# 세미나 및 콜로퀴엄

구글 Calendar나 iPhone 등에서 구독하면 세미나 시작 전에 알림을 받을 수 있습니다.

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.

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.

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.

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.

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.

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자연과학동(E6-1) Room 3435
KAIST CMC 정오의 수학산책
임선희 (서울대학교)
On the work of Mirzakhani : from counting geodesics to the classification of measures

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.

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E6-1, ROOM 3433
Discrete Math
Xavier Goaoc (Université Paris-Est, Marne-la-Vallée, France)
Limits of order types

The limits of converging sequences of graphs are natural objects from extremal graph theory that are also representable as measure-theoretic objects (graphons) or as algebraic objects (flag algebra homomorphisms). I will give an introduction to this theory via a geometric relative: its adaptation from graphs to a combinatorial encoding of planar point sets. This is based on joint work with Alfredo Hubard, Rémi de Joannis de Verclos, Jean-Sébastien Sereni and Jan Volec (http://drops.dagstuhl.de/opus/volltexte/2015/5126/). The talk will not assume any specific knowledge.

The main aim of the lectures is to show some recent applications of

positive characteristic version of non-abelian Hodge theory to study

of algebraic varieties and their invariants in positive and zero characteristics.

I will also review classical Simpson's correspondence and relate

it to recent developments.

빅데이터(Big-Data), 단어는 많이 들어보았는데 과연 빅데이터의 본질은 무엇일까? Volume, Velocity, Variety의 3V로 정의되는 빅데이터는 과연 우리의 미래를 예측할 수 있을까? 이번 강연에서는 우리가 살고 있는 복잡한 사회를 이해하는 도구로서의 빅데이터와 네트워크 과학에 관한 설명과 함께, 그 성공적 응용사례로 구글의 검색엔진을 이용한 선거의 예측, N-gram 프로젝트를 통한 과거의 검색엔진 개발, 알파고의 AI 알고리즘, 대규모 특허분석을 통한 기술예측, 인문학 및 예술에서의 빅데이터 분석사례 등을 소개하고자 합니다. 또한 성공의 이면에 숨겨있던 빅데이터의 한계점과 어두운 그림자까지 빅데이터의 모든 것을 함께 고민하고 이를 통해 우리의 미래의 가능성을 살짝 엿보고자 합니다.

The main aim of the lectures is to show some recent applications of

positive characteristic version of non-abelian Hodge theory to study

of algebraic varieties and their invariants in positive and zero characteristics.

I will also review classical Simpson's correspondence and relate

it to recent developments.

The main aim of the lectures is to show some recent applications of

positive characteristic version of non-abelian Hodge theory to study

of algebraic varieties and their invariants in positive and zero characteristics.

I will also review classical Simpson's correspondence and relate

it to recent developments.

In these two talks, I will introduce geometric problems related to mass in general relativity, such as a series of geometric inequalities, and

conjectures regarding a notion of quasi-local mass proposed by Bartnik.

The geometric inequalities we consider include the angular momentum-mass

inequality for axially symmetric initial data for the Einstein equations.

Note that the special cases treating maximal data have been proved by Dain

et al. Here I will explain how to reduce the general (non-maximal) case to

the known maximal case, and then discuss the solvability of the system of

Elliptic PDEs arose in the process, for near maximal case.

The second part of the talk will mainly provide an introduction to the

static/stationary metric extension conjectures, related to Bartnik

quasi-local mass. I will briefly discuss some known results for the static

metric extension conjecture by Anderson, Anderson/Khuri, Miao et al., and

show a local existence theorem for the solutions of axially symmetric,

stationary vacuum Einstein equations.

We show how to compute for n-vertex planar graphs in roughly O(n^(11/6)) expected time the diameter and the sum of the pairwise distances. These are the first algorithms for these problems using time O(n^c) for some constant c<2, even when restricted to undirected, unweighted planar graphs.

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E6-1, ROOM 3433
Discrete Math
Xavier Goaoc (Université Paris-Est, Marne-la-Vallée, France)
Shatter functions of (geometric) hypergraphs

In combinatorial and computational geometry, the complexity of a system of sets is often studied via its shatter function. I will introduce these functions, and discuss how their asymptotic growth rate is governed from a single of its values, in the spirit of the classical notion of “Vapnik-Chernonenkis dimension” of hypergraphs. In particular, I will describe a probabilistic construction that refutes a conjecture of Bondy and Hajnal. This is joint work with Boris Bukh (https://arxiv.org/abs/1701.06632). The talk will start from first principles.

대표적인 심혈관 질환인 관상동맥질환은 유럽에서는 5명중 1명이, 미국에서는 6명중 1명이

사망에 이르는 고위험 질병으로 선진국병으로 불린다. 대한민국에서도 생활수준의 향상과

서구화된 식습관에 따라 관상동맥질환이 중장년층에 크게 발병하고 있으며, 3대

사망원인(뇌혈관질환, 심혈관질환, 암)중에 하나이다. 심장질환의 경우 2006년에는 인구

10만명당 41.1명이던 사망률이 2017년에는 55.6명으로 증가하는 추세에 있다. 이러한

증가추세를 둔화시키기 위해서는 다각도적인 접근이 필요하겠으나, 진단과 치료의 영역에서는

정확한 혈관상태를 가시화된 영상정보로 의료진에게 제공할 수 있는 것이 필수적이다. 하지만

현재까지 표준화된 기술(gold standard)은 C-arm을 이용한 2차원 혈관 영상이 유일하다. X-ray를

이용한 실시간성이 가장 큰 장점인 2차원 영상이지만, 투사영상이 가지고 있는 궁극적인

한계(투사각도, 혈관의 중첩상태, 비틀림, 이미지 왜곡등)는 피할 수 없다. 이러한 문제점 때문에,

혈관의 스텐트 확장술등에 있어서 적정한 스텐트의 크기를 찾는데 어려움이 따르게 되며, 이는

수술시의 확정할 수 없는 위험요소가 된다.

본 발표에서는 C-arm 기반 2차원 조영영상을 이용하여 3차원 또는 시간의 움직임까지 고려한

4차원 관상동맥 영상복원방법에서 요구되는 수학적인 문제들에 대해서 이야기하고, 현재까지

연구된 결과를 논의 하고자 한다.

A coherent sheaf F on a projective variety X is Ulrich if its pushforward by a finite degree map is trivial. Since they naturally appears in several different theories, the study of Ulrich bundles becomes important. In this talk, I will discuss two different approaches to construct Ulrich bundles on the intersection of two 4-dimensional quadrics: via Serre correspondence and via derived categories. I will also briefly explain an unexpected connection between generalized theta series. This is a joint work with Y. Cho and K.-S.Lee.

In these two talks, I will introduce geometric problems related to mass in general relativity, such as a series of geometric inequalities, and

conjectures regarding a notion of quasi-local mass proposed by Bartnik.

The geometric inequalities we consider include the angular momentum-mass

inequality for axially symmetric initial data for the Einstein equations.

Note that the special cases treating maximal data have been proved by Dain

et al. Here I will explain how to reduce the general (non-maximal) case to

the known maximal case, and then discuss the solvability of the system of

Elliptic PDEs arose in the process, for near maximal case.

The second part of the talk will mainly provide an introduction to the

static/stationary metric extension conjectures, related to Bartnik

quasi-local mass. I will briefly discuss some known results for the static

metric extension conjecture by Anderson, Anderson/Khuri, Miao et al., and

show a local existence theorem for the solutions of axially symmetric,

stationary vacuum Einstein equations.

It was shown by Schaffer that for a dense class of compactly supported smooth initial data,the number of shock curves is finite for large time. It was not known if there is a smooth data for which the number of shock curves is unbounded. In this talk, using the structure of entropy solutions, one can construct a smooth initial data for which the number of shocks curves is infinite.

For a motivation of stochastic integrals, we start with stochastic (random) versions of deterministic systems and then we discuss the Ito integral as a non-anticipating (adapted) stochastic integral.

Based on the quantum decomposition of Brownian motion, we study the Skorohod integral and Stratonovich integral as anticipating (non-adapted) stochastic integrals.

Note that the Skorohod integral is the adjoint action of the Malliavin derivative.

I discuss recent progress in the classification of simply

connected Godeaux surfaces. There are two parts. The first

describes remarkable recent work of Isabel Stenger

(Kaiserslautern). She constructs an 8-dimensional rationally

parametrised moduli family of "general" surfaces, for which

the bicanonical pencil has no hyperelliptic fibres. Her

family contains a 7-dimensional subfamily with a single

hyperelliptic fibre and a 6-dimensional subfamily with two

hyperelliptic fibres. The second part describes my different

attack on the same problem, on which there has been some

progress over the last 25 years.

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)

I will explain the construction of coordinates on the moduli spaces of maximal globally hyperbolic 3 dimensional spacetimes generalising the shear coordinates on the Teichmüller space. I will then discuss the Poisson structure and the mapping class group action on these moduli spaces. These constructions are motivated by the problem of quantum gravity, so I’ll also give a brief description of some of the ideas behind quantisation and how they apply in the context of 3d gravity. This talk is based on joint work with Catherine Meusburger and ongoing work with Hyun Kyu Kim.

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)

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.

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.

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.

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.

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.