Department Seminars & Colloquia




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Flat sides on a closed convex hypersurface shrink with finite speed under the Gauss curvature flow as a free boundary propagation, which describes the shape of worn stones.
On the flat sides, the Gauss curvature vanishes to zero, while the mean curvature tends to infinity in dimension n > 2.
In this talk, we will study the ratios of one singular and five degenerate values on the Gauss curvature flow so that we obtain the optimal C^{1,1/(n-1)} regularity.
In addition, by using one of the ratio, we will classify the closed self-similar solutions to the Gauss curvature flow.
Host: 강완모     Korean English if it is requested     2017-01-06 10:06:10

Data assimilation or filtering of nonlinear dynamical systems combines
numerical models and observational data to provide the best statistical
estimates of the systems. Ensemble-based methods have proved to be
indispensable filtering tools in atmosphere and ocean systems that are
typically high dimensional turbulent systems. In operational applications,
due to the limited computing power in solving the high dimensional systems,
it is desirable to use cheap and robust reduced-order forecast models to
increase the number of ensemble for accuracy and reliability. This talk
describes a multiscale data assimilation framework to incorporate
reduced-order multiscale forecast methods for filtering high dimensional
complex systems. A reduced-order model for two-layer quasi-geostrophic
equations, which uses stochastic modeling for unresolved scales, will be
discussed and applied for filtering to capture important features of
geophysical flows such as zonal jets. If time permits, a generalization of
the ensemble-based methods, multiscale clustered particle filters, will be
discussed, which can capture strongly non-Gaussian statistics using
relatively few particles.

Host: 이창옥     Korean English if it is requested     2017-01-05 10:03:04

Fourier restriction norm method, more precisely X^{s,b} spaces
introduced by Bourgain, was an efficient tool for low regularity
problems of nonlinear dispersive equations. In the well-posedness
problems, we are often interested in proving multilinear estimates in
X^{s,b} spaces. These estimates are in turn reduced to weighted convolution
estimates of L^2 functions. In [Ta], Tao systematically studies this
type of weighted L^2 convolution estimates.

In the lectures, I will roughly follow [Ta]. After introducing preliminary
reduction and fundamental tools, I will cover some selected topics toward the
orthogonal interaction of Schrödinger waves. Although this is motivated
by a well-posedness problem in PDEs, I will mostly focus on desired
bilinear estimates of harmonic analytic nature. I will assume familiarity to Fourier transform and Littlewood-Paley decomposition. (In particular, MAS 640 is sufficient.)

[Ta]  T. Tao, Multilinear weighted convolution of L^2 functions and
applications to nonlinear dispersive equations, Amer. J. Math.
123(2001), 839-908.

To be announced     2016-12-30 10:03:29

Previously, no global well-posedness and stability theory for the hyperbolic system of (viscous) conservation laws in the class of arbitrarily large initial datas have been developed. 

Recently, we have developed a new theory about that large perturbations of shock waves to the system of conservation laws are contractive, up to shift, in time. 
In this talk, we present a very new result on the contraction (up to shift) of large perturbations of shocks to the isentropic Navier-Stokes systems.
The perturbation is measured by a weighted relative entropy, and the shift is constructed from dynamics of the perturbation.
Host: 권순식     To be announced     2017-01-02 09:28:02

Fourier restriction norm method, more precisely X^{s,b} spaces
introduced by Bourgain, was an efficient tool for low regularity
problems of nonlinear dispersive equations. In the well-posedness
problems, we are often interested in proving multilinear estimates in
X^{s,b} spaces. These estimates are in turn reduced to weighted convolution
estimates of L^2 functions. In [Ta], Tao systematically studies this
type of weighted L^2 convolution estimates.

In the lectures, I will roughly follow [Ta]. After introducing preliminary
reduction and fundamental tools, I will cover some selected topics toward the
orthogonal interaction of Schrödinger waves. Although this is motivated
by a well-posedness problem in PDEs, I will mostly focus on desired
bilinear estimates of harmonic analytic nature. I will assume familiarity to Fourier transform and Littlewood-Paley decomposition. (In particular, MAS 640 is sufficient.)

[Ta]  T. Tao, Multilinear weighted convolution of L^2 functions and
applications to nonlinear dispersive equations, Amer. J. Math.
123(2001), 839-908.

To be announced     2016-12-30 10:01:50

Fourier restriction norm method, more precisely X^{s,b} spaces
introduced by Bourgain, was an efficient tool for low regularity
problems of nonlinear dispersive equations. In the well-posedness
problems, we are often interested in proving multilinear estimates in
X^{s,b} spaces. These estimates are in turn reduced to weighted convolution
estimates of L^2 functions. In [Ta], Tao systematically studies this
type of weighted L^2 convolution estimates.

In the lectures, I will roughly follow [Ta]. After introducing preliminary
reduction and fundamental tools, I will cover some selected topics toward the
orthogonal interaction of Schrödinger waves. Although this is motivated
by a well-posedness problem in PDEs, I will mostly focus on desired
bilinear estimates of harmonic analytic nature. I will assume familiarity to Fourier transform and Littlewood-Paley decomposition. (In particular, MAS 640 is sufficient.)

[Ta]  T. Tao, Multilinear weighted convolution of L^2 functions and
applications to nonlinear dispersive equations, Amer. J. Math.
123(2001), 839-908.

To be announced     2016-12-16 10:12:24
We prove that if T1,…, Tn is a sequence of bounded degree trees so that Ti has i vertices, then Kn has a decomposition into T1,…, Tn. This shows that the tree packing conjecture of Gyárfás and Lehel from 1976 holds for all bounded degree trees.
We deduce this result from a more general theorem, which yields decompositions of dense quasi-random graphs into suitable families of bounded degree graphs.
In this talk, we discuss the ideas used in the proof.
This is joint work with Felix Joos, Daniela Kühn and Deryk Osthus.

 

Host: 엄상일     To be announced     2016-12-21 09:34:46

Micromagnetics studies magnetic behavior of ferromagnetic materials at sub-micrometer length scales. These scales are large enough to use a continuum PDE model and are small enough to resolve important magnetic structures such as domain walls, vortices and skyrmions. The dynamics of the magnetic distribution in a ferromagnetic material is governed by the Landau-Lifshitz equation. This equation is highly nonlinear and has a non-convex constraint that the magnitude of the magnetization is constant. We present explicit and implicit mimetic finite difference schemes for the Landau-Lifshitz equation, which preserve the magnitude of the magnetization. These schemes work on general polytopal meshes, which provide enormous flexibility to model magnetic devices with various shapes. We will present rigorous convergence tests for the schemes on general meshes that includes distorted and randomized meshes. We will also present numerical simulations for the NIST standard problem #4 and the formation of the domain wall structures in a thin film. This is a joint work with Konstantin Lipnikov.

 

Host: 김용정     English     2016-12-13 08:53:33

In recent years, there have been great research activities for PDEs and functionals with non-standard growth, and nonlocal equations with p-growth. I will introduce recent progresses in regularity theory for those, together with related results obtained by myself and further researches.

Korean     2016-12-05 14:01:49
Field patterns occur in space-time microstructures such that a disturbance propagating along a characteristic line, does not evolve into a cascade of disturbances, but rather concentrates on a pattern of characteristic lines. This pattern is the field pattern. In one spatial direction plus time, the field patterns occur when the slope of the characteristics is, in a sense, commensurate with the space-time microstructure. Field patterns with different spatial shifts do not generally interact, but rather evolve as if they live in separate dimensions, as many dimensions as the number of field patterns. Alternatively one can view a collection as a multicomponent potential, with as many components as the number of field patterns. Presumably if one added a tiny nonlinear term to the wave equation one would then see interactions between these field patterns in the multidimensional space that one can consider them to live, or between the different field components of the multicomponent potential if one views them that way. As a result of PT-symmetry many of the complex eigenvalues of an appropriately defined transfer matrix have unit norm and hence the corresponding eigenvectors correspond to propagating modes. There are also modes that blow up exponentially with time. This is joint work with Ornella Mattei, Alexanderand Natalia Movchan, and Hoai Minh Nguyen
Host: 임미경     English     2016-12-05 13:25:52

The p-Laplace equation is a nonlinear generalization of the Laplace equation in the Sobolev space and appears in various physical phenomena. I will give an overview over regularity properties for its weak solutions, in particular, Holder regularity, (nonlinear) Calderon-Zygmund theory and pointwise estimates with nonlinear potentials.

English     2016-12-05 13:59:52

For counting weighted independent sets weighted by a parameter λ (known as the hard-core model) there is a beautiful connection between statistical physics phase transitions on infinite, regular trees and the computational complexity of approximate counting on graphs of maximum degree D. For λ below the critical point there is an elegant algorithm devised by Weitz (2006). The drawback of his algorithm is that the running time is exponential in log D. In this talk we’ll describe recent work which shows O(n log n) mixing time of the single-site Markov chain when the girth>6 and D is at least a sufficiently large constant. Our proof utilizes BP (belief propagation) to design a distance function in our coupling analysis. This is joint work with C. Efthymiou, T. Hayes, D. Stefankovic, and Y. Yin which appeared in FOCS ’16.

Host: 엄상일     English     2016-12-09 09:44:48

Mathematical models can be equipped with input data that is
affected by a relatively large amount of uncertainty due to intrinsic
variability in the physical system or difficulties in accurately
characterizing the system under investigation. In this talk, we
discuss numerical methods developed for stochastic PDEs subject to
high-dimensional random input. The algorithms are based on
high-dimensional model representations, e.g., the ANOVA decomposition
and separated series expansions combined with probabilistic
collocation methods. In addition, we employ these methods to reduced
basis algorithms to further enhance the efficiency. These approaches
overcome the curse of dimensionality and can compute the stochastic
solutions of extremely high-dimensional systems. Here, we demonstrate
the effectiveness of these methods to the Poisson and advection
equation with random coefficient and random forcing.

Host: 이창옥     To be announced     2016-11-29 11:47:59

The theory of inhomogeneous analytic materials is developed. These are materials where the coefficients entering the equations involve analytic functions. Analytic materials provide a large class of inhomogeneous bodies for which one can exactly solve for the fields. This adds to our palette of other tools for constructing exact solutions, that includes Dolin's transformation optics and the use of null-Lagrangians. Three types of analytic materials are identified. The first two types involve an integer p. If p takes its maximum value then we have a complete analytic material. Otherwise it is incomplete analytic material of rank p. For two-dimensional materials further progress can be made in the identification of analytic materials by using the well-known fact that a 90◦rotation applied to a divergence free field in a simply connected domain yields a curl-free field, and this can then be expressed as the gradient of a potential.

Host: 임미경     English     2016-12-05 11:55:49

 I will start from a basic formulation of deformation quantization of Poisson manifolds. Roughly speaking, it is to replace smooth functions on a Poisson manifold by certain self-adjoint operators on a Hilbert spaces according to some rules. In case of a variety with a Poisson structure, for example a cluster variety, one tries to quantize the transition functions between charts in a consistent manner. I will try to explain the consequences, and how it is solved.

Host: 서의린     To be announced     2016-12-05 16:03:29

 

Langlands program is a set of conjectures that construct the bridge between two different areas: Number Theory (Galois representations) and Representation Theory (Automorphic forms). Recently, Heiermann and I have constructed the generic local Langlands correspondence for GSpin groups (One main conjecture in the Langlands program). More precisely, we construct the local Langlands parameter that corresponds to an irreducible admissible generic representation of GSpin groups. I further define generic L-packet (an L-packet that contains a generic representation) under one assumption and study the structure of the generic L-packet. As an application, I prove the strong version of the generic Arthur packet conjecture in the case of GSpin groups. The strong version of the generic Arthur packet conjecture states that if the L-packet attached to an Arthur parameter has a generic member, then it is a tempered L-packet. Furthermore, we can apply our idea to the case of classical groups. This case is still in progress.

If time permits, I will explain the result on the classification of strongly positive representations, which is one main tool to construct the LLC for GSpin groups. 

Host: 박진현 2734     To be announced     2016-11-28 11:31:43
We consider weakly dependent stationary and regularly varying stochastic processes in discrete time and prove a sequence of results about their long term behavior. After describing the limiting distribution of individual clusters of extremes, we show a new type of point process convergence theorem. Using it, we are able to prove a new functional limit theorem, which covers several time series models relevant in applications, for which standard limiting theory in the space D of cadlag functions fails. To describe the limit of partial sums in this more general setting we use the concept of so–called decorated cadlag functions. Our method can be further applied to analyze asymptotics of record times in a stationary sequence. The talk is based on the joint work with Hrvoje Planinić and Philippe Soulier
Host: Paul Jung     English     2016-11-30 13:00:26

We investigate the well-documented underperformance of delta-hedged option portfolios in relation to ex ante moments of the stock market’s return distribution. Using a sample of S&P 500 index options, we find that delta-hedged option gains decrease with ex ante volatility, in support of negative volatility risk premium. Moreover, the delta-hedged gains are negatively associated with skewness and kurtosis among call options, but positively associated with the higher moments among put options. These results suggest that investors pay premium for call options in anticipation of a positive jump, while they pay premium for put options in anticipation of a negative jump.

Host: 최건호     Korean     2016-10-28 16:17:55

Wavelets are introduced as an alternative to the classical Fourier analysis in late 80s, and since then, they have been used in various applications including signal and image processing. In this talk, I will review some of basics and challenges of wavelets, especially from the point of view of wavelet constructions in multidimensional setting. I will then present some new methods of construction.

 

 

Host: 임미경     Korean     2016-08-24 17:56:59

This talk is about the interplay between analysis, conformal geometry, and algebra. In the first part of the talk we will present an explicit computation of the logarithmic singularities of the Green functions of the conformal powers of the Laplacian. This is partly motivated by the program of Fefferman in conformal and Cauchy-Riemann geometry. As an application we obtain a spectral theoretic characterization of the conformal class of the round sphere. In the second part, we will present applications of techniques of noncommutative geometry to conformal geometry. The main results are a version in conformal geometry of the inequality of Vafa-Witten, a reformulation of local index formula in Atiyah-Singer in conformal geometry, and the construction of a new family of conformal invariants and their calculations in terms of equivariant characteristic classes. This last part involves a significant amount of homological algebra.

Host: 박진현     English     2016-11-22 15:59:02
 
The Allee effect is a phenomenon that a species gets extinct if the population
 
density is small. Allen-Cahn type bistable nonlinearities are often used to model such
 
a phenomenon. However, such a model never gives an extinction in a finite time.
 
In this talk we consider a different approach of simply subtracting a small constant
 
from a logistic model. This model will give us finite time extinction and compactly
 
supported peak solution. This peak solution provide us a criterion for extinction or
 
population expansion.
 
This is the joint work with J. Chung, Y.-J. Kim and X. Pan.
Host: 권순식     To be announced     2016-12-05 09:38:24

Speaker: Jae Choon Cha (POSTECH)

 

Title: 4차원 공간의 수학 (Mathematics of Dimension 4)

Abstract: 수학적 공간을 연구하는 위상수학의 가장 큰 목표는 어떤 종류의 공간이 존재하고 어떻게 이들을 식별할 수 있는지를 밝혀내어 공간을 완전히 분류하는 것이다. 3차원 이하 또는 5차원 이상의 공간에 대해서는 그 본질적 특성이 이미 상당 부분 규명되어 있는 데에 반해, 매우 흥미롭게도 4차원 공간에 대해서는 여러 핵심적 난제가 아직도 해결되지 않은 채로 남겨져 있다. 이 강연에서는 왜 4차원에서만 다른 차원에서 나타나지 않는 특이한 현상이 발생하는지 살펴보고, 4차원의 비밀을 이해하기 위해 오늘날 수학자들이 도전하고 있는 근본적 문제에 대하여 소개하고자 한다.

Host: Ji Oon Lee     To be announced     2016-09-05 14:29:40

On an open Riemannian manifold of negative curvature, the L^2-spectrum and the positive spectrum of the Laplace-Beltrami operator are closely related by a theorem of Sullivan. Positive spectrum are used to investigate the behavior of Green function at the bottom of the L^2-spectrum. We show that Martin boundary at the bottom of the spectrum coincides with the geometric boundary, and we will explain how ergodic theory of the geodesic flow on a closed Riemannian manifold M of negative curvature can be used to give an asymptotics of the heat kernel on the universal cover of M. This is a joint work with François Ledrappier.

Host: Paul Jung     English     2016-11-25 16:01:10

 We consider the dynamical system of Sinai billiards with a single cusp where two walls meet at the vertex of a cusp and have zero one-sided curvature, thus forming a flat point at the vertex. For Holder continuous observables, we show that properly normalized Birkhoff sums, with respect to the billiard map, converge in law to a totally skewed alpha-stable law.

Host: 수리과학과     English     2016-08-24 17:53:47