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.

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.

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.

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.

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.

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.

 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.

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.

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.

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.

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.

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.

 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.

The k disjoint paths problem, which was shown to be efficiently solvable for fixed k in undirected graphs in a breakthrough result by Robertson and Seymour, is notoriously difficult in directed graphs. In directed graphs, he problem is NP-complete even in the case when k=2. In an attempt to make the problem more tractable, Thomassen conjectured that if a digraph G were sufficiently strongly connected, then every k disjoint paths problem in G would be feasible. He later answered his conjecture in the negative, showing that the problem remains NP-complete when k=2, even when we assume that the graph is arbitrarily highly connected.

We consider the following further relaxation of the problem: a half-integral solution to a k disjoint paths problem is a set of paths linking the desired vertices such that each vertex of the graph is in at most two of the paths. We will present the new result that the half-integral k disjoint paths problem can be efficiently solved (even when the parameter k is included as part of the input!) if we assume the graph is highly connected.

This is joint work with Irene Muzi and Katherine Edwards.

Speaker: 임보해 (KAIST)

Title: 타원곡선의 군의 구조

Abstract: 본 강연에서는 타원곡선의 정의 및 대수적 구조를 소개하고 현재 관련 연구 문제들을 소개하겠습니다.  타원곡선과 꼬임곡선, 그리고 congruent number 와의 관계 등을 소개하겠습니다.

Registration: https://goo.gl/forms/AeWL5bCeZbInh6d73

It has long been known that Feedback Vertex Set can be solved in time 2^O(w log w) n^O(1) on graphs of treewidth w, but it was only recently that this running time was improved to 2^O(w) n^O(1), that is, to single-exponential parameterized by treewidth. We consider a natural generalization of this problem, which asks given a graph G on n vertices and positive integers k and d, whether there is a set S of at most k vertices of G such that each block of G-S has at most d vertices. The central question of this talk is: “Can we obtain an algorithm that runs in single-exponential time parameterized by treewidth, for every fixed d?” The answer is negative. But then, one may be curious which properties of Feedback Vertex Set that make it allow to have a single-exponential algorithm. To answer this question, we add an additional condition in the general problem, and provide a dichotomy result.
Formally, for a class

A quantum walk is a (rather imperfect analog) of a random walk on a graph.

They can be viewed as gadgets that might play a role in quantum computers, and have been

used to produce algorithms that outperform corresponding classical procedures. Physical

questions about these walks lead to problems in spectral graph theory, and they also provide

interesting new graph invariants. In my talk I will present some of the background,

and some of the many open problems that they have given rise to.

 BRST method was introduced in physics to quantize classical field theories with gauge symmetries, to make sure these symmetries are preserved even after quantization procedures. This requires some preparation at the classical level, such as introducing ghosts and antifields. These can be formulated in precise mathematical terms, using Koszul-Tate chain complex that incorporates antifields, and BRST cochain complex that introduces ghosts. If BRST complex can be constructed, we can find a solution to the classical master equation that extends the original Lagrangian with antifields and ghosts. 

A homomorphism is an adjacency preserving map between the vertex sets of two graphs. A n-vertex, k-regular graph is strongly regular, with parameters (n,k,λ, μ), if there exist numbers λ and μ such that every pair of adjacent vertices share λ common neighbors and every pair of non-adjacent vertices share μ common neighbors. A strongly regular graph is primitive if neither it nor its complement is a disjoint union of complete graphs. We prove that if G and H are primitive strongly regular graphs with the same parameters and φ is a homomorphism from G to H, then φ is either an isomorphism or a coloring (homomorphism to a complete subgraph). Moreover, any such coloring is optimal for G and its image is a maximum clique of H. Therefore, the only endomorphisms of a primitive strongly regular graph are automorphisms or colorings. This confirms and strengthens a conjecture of Peter Cameron and Priscila Kazanidis that all strongly regular graphs are cores or have complete cores. The proof of the result is elementary, mainly relying on linear algebraic techniques.

Describing the eigenvalue distribution of the sum of two general
Hermitian matrices is basic question going back to Weyl. If the matrices
have high dimensionality and are in general position in the sense that one
of them is conjugated by a random Haar unitary matrix, the eigenvalue
distribution of the sum is given by the free additive convolution of the
respective spectral distributions. This result was obtained by Voiculescu on
the macroscopic scale. In this talk, I show that it holds on the microscopic
scale all the way down to the eigenvalue spacing with an optimal error bound.
Joint work with Zhigang Bao and Laszlo Erdos.     

Describing the eigenvalue distribution of the sum of two general
Hermitian matrices is basic question going back to Weyl. If the matrices
have high dimensionality and are in general position in the sense that one
of them is conjugated by a random Haar unitary matrix, the eigenvalue
distribution of the sum is given by the free additive convolution of the
respective spectral distributions. This result was obtained by Voiculescu on
the macroscopic scale. In this talk, I show that it holds on the microscopic
scale all the way down to the eigenvalue spacing with an optimal error bound.
Joint work with Zhigang Bao and Laszlo Erdos.

  In this talk I will start by describing a certain subset of real numbers which contain all the numbers which are of interest to arithmeticians. These numbers are called periods, and they form a countable set and in principle are much more easy to handle than a general real number. Then I will specialize to a certain subset of periods which are of mixed Tate type. The ones which have good reduction for all primes have periods called multi-zeta values, which were first defined by Euler. The remainder of the talk will be about a p-adic version of these numbers called p-adic multi-zeta values. 

In this talk, the Burgers-alphabeta equation, which was first introduced by Holm and Staley, is considered in the special case where nu= 0 and b = 3. Traveling wave solutions are classified to the Burgers-alphabeta equation containing
four parameters b, alpha, nu, and beta, which is a nonintegrable nonlinear partial differential equation that
coincides with the usual Burgers equation and viscous b-family of peakon equation, respectively, for two specific
choices of the parameter beta = 0 and beta = 1. Under the decay condition, it is shown that there are smooth,
peaked and cusped traveling wave solutions of the Burgers-alphabeta equation with nu= 0 and b = 3 depending on
the parameter beta. Moreover, all traveling wave solutions without the decay condition are parametrized by the
integration constant k1 in R. In an appropriate limit beta= 1, the previously known traveling wave solutions of
the Degasperis–Procesi equation are recovered.

A reaction network is the system in which various reactions occur between several physical, chemical or biological species. To investigate the dynamics of the reaction networks such as genetic networks, metabolic networks and epidemic compartment systems, researchers use mathematical models based on the principle of mass-action kinetics or other kinetics. In this talk, we present the ways of mathematical modeling and computational methods for describing the time-evolution of the reaction networks. We show some simulation results obtained by applying the methods to motivating examples including epidemic models such as H1N1 influenza and foot-and-mouse disease spread.

Helly's theorem, a classical result in discrete geometry, asserts that if n>d convex subsets of R^d have empty intersection, some d+1 of them must already have empty intersection. I will discuss some topological generalizations of Helly's theorem, where convexity is replaced by connectivity assumptions on the nonempty intersections, that lead to non-embeddability results of Borsuk-Ulam type and to variations on Leray's acyclic cover theorem (or the Nerve theorem).

Part 2 (Monday): The Nerve theorem.

 MUltiple SIgnal Classification (MUSIC) is a famous non-iterative algorithm for detecting various kind of anomaly in inverse scattering problem. Recently, it was applied to the various problems, for example, detection of antipersonnel mines buried in the ground, location search for small inclusions, internal corrosion in pipelines, and shape reconstruction of arbitrary shaped thin inclusions, cracks, and extended targets, etc. Throughout these results, it has been confirmed that MUSIC is fast, stable, and effective algorithm but some phenomena cannot be explained in the traditional approach. For example, the unexpected appearance of some artifacts or two curves along the boundary of targets instead of the true shape, an image with poor resolution with small number of incident/observation directions, and so on. In this presentation, basic concept of MUSIC algorithm is surveyed and a relationship between Bessel function of integer order of the first kind is introduced. This relationship will explain various phenomena that mentioned previously. Time permitting, a condition of incident/observation direction will be considered for a successful shape detection of small or arc-like cracks in limited-view inverse scattering problem.

Helly's theorem, a classical result in discrete geometry, asserts that if n>d convex subsets of R^d have empty intersection, some d+1 of them must already have empty intersection. I will discuss some topological generalizations of Helly's theorem, where convexity is replaced by connectivity assumptions on the nonempty intersections, that lead to non-embeddability results of Borsuk-Ulam type and to variations on Leray's acyclic cover theorem (or the Nerve theorem).

Synchronization of oscillators denotes a phenomenon for the adjustment of rhythms  among weakly coupled oscillators, andis one of the collective modes appearing in oscillatory complex systems such as ensembles of  Josepson junctions, pacemaker cells and  fireflies etc.  In this talk, I will briefly discuss some challenging mathematical problems for synchronization via the Kuramoto and Lohe models, andpresent a survey of mathematical developments for these models in recent years.

The fact that the complete graph K5 does not embed in the plane has been generalized in two independent directions. On the one hand, the solution of the classical Heawood problem for graphs on surfaces established that Kn embeds in a closed surface M if and only if (n − 3)(n − 4) ≤ 6b1(M), where b1(M) is the first Betti number of M. On the other hand, van Kampen and Flores proved that the k-skeleton of the n-dimensional simplex (the higher-dimensional analogue of Kn+1 embeds in R2k if and only if n ≤ 2k + 1.

I will discuss a conjecture of Kuhnel that generalizes both the Heawood inequality and the van Kampen-Flores theorem, and present some partial results toward this conjecture.

Agenda

17:00~17:20  Mathematica Campus License 사용방법 및 Tip

             강사 : 황지원 차장 (다한테크&Wolfram Certified Instructor)

17:20~17:50  Mathematica New Feature

             강사 : Farid Pasha (International Business Manager, Wolfram Research Inc.)

17:50~18:00 Break

18:00~18:50  Mathematica Technology – Graphics & Machine Learning

             강사 : 정재범 (Kernel개발자, Wolfram Research Inc.)

Mathematica의 개발사인 Wolfram Research Inc에서 Graphics Visualization 관련 Kernel 개발자인

한국인 박사를 모시고, 현재 최신 기술의 동향과 이와 관련된 Mathematica 기능을 볼 수 있는 기회입니다.

특히 Geometry와 3D Printing에 대해 자세한 설명을 들어볼 수 있으며, 최근 제일 관심을 끌고 있는

Machine Learning에 대한 Mathematica의 예제들을 살펴볼 수 있습니다.

또한 이번 세미나에서는 KAIST에서 보유하고 있는 Mathematica의 사용방법과 안내를 받아볼 수 있으니,

학생들을 비롯하여 많은 교수님들의 관심 부탁 드립니다.

The transition density (if it exists) of Markov process is the heat kernel of the generator of the process. The transition density of a general Markov process rarely admits an explicit expression. Thus obtaining sharp estimates on transition density is a fundamental problem both in probability theory and in analysis. In this talk, we discuss the behavior of transition density (Dirichlet heat kernel) for subordinate Brownian motions in $C^{1,1}$-open subsets whose scaling orders are not necessarily strictly less than $2$. Our estimate is sharp and explicitly expressed in terms of the distance to the boundary, Laplace exponent of subordinator and its derivative. This is a joint work with Ante Mimica.

We consider a parabolic-parabolic equations which describes some chemotactic model with advection and absorbing reaction.  We establish the local and global well-posedness of regular solutions for the model. We also prove that the total mass of the cell density asymptotically approaches a strictly positive constant, provided that efficiency of absorbing reaction is strong enough. This talk is based on the joint work with Jaewook Ahn(Kyushu U.), Kyunkeun Kang(Yonsei U.) and Junha Kim(Chung-Ang U.).