2, various arithmetic properties of the traces of singular values of modular functions on the full modular group have been found. In this talk we show some arithmetic properties of cycle integrals of a sesqui-harmonic Maass form of weight zero whose image under hyperbolic Laplacian is the modular invariant $j$ up to some constant.
16:10--16:50
Shuji SAITO (Tokyo Inst. of Technology)
Higher dimensional Hasse principle and resolution of quotient singularities
Summary
Quite a few examples have been observed which show that an arithmetic method
can play a significant role for a geometric question. In this talk we
present such a new example.
The issue is a mysterious link between two principles governing
apparently unrelated areas of mathematics. One is Hasse principle that is a
global-local principle in number theory.
Another is McKay principle in geometry which concerns resolutions of
quotient singularities.
We deduce new information on the configuration of the exceptional divisor of
a resolution of a quotient singularity from Hasse principle for higher
dimensional arithmetic varieties.
17:00--17:40
Yongnam LEE (KAIST)
$\mathbb Q$-Gorenstein deformation theory and its applications
Summary
In this presentation, I will review singularities of class T and $\mathbb Q$-Gorenstein deformation theory. Construction of simply connected surfaces of general type with $p_g=q=0$ via $\mathbb Q$-Gorenstein smoothings over the field of any characteristic will be also reviewed. Most works in this talk have been carried out by the joint research with Jongil Park and Noboru Nakayama.
Geometry and Topology Session at Hall 1
14:00--14:40
Shin-ichi OHTA (Kyoto Univ.)
Ricci curvature in Finsler geometry and applications
Summary
In this talk, I discuss the notion of weighted Ricci curvature for a pair of a Finsler manifold and a measure on it. Bounding this curvature from below is equivalent to Lott, Sturm and Villani's curvature-dimension condition. This characterization has a number of analytic and geometric applications such as the log-Sobolev inequality and the Bishop-Gromov volume comparison. Moreover, beyond the general theory of the curvature-dimension condition, we can generalize the Bochner-Weitzenbock formula (in terms of a natural nonlinear Laplacian) and the Cheeger-Gromoll splitting theorem. Partly based on joint work with Karl-Theodor Sturm (University of Bonn).
14:50--15:30
Sang-hyun KIM (KAIST)
Embeddability between right-angled Artin groups
Summary
Let $\Gamma$ be a finite simplicial graph. In this talk, we study the question of which right-angled Artin groups are embedded in the right-angled Artin group $A(\Gamma)$. For this, we define two infinite graphs $\Gamma^e$ and $\Gamma^e_k$. The graph $\Gamma^e$ is called the \emph{extension graph} of $\Gamma$ and realized as an induced subgraph of a certain curve complex. Using properties of pseudo-Anosov maps on hyperbolic surfaces, we prove that another right-angled Artin group $A(\Lambda)$ embeds into $A(\Gamma)$ only if $\Lambda$ is an induced subgraph of $\Gamma^e_k$. If $\Gamma$ is triangle--free, then $A(\Lambda)$ embeds into $A(\Gamma)$ if and only if $\Lambda$ is an induced subgraph of $\Gamma^e$. We explain group theoretic, combinatorial and geometric aspects of this extension graph.
16:10--16:50
Shigeyuki MORITA (Univ. of Tokyo, Emeritus)
Characteristic classes in low dimensional topology --Prospects and computations--
Summary
First we propose a series of characteristic classes which would shed light on a mysterious relationship between $3$ and $4$-dimensional topology, both in the smooth and the topological categories. The group $\boldsymbol\Theta^3$ consisting of homology cobordism classes of homology $3$-spheres will play a crucial role. We then consider the ``Lie algebra version'' of these classes which were defined earlier and discuss the known results about them. These classes were developed in the framework of the Lie version of the {\it formal symplectic geometry} which is a very deep theory due to Kontsevich. This theory has three versions and we also mention our recent results concerning the other two cases, commutative and associative ones, as well. They are related to the theory of finite type invariants of homology $3$-spheres, characteristic classes of transversely symplectic foliations and the cohomology of the moduli spaces of curves. Finally we present a few prospects for future research which include clarifying possible relations of these characteristic classes with number theory and also the ``ultimate goal'' of enhancing our study to the group level rather than the Lie algebra context. This talk is based on joint work with Takuya SAKASAI and Masaaki SUZUKI.
17:00--17:40
Jaigyoung CHOE (KIAS)
Higher dimensional versions of the Enneper surface, catenoid and helicoid
Summary
The Enneper surface, catenoid and helicoid are the three simplest complete minimal surfaces in $\mathbb R^3$. In $\mathbb R^3$ one can use the Weierstrass representation formula to construct complete minimal surfaces. But there is no such systematic method of constructing minimal subamnifolds in $\mathbb R^n$. However, using some geometric arguments, we are going to construct the higher dimensional Enneper surface, catenoid and helicoid in $\mathbb R^n$.
Analysis Session at Hall 2
14:00--14:40
Kenji NAKANISHI (Kyoto Univ.)
Global dynamics of nonlinear dispersive equations
Summary
This talk reviews the recent progress in the analysis of global dynamics for PDE's of nonlinear dispersive type, mainly about the speaker's joint work with Wilhelm Schlag and Joachim Krierger.
14:50--15:30
Ki-Ahm LEE (SNU)
Homogenization of the oscillating data on a lower dimensional surface
Summary
In this talk, I would like to discuss the homogenization of the partial differential equations with a oscillating data on a lower dimensional surfaces. Such problems arise on standard Dirichlet or Neumann problem with oscillating boundary data and Free boundary problems. The lower dimensional character creates similar issues: oscillating of boundary in mod 1 , multiple limits on rational direction and unique behavior on the irrational direction, which will be discussed through examples. And we find out the effective data by some estimates on the so-called correctors which will be used to correct super- or sub-solutions of the homogenized equation to be a super- or sub-solutions of $\epsilon$-problems. And then we will discuss the general lower dimensional surface.
16:10--16:50
Yoshikazu GIGA (Univ. of Tokyo)
Blow-up arguments and the Navier-Stokes equations
Summary
In the theory of partial differential equations it is often important to establish an a priori $L^\infty$-bound for solutions. A blow-up argument is a powerful indirect argument to obtain $L^\infty$-bounds. In this note as a short summary we give two applications of the blow-up method to the Navier-Stokes and Stokes equations. As the first application we give a geometric blow-up criterion for a possible singularity of the Navier-Stokes equations proved by Hideyuki Miura and the author. We also show a sketch of the proof of analyticity of the Stokes semigroup in $L^\infty$-type spaces which was recently established by Ken Abe and the author.
17:00--17:40
Kang Tae KIM (POSTECH)
On a generalization of Forelli's theorem
Summary
The purpose of this talk is to present the recent results on various generalizations and variations of Forelli's theorem (1977) on complex analyticity of complex-valued functions in several complex variables. This talk will include recent theorems by E.M. Chirka (2006), Kim-Poletsky-Schmalz (2008), Joo-Kim-Schmalz (2012) as well as Kim-Schmalz (in progress).
Probability Theory and Applied Mathematics Session at Hall 3
14:00--14:40
Makiko SASADA (Keio Univ.)
Microscopic dynamics for the porous medium equation and other degenerate parabolic equations
Summary
I present some recent developments on the rigorous derivation of the porous medium equation and other degenerate parabolic equations from stochastic microscopic dynamics. I discuss two types of stochastic models whose empirical densities (or energies) obey the PME in the hydrodynamic limit.
14:50--15:30
Jung Hee CHEON (SNU)
Discrete logarithm with auxiliary inputs
Summary
The discrete logarithm problem with auxiliary inputs (DLPwAI) is asked to find $\alpha\in F_p$ with auxiliary inputs $g, g^{\alpha}, ..., g^{\alpha^{d}}$ where $g$ is a generator of group of order $p$. In Eurocrypt 2006, an algorithm is proposed to solve DLPwAI in $O(\sqrt{p/d})$ when $d | p-1$ or $d|p+1$. In this paper, we reduce the DLPwAI to the problems to find polynomials with small value sets or to find efficiently computable bivariate function so called the inverse oracle which returns $h^{a/b}$ with inputs $g^a$ and $g^b$ for some group elements $g, h$.
16:10--16:50
Takashi KUMAGAI (Kyoto Univ.)
Random walks on disordered media and their scaling limits
Summary
The problem of random walk on a percolation cluster `the ant in the labyrinth' has received much attention both in the physics and the mathematics literature. In 1982, Alexander and Orbach made a stimulating conjecture concerning the anomalous behavior of random walk on the critical percolation cluster. In 1986, Kesten proved such an anomalous behavior of the random walk on the critical percolation for trees and for ${\mathbb Z}^2$ in a weak sense. Kesten's work triggered intensive research on diffusions and analysis on fractals. After a few decades, the area is mature enough to attack the original problems on disordered media. Recently, it has been proved that random walk on the critical percolation cluster is sub-diffusive on ${\mathbb Z}^d$ when $d$ is high enough, resolving the Alexander-Orbach conjecture affirmatively for the cases. In the talk, we will summarize recent developments on the behavior of random walks on disordered media. Because of the inhomogeneity of the media, we can observe anomalous behavior of the random walk. For simplicity, we will mainly consider random walk on the percolation cluster. We estimate long time behavior of the heat kernel and obtain a scaling limit of the random walk for the supercritical percolation cluster (and more generally, for random conductance models). For random walk on the critical percolation cluster (conditioned on non-extinction), we will show the anomalous heat kernel behavior and explain core ideas of resolving the Alexander-Orbach conjecture. We will also briefly mention some related results for random walk on the Erd\H{o}s-R\'enyi random graph in the critical window.
17:00--17:40
Hyeong In CHOI (SNU)
Commodity Futures model and applications
Summary
In this talk we present a brief survey of our recent work on the commodity futures model. The most salient aspect of our work is that in our model, once the interest rate model is given, the drudgeries of having to deal with the convenience yield disappear altogether, which has been the most thorny point in this theory so far. In particular, given the HJM forward rate volatility or alternatively the volatility of the bond price dynamics, the whole thing simply boils down to the commodity futures price volatility, from which all other model parameters follow. In particular, it lets us do away with many excessive baggages like having to model the dynamics of the spot price or the convenience yield. We then show that, adopting Dupire's method, the commodity futures price volatility is completely determined once all the call option price is given for all maturities and strike prices. We also introduce Jung's work on foreign futures price model.
