Department Seminars & Colloquia
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Computational electromagnetics (CEM) that develops efficient algorithms for solving Maxwell's equations has been a popular subfield of applied mathematics since the invention of computers. Computational nanophotonics applies the CEM techniques to studying nanophotonics, a field of optics that concerns the interaction of light with nanometer-scale objects. However, nanophotonics poses new challenges to CEM, because physical conditions of interest in nanophotonics are very different from those in micro- and radio-wave engineering, which have been the main application areas of CEM. In this talk, I will discuss three exemplary nanophotonic contexts whose unique physical conditions give rise to unexpected computational challenges: plasmonics, dynamic modulation, and nanophotonic laser design. I will demonstrate that it is possible to overcome such challenges by developing new computational techniques.
The first example is plasmonics. Plasmonics is a subfield of nanophotonics that studies the interaction of light with metals. The unique physical condition faced in plasmonics is the significant reduction of wavelength inside metals, which makes Maxwell's equations spatially multiscale. I will explain how such spatial multiscaling leads to ill-conditioned Maxwell's equations that deteriorates the convergence rate of Krylov subspace solvers. Then, I will demonstrate that the conditioning of Maxwell's equations can be significantly improved by simply choosing the correct absorbing boundary condition. Additionally, I will introduce a technique to engineer the eigenvalue spectrum of the Maxwell operator, which gives us insight into the relationship between the eigenvalue spectrum and the convergence rate of Krylov subspace solvers.
The second example is dynamic modulation. In the context of nanophotonics, dynamic modulation refers to a technique to modulate the electric permittivity of media in time. This technique is useful for realizing novel nanophotonic components such as optical isolators by breaking time-reversal symmetry. The unique physical condition faced in dynamic modulation is the huge contrast in frequency between optical signals (~THz) and modulation (~GHz), which makes Maxwell's equations temporally multiscale. I will explain how such temporal multiscaling makes it practically impossible to use the popular time-domain solvers of Maxwell's equations in modeling dynamic modulation. Then, I will introduce a new frequency-domain solution technique that models dynamic modulation by incorporating multiple frequency components.
The third and last example is nanophotonic laser design. In order to design and test nanometer-scale lasing devices computationally, it is necessary to solve the equation dictating the behavior of laser modes. Such an equation is a nonlinear eigenvalue equation described by the steady-state ab initio laser theory (SALT). The computational difficulty in solving the SALT equation is that the equation is nonholomorphic (i.e., nondifferentiable in the complex domain) and therefore the Newton–Raphson method, which is the method of choice for solving large-scale nonlinear eigenvalue equations, is inapplicable. I will explain how to overcome this difficulty with my implicit Newton step calculation technique that allows us to apply the Newton–Raphson method even to nonholomorphic equations.
The techniques used to overcome computational obstacles in these three examples have been successfully applied to real-world engineering problems. I will showcase several nanophotonic components designed with my computational techniques.
During the past 10 years, most of acoustic metamaterial research has been done within a theoretical frame in which the medium is at rest. However, such acoustic metamaterials cannot preserve their unique properties or functions in the presence of ow. For example, the well-known acoustic cloak for a cylindrical object fails even at low subsonic flow. In a previous study, the wave operator couldn't take into account the effect of non-uniformity of the flow around the metamaterial as well as the density inhomogeneity due to the compressibility of fluid. Therefore, in this study, we propose a theoretical framework to consider the effect of non-uniform mean flow on acoustic metamaterials aiming at understanding the underlying physics and designing a new type of acoustic metamaterial.
Cellular adhesion is one of the most important interaction force between cells and other tissue components. In 2006, Armstrong, Painter and Sherratt introduced a non-local PDE model for cellular adhesion, which was able to describe known experimental results on cell sorting and cancer growth. While the numerical implementation leads to nice results, the analysis of this non-local model is challenging. In this talk I will present a random walk derivation of the Armstrong-Painter-Sherratt adhesion model, I will discuss local and global existence of solutions, and the underlying bifurcation structure of steady states. (joint work with A. Buttenschoen, K. Painter, A. Gerisch, M. Winkler).
In this talk, we study a rumor spreading model. We employ a steady state analysis to obtain the
final size of the rumor. From the analysis, we can show the existence of a threshold on a momentum
type initial quantity related to rumor outbreak occurrence regardless of the total initial population. Moreover, we consider several cases of generalization of the rumor spreading model and rumor outbreak.
2017년 제7회 정오의 수학산책
일시: 11월 17일(금) 12:00 - 13:15
장소: KAIST 수리과학과 E6-1 3435호
강연자: 양현미 교수 (서울대학교)
제목: 4th Industrial Revolution and the New Talents We Need
내용: TBA
등록: https://docs.google.com/forms/d/1l1px6mJC7YtDeZspNpn0f0MmRlfJm-j_JnkCCl--xQk/edit?ts=59fbeb7e
* 11월 24일(금) 채동호 교수님 강연은 연사분 사정으로 취소되었습니다.
I will introduce the dual of the formal affine Demazure algebra, which is the algebraic replacement of T-equivariant oriented cohomology of complete flag variety. I will also mention the proof of the generalized Borel Isomorphism. In the second half of this talk, I will define the two Hecke actions on the dual of the formal affine Demazure algebra. Then I will define the push-pull operators of the oriented cohomology and define perfect pairings on the equivariant cohomology of complete and partial flag varieties. I will also mention hyperbolic cohomology and its relation with Kazhdan-Lusztig basis.
Room 3433, Bldg. E6-1
Discrete Math
Noel Jonathan (University of Warwick, UK)
The Best Way to Spread a Rumour in a High-Dimensional Grid
Room 3433, Bldg. E6-1
Discrete Math
Bootstrap percolation is a simple process which is used as a model of propagation phenomena in real world networks including, for example, the spread of a rumour in a social network, the dynamics of ferromagnetism and information processing in neural networks. Given a graph G and an integer r, the r-neighbour bootstrap process begins with a set of “infected” vertices and, in each step, a “healthy” vertex becomes infected if it has at least r infected neighbours. A central problem in the area is to determine the size of the smallest initial infection which will spread to every vertex of the graph. In this talk, I will present a trick for obtaining lower bounds on this quantity by transforming the problem into an infection problem on the edges of the graph and applying some basic facts from linear algebra. In particular, I will outline a proof of a conjecture of Balogh and Bollobás (2006) on the smallest infection which spreads to every vertex of a high-dimensional square lattice and mention some potential applications to analysing the behaviour of a random infection in this setting. This talk is based on joint work with Natasha Morrison.
I will introduce the concept of oriented cohomology in the sense of Levine and Morel, and the work of Kostant and Kumar on algebraic construction of singular cohomology and Grothendieck group of flag varieties. Then I will introduce the formal group algebra of Calmes-Petrov-Zainoulline, which is the algebraic replacement of T-equivariant oriented cohomology of a point. I will introduce the definition of formal affine Demazure algebra and sketch the proof of the structure theorem.
본 강연에서는 산업계에서 실제 적용되고 있는 산업수학의 예제들을 소개하고 계산 유체역학 분야의 특허와 관련하여 제안할 수 있는 아이디어를 논하고자 한다. 주된 논의점은 어떻게 산업수학이 산업계가 대면하고 있는 실질적인 문제들에 의미있는 해법을 제시할 수 있는지에 있다. 이를 위해서, 첫 번째로 수치 편미방 분야의 예제를 설명한다. 수치 편미분 방정식에서 가장 기본적으로 사용되는 방법론을 설명하고 계산 유체 역학의 실무적 관점에서 이해될 수 있는 근본적 문제들을 토론해본다. 더불어 현실적인 측면을 반영한 알고리즘을 생각해 보고, 산업수학자에게 의미있는 문제들을 제시하고자 한다. 두 번째 예제에서는 최적화 기법을 이용해서 간단한 2차원 스케치로부터 3차원 모델을 구성하는 모형을 논의한다. 실제로 만화 제작에 쓰이는 이 알고리즘으로부터 응용될 수 있는 문제들도 다룰 예정이다. 셋째, 영상처리분야에서의 Euler’s elastica를 효율적으로 계산하는 방법론을 간단히 설명하고, 이를 바탕으로 실무에 적용 가능한 연구 방향을 논의해 본다.
Hyperbolic dynamical systems are nowadays fairly well understood from the topological and ergodic point of view. In this talk, we discuss some recent and ongoing works on the dynamics beyond hyperbolicity. In the first part, we will provide a characterization of robustly shadowable chain transitive sets for C1-vector fields on compact smooth manifolds. In the second part, we extend the concepts of topological stability and pseudo-orbit tracing property from homeomorphisms to Borel measures, and prove that every expansive measure with the pseudo-orbit tracing property is topologically stable. This represents a measurable version of the stability theorem by Peter Walters. The first part is joint work with M. Reza, and the second part is joint work with C.A. Morales.
The geometry of the moduli space of sheaves on $mathbb{P}^2$ has been studied in various viewpoints, for instance curve counting, the strange duality conjecture, and birational geometry via Bridgeland stability. For small degree cases, it was possible to classify all rational contractions and compute the cohomology ring of the moduli space.
In this talk, we consider the next simplest case of a quadric surface. We construct a flip between the moduli space of sheaves and a projective bundle, and show that their common blown-up space is the moduli space of stable pairs. In principle, this enable us to compute the cohomology ring of the moduli space from that of the projective bundle.
In the absence of the entropy condition, we construct an $L^infty$ solution to the Cauchy problem of a scalar conservation law in one space dimension that exhibits microscopic oscillation in the interior of its support when the initial function is non-constant, continuous and compactly supported. As a result, such a solution is nowhere continuous in the interior of the support. Our method of proof is to convert the main equation into a suitable partial differential inclusion and to rely on the convex integration method of M"uller and v{S}ver'ak. In doing so, we find an appropriate subsolution by solving certain ordinary differential equations and make use of it to tailor an in-approximation scheme that reflects persistence of oscillations.
자연과학동(E6-1) Room 3435
KAIST CMC noon lectures
Byeong-Kweon Oh (Seoul National University)
Representations of quadratic forms
자연과학동(E6-1) Room 3435
KAIST CMC noon lectures
2017년 제6회 정오의 수학산책
일시: 11월 3일(금) 12:00 - 13:15
장소: KAIST 수리과학과 E6-1 3435호
강연자: 오병권 교수 (서울대학교)
제목: Representations of quadratic forms (이차형식의 표현)
내용: 이 강연에서는 가우스(K. F. Gauss)이래 발전을 거듭하여 온 정수 계수의 이차형식에 의한 정수 표현에 대하여 살펴본다. 특히, 주어진 이차형식의 차원이 4이상인 경우, 표현되는 정수를 모두 구하는 방법을 알아본다. 또한 이차형식의 차원이 3인 경우 표현되는 정수를 모두 구하기 위하여 시도되고 있는 다양한 방법을 소개한다.
등록: https://goo.gl/forms/hFwMk8xiDJVBPgD03
In a constantly changing world, animals must account for fluctuations and changes in their environment when making decisions. They must make use of recent information, and appropriately discount older, irrelevant information. But to do so they need to learn the rate at which the environment changes. Recent experimental studies show that humans and other animals can indeed do so. However it it is unclear what underlying computations they use to achieve this. Developing normative models of evidence accumulation is a first step in quantifying such decision-making processes. While optimal, these algorithms are computationally intensive. To address this problem we developed an approximation of the normative inference process, and show how this approximate computation can be implemented in neural circuits. In the second part of the talk I will discuss evidence accumulation on networks where private information can be shared between neighboring nodes.
Bestvina, Kleiner and Sageev showed that every 2-quasiflat in a 2-dimensional CAT(0) cube complex is at finite Hausdorff distance from a finite union of 2-dimensional quarter-plane and Huang generalized it to $n$. Using this invariant, several quasi-isometric classification problems in right-angled Artin groups and graph braid groups are solved. In this talk, We discuss how this invariant works when we classify planar graph 2-braid groups up to quasi-isometries.
Let K be a finite extension of Qp. It is believed that one can attach a smooth Fp-representation of GLn(K) (or a packet of such representations) to a continuous Galois representation Gal(Qp/K) → GLn(Fp) in a natural way, that is called mod p Langlands program for GLn(K). This conjecture is known only for GL2(Qp): one of the main difficulties is that there is no classification of such smooth representations of GLn(K) unless K = Qp and n = 2. However, for a given continuous Galois representation ρ0 : Gal(Qp/Qp) → GLn(Fp), one can define a smooth Fp-representation Π0 of GLn(Qp) by a space of mod p automorphic forms on a compact unitary group, which is believed to be a candidate on the automorphic side corresponding to ρ0 for mod p Langlands correspondence in the spirit of Emerton. The structure of Π0 is very mysterious as a representation of GLn(Qp), and it is not known that ρ0 and Π0 determine each other. In this talk, we discuss that Π0 determines ρ0 , provided that ρ0 is ordinary and generic. More precisely, we prove that the tamely ramified part of ρ0 is determined by the Serre weights attached to ρ0 , and the wildly ramified part of ρ0 is obtained in terms of refined Hecke actions on Π0. The talk is based on a joint work with Zicheng Qian.
Right-angled Artin groups (RAAGs) are defined from finite simplicial graphs.
It is a fundamental question whether or not, given two RAAGs, there is an embedding from one group to the other.
Extension graphs are useful in solving this problem.
In this talk, I will briefly review RAAGs and extension graphs,
and show some results on solving the embeddability problem in RAAGs by using extension graphs.
For a two-dimensional irreducible Galois representation arising from a cusp form, there is a notion of the optimal level introduced by Jean-Pierre Serre. His epsilon conjecture, proved by Kenneth Ribet, concerns about this. Later, Diamond and Taylor classified non-optimal levels of irreducible Galois representations. In this talk, I will discuss their counterpart, namely, non-optimal levels of a reducible mod l Galois representation arising from a weight 2 cusp form.
We discuss the set of critical exponents of discrete groups acting on a regular tree. If the quotient graph is finite, then the critical exponent is an algebraic number.
In general, given an arbitrary real number between 0 and the volume entropy of the regular tree, we discuss how we can construct a discrete group whose critical exponent realizes the number.
We also study the minimal polynomials of Schottky free discrete groups of rank 2.
In this talk, we will start with a very famous problem in number theory, Fermat's last theorem. In its proof given by Andrew Wiles, modular forms and Galois representations play very crucial roles, and they are still very important subjects in modern number theory. We will discuss several aspects of them in problems of number theory.
Starting with introducing a general Newtonian Boltzmann theory, we will establish global-in-time well-posedness and stability results for solutions nearby the relativistic Maxwellian to the special relativistic Boltzmann equation without angular cutoff. We assume the generic soft-potential conditions on the collision kernel in that were derived by Dudynski and Ekiel-Jezewska (Commun Math Phys 115(4):607--629, 1985). In this physical situation, the angular function in the collision kernel is not locally integrable, and the collision operator behaves like a non-isotropic fractional diffusion operator.
The local density is a local factor of the celebrated Smith-Minkowski-Siegel mass formula, which is an essential tool in the classification of quadratic or hermitian forms. In this talk, I will first explain how the local density is reformulated in terms of arithmetic geometry. I will then explain a simple and uniform construction of smooth group schemes, associated to any bilinear form defined over any local field, which is a desingularization of group schemes over DVR. This construction yields the precise and new local density formulas in many unknown cases.
In 1997, S. Kudla announced a paper to propose a program to match certain arithmetic intersection numbers on Shimura varieties with the sum of (the derivative of) the Fourier coefficients of certain modular forms. This, called Kudla's program, is nowadays considered as one of the central areas in number theory. In this talk, I will introduce the motivation of Kudla's program, starting from Hurwitz (1885) and Gross-Keating's work (1993). I will then explain that both sides have the same inherent structures beyond matching values, and propose the mod p version of Kudla's program at the special fiber. This talk is based on a joint work with T. Yamauchi.
One of the most canonical ways to define an invariant of a singular variety is to take a resolution of its singularities, and use invariants of the resulting smooth variety. We will begin this talk with a quick overview of Grothendieck's, and then Voevodsky's theory of motives. We will explain how to use resolutions to extend the theory to singular varieties, even when resolutions don't exist. Then, if there is time, we will discuss transportation of these techniques to differential forms leading to potential applications in birational geometry.
In this talk, we will introduce a notion of a noncommutative probability space and useful properties.
Then we will discuss various convergence results of weighted sums in a noncommutative probability space, e.g., weak law of large numbers, convergence rates and precise asymptotics, etc.
Also, we will discuss some noncommutative inequalities, e.g., Fuk-Nagaev inequalities, Bennett inequality and Rosenthal inequality, etc
One of the major developments in 20th century mathematics is homotopy theory, which studies topological spaces up to stretching and bending. Basic tools in homotopy theory, like singular homology and homotopy groups, are constructed using paths from the unit interval. Such techniques were unavailable for a long time in algebraic geometry, due to polynomials being too rigid. In this talk we will discuss ways around this developed by Morel, Suslin, Voevodsky, et al.
Complex dynamical systems such as turbulent flows are well known for a wide range of active spatio-temporal scales, which makes it difficult to compute them numerically due to a huge degree of freedom. Multiscle numerical methods aim to achieve reduced-order models, which are computationally cheap and robust, instead of solving full resolution models. We compare several multiscale methods such as Multiscale FEM, Heterogeneous Multiscale Method and Superparameterization and proposes a stochastic Superparameterization as a seamless multiscale method for hyperbolic partial differential equations without scale separation. We show how stochastic modeling for unresolved scales affect coarse-graining in multiscale modeling and apply the stochastic method to a two-layer quasi-geostrophic equation, which is a standard model for ocean turbulent flows. We then briefly discuss applications in data assimilation and parameter estimation for Physics-constrained problems. A part of this talk includes joint work with A Majda (Courant) and I Grooms (U of Colorado Denver).
Observational data along with mathematical models play a crucial role in improving prediction skills in science and engineering. In this talk we focus on the recent development of Bayesian inference techniques, data assimilation and parameter estimation, for Physics-constrained problems that are often described by partial differential equations. We discuss the similarities shared by the two methods and their differences in mathematical and computational points of view and future research topics. As applications, numerical weather prediction for geophysical flows and parameter estimation of kinetic reaction rates in the hydrogen-oxygen combustion are provided. This talk aims for researchers and students in all disciplines of science and engineering and only a minimum level of undergraduate mathematics is required.
자연과학동(E6-1) Room 3435
KAIST CMC noon lectures
Seon Hee Lim (Seoul National University)
On the work of Mirzakhani : from counting geodesics to the classification of measures
자연과학동(E6-1) Room 3435
KAIST CMC noon lectures
2017년 제5회 정오의 수학산책
일시: 10월 13일(금) 12:00 - 13:15
장소: KAIST 수리과학과 E6-1 3435호
강연자: 임선희 교수 (서울대)
제목: On the work of Mirzakhani : from counting geodesics to the classification of measures
(미르자카니의 결과들 : 측지선의 개수부터 측도의 분류까지)
내용: 본 강연에서는 미르자카니의 업적들에 대해 살펴볼 예정입니다. 곡면에서 단순 측지선의 갯수에 대한 문제, 곡면들의 공간 (Moduli space of Riemann surfaces)에서 측지선의 갯수 세기, 부피 변화, 그리고 특정한 군의 작용에 대한 불변 측도의 분류 문제 등에서 미르자카니가 얻은 결과들을 음미하고, 그러한 문제들이 서로 어떻게 연결되어 있는지를 쌍곡 동역학 (dynamics on hyperbolic spaces) 에서 이해해 보고자 합니다.
등록: 아래 링크를 통해 사전등록 바랍니다.
https://goo.gl/forms/DbAazUTgVwyc7vQp1
We calculate the global log canonical thresholds of log del Pezzo surfaces embedded in weighted projective spaces as codimension two. As important applications, we show that most of them are weakly exceptional and admit K"ahler-Einstein metrics. This is a joint work with Joonyeong Won.
The spatially varying coefficient process model is a nonstationary approach to explaining spatial heterogeneity by allowing coefficients to vary across space. To accommodate geographically hierarchical data, we develop a methodology for generalizing this model. We consider two-level hierarchical structures and allow for the coefficients of both low-level and high-level units to vary over space. We assume that the spatially varying low-level coefficients follow the multivariate Gaussian process, and the spatially varying high-level coefficients follow the multivariate simultaneous autoregressive model that we develop by extending the standard simultaneous autoregressive model to incorporate multivariate data. We apply the proposed model to transaction data of houses sold in 2014 in a part of the city of Los Angeles. The results show that the proposed model predicts housing prices and fits the data effectively.