# Seminars & Colloquia

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We study the dynamical properties of the topological generalized beta transformations, which generalizes the concept of generalized beta transformations defined by Gora. In particular, we generalize the result on admissible sequence for unimodular maps to the case of generalized beta maps, and also study the properties of the topological entropy and its Galois conjugates, generalizing some results by Tiozzo. This talk represents an ongoing collaboration with Diana Davis, Kathryn Lindsey and Harry Bray.

Host: 백형렬     To be announced     2017-08-17 08:59:06

In this talk, I will talk on transverse stability line solitary waves for KP-II and the Benney Luke equation. Both equations are long wave long wave models for 3D water waves with weak surface tension. I will explain that the resonant continuous eigenmodes which we can find in an exponentially weighted space have to do with modulation of line solitary waves.

Host: Soonsik Kwon     Contact: Hyeonsu Kim (8545)     English     2017-08-21 10:31:40
In this talk, we consider continuous and discrete nonlinear Schr\"odinger equations (NLS) with potential.
Under the assumption that Schr\"odinger operator has several eigenvalues, it is well known that there exist small bound states (localized in space, periodic in time solutions).
We will show that for the continuous NLS, even though the linear Schr\"odinger equaiton has quasi-periodic (in time) solutions, there exists no quasi-periodic (in time) solutions and for discrete NLS, there can exists quasi-periodic (in time) solutions.
Similar result holds for nonlinear Klein Gordon equations.
Host: Soonsik Kwon     Contact: Hyeonsu Kim (8545)     English     2017-08-21 10:32:39
Barycentre is the geometric mean of a distribution on a metric space; it is a point that minimizes its average distance squared to the given distribution.  Such a point is highly non-unique in general, though we have uniqueness when the underlying space is the Euclidean space or more generally a space of nonpositive curvature with trivial topology. We consider such a notion from the viewpoint of optimal transport which gives a natural distance structure between probability measures.
This allows us to uniquely interpolate many probability distributions, called the Wasserstein barycentre, as initiated by Agueh and Carlier. It also leads to a uniquely defined canonical barycentre, which is obtained by relaxing the notion of barycentre point to a barycentre measure. We will explain these developments, based on join work with Brendan Pass
Host: 강완모     Korean English if it is requested     2017-08-09 12:57:10

What is the probability that in a telecommunication system, an atypically large proportion of users experiences a bad quality of service induced by lack of capacity? Although today's wireless networks exhibit nodes embedded in the Euclidean plane, large deviations have so far predominantly been analyzed in mean-field approximations. We explain how to prove a large deviation principle for a random spatial telecommunication network featuring capacity-constrained relays. This talk is based on joint work with Benedikt Jahnel and Robert Patterson (WIAS Berlin)

Host: Paul Jung     English     2017-08-23 10:09:52

Backward stochastic differential equation (BSDE) is a generalization of martingale representation theorem  and it has been widely used for financial derivative pricing and stochastic optimization. Traditionally, most of well-posedness result of BSDE were based on contraction mapping theorem on the space of stochastic processes.
In our work, we were able to transform BSDEs into fixed point problems in the space of Lp random variables. The simplicity of our framework enables us to apply various kind of fixed point theorems which have not been tried in previous literature. In particular, this enables to remove infinite dimensionality arise from time and we were able to use white noise analysis to use topological fixed point theorems. As a result, we were able to generalize previous well-posedness results: e.g. time-delayed type, mean-field type, multidimensional super-linear type.
The talk is aimed for those who are not familiar with BSDE and it is based on BSE's, BSDE's and fixed point theorem (joint work with Patrick Cheridito)

Host: 강완모     Korean English if it is requested     2017-08-09 12:59:07

We classify minimal projective 3-folds of general type with the geometric genus $p_g= 1, 2$ or 3 by studying the birationality of $varphi_{16}, varphi_{6}$ and $varphi_{5}$. This is a joint work with Meng Chen and Matteo Penegini.

Host: 신용주     English     2017-08-21 08:58:33
We will study relations between Fully nonlinear PDEs, Embedding problems, Isoperimetric inequality, Curvature flows, and Optimal transportation.
In particular, we will discuss about fully degenerate or singular elliptic equations in geometric problems and its applications curvature derivative controls in singularity analysis.
We will also divergence structures in certain fully nonlinear PDEs and their geometric background.
Host: 강완모     Korean English if it is requested     2017-08-09 12:55:41

A novel high-order numerical scheme is proposed to solve the shallow water equations (SWEs) on arbitrary rotating curved surfaces. Based on the method of moving frames (MMF), the proposed scheme not only has the smallest dimensionality of two in space, but also does not require either of (i) metric tensors, (ii) composite meshes, or (iii) the surrounding space. The MMF-SWE formulation is numerically discretized using the discontinuous Galerkin method of arbitrary polynomial order p in space and an explicit Runge-Kutta scheme in time. In this talk, we start with the fundamental concepts of the innovational moving frames for Riemannian geometry developed by the famous French mathematician Elie Cartan in the early 20th century. Then, we discuss its adaptation and validity in the discrete space for scientific computing by overviewing the past works on conservational laws and diffusion equations. Applications to SWEs  will be explained in details in views of algorithmic novelty to overcome the classical issues of PDEs on the closed surface such as geometric singularities and rotational effects. Results of six standard tests on the sphere will be displayed with the optimal order of convergence of p+1. Also, its general applicability and stability on arbitrary rotating surfaces such as ellipsoid, irregular, and non-convex surfaces will be demonstrated.

Host: 김용정     Korean     2017-08-09 16:26:38

We proved a Kazhdan type theorem for the canonical metrics of finite graphs. Namely, we show that the canonical metric of finite normal coverings of the graph converges when the covering converges, and the limit depends only on the limit of the coverings. We also generalize the argument to higher dimensional simplicial complexes. The proof is mostly based on an analogous argument in the case of Riemann surfaces and Lück's approximation theorem for L^2 cohomology. This is joint work with Farbod Shokrieh.

Host: 백형렬     English     2017-08-04 09:30:46

High-dimensionality is one of the major challenges in stochastic simulation of realistic physical systems. The most appropriate numerical scheme needs to balance accuracy and computational complexity, and it also needs to address issues such as multiple scales, lack of regularity, and long-term integration.

In this talk, I will review state-of-the-art numerical techniques for high-dimensional systems including low-rank tensor approximation, sparse grid collocation, and ANOVA decomposition. The presented numerical methods are tested and compared in the joint response-excitation PDF equation that generalizes the existing PDF equations and enables us to do kinetic simulations with non-Gaussian colored noise. The alternative to the numerical approach, I will discuss dimension reduction techniques such as Mori-Zwanzig approach and moment closures that can obtain reduced order equations in lower dimensions. I will also present numerical results including stochastic Burgers equation and Lorenz-96 system.

English     2017-07-21 11:30:09