Department Seminars & Colloquia
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This talk is concerned with solving some structured multi-linear systems, especially focusing on the equations whose coefficient tensors are M-tensors, or called M-equations for short. We prove that a nonsingular M-equation with a positive right-hand side always has a unique positive solution. Several iterative algorithms are proposed for solving multi-linear nonsingular M-equations, generalizing the classical iterative methods and the Newton method for linear systems. Furthermore, we apply the M-equations to some nonlinear differential equations and the inverse iteration for spectral radii of nonnegative tensors.
In statistical methods for language and document modeling, there are
two major perspectives: representation at the document level, and
representation at the word level. At the document level, topic models
such as latent Dirichlet allocation (LDA) and hierarchical Dirichlet
process (HDP), based on the word-document matrix, aim to discover
topics whose dimensionality is much lower than the size of the
vocabulary. At the word-level, language models such as n-grams and
neural word embedding, based on the word co-occurrence matrix, aim to
represent each word in a high-dimensional vector space. In this work,
we develop Dual Representation Topic Model (DRTM), a novel topic model
which combines the advantages of the two approaches. DRTM models
documents and words based on the locations of the individual words
within documents, as well as the local contexts of the words. DRTM
transforms each document into a network of words by generating edges
when words of near proximity have high semantic similarity. Then it
infers the topic for each edge - a pair of words - rather than
assigning topics for individual words as in traditional topic models.
This enables the model to learn a better document representation by
inferring the global topics while considering the local contexts of
individual words.
- Speaker: Professor Lawrence Ein
(LAS Distinguished Professor, University of Illinois at Chicago)
- Title: Secant varieties and asymptotic syzygies
- Abstract: We would a give a simple proof for the theorems of Ran and Behesthi - Eisenbud on giving an upper bound of on the dimensions of the higher secant varieties, which will generalize to curves that are not just lines.
In the second half, I would like to discuss joint work with Erman and Lazarsfeld on giving a simple proof for a non-vanishing theorem for the syzygies of the Veronese embedding of the projective space.
E6-1, ROOM 1409
Discrete Math
Cyril Nicaud (Université Paris-Est, France)
Introduction to analytic combinatorics
E6-1, ROOM 1409
Discrete Math
In classical combinatorics, sequences of positive integers are usually studied through their generating series. These formal power series can be used to classify the sequences, to obtain closed formulas for the number of object of a given size, …
Seeing the generating series as analytic functions, we can use tools of complex analysis (such as the residue theorem) to obtain, typically, an asymptotic equivalent to the sequence under consideration.
In this talk I will give a quick overview of the main results obtained in this field, from the automatic construction of generating series to some theorems coming from the theory of functions of a complex variable.
The talk will not assume any specific knowledge in combinatorics or complex analysis.
E6-1, 1409
Discrete Math
Doowon Koh (Chungbuk National University)
Introduction of the finite field Erdős distance problem
E6-1, 1409
Discrete Math
The purpose of this talk is to study the finite field analog of the Erdős distance problem. First, the conjecture and known results on the problem are reviewed. Second, we introduce basic skills to deduce results on the problem. Finally, we address how to improve currently known results on the Erdős distance problem.
Sponsored by KAIST BK21 Plus.
2000년 초부터 저금리기조가 유지되면서, 주가연계증권(ELS), 파생결합증권(DLS), 구조화 채권(Structured Notes) 등의 구조화상품으로 대변되는 금융혁신은 급격한 양적 성장을 이루었다. 그러나 이러한 금융의 혁신이 금융기관의 성과 및 금융안정에 미치는 영향에 대해서는 거의 분석이 이루어지지 못하였다. 물론, 구본일, 엄영호, 지현준(2007), 엄영호(2015), 강병진(2016) 등의 몇몇 연구에서 구조화 파생상품의 가격적정성 및 기초자산에 미치는 영향, 그리고 구조화상품 투자에 따른 투자자의 효용에 대한 분석은 이루어졌으나, 구조화 상품 발행에 따라 금융기관의 경영 안정성에 미친 영향에 대해 국내의 연구는 전무한 현실이다. 해외의 사례에서도 Breuer and Perst(2007), Branger and Breuer(2008), Henderson and Pearson(2011) 등의 연구들이 구조화상품의 발행가격의 적정성 및 가격결정에 미치는 영향 등에 대한 분석은 수행하였으나 소비자를 대상으로 하는 금융혁신(파생결합증권)의 금융기관 안정성에 미치는 영향에 대한 분석은 매우 제한되어 있다. 다만, 2007-2008년 글로벌 금융위기를 겪은 후, 금융기관 사이에서 이루어진 복잡한 금융혁신(파생결합계약)이 위기를 증폭시켰다는 이론적 및 실증적 분석을 제시하였으나, 국내의 상황과는 큰 차이가 있다. 해외 시장에서 이루어지는 금융혁신이 대부분 금융기관 간 거래에서 발생하는 반면, 국내에서는 대부분의 파생결합증권이 소비자 대상 판매상품인 경우가 많기 때문에, 해외시장의 사례가 국내 금융환경에 그대로 적용하는데 무리가 따른다. 본 연구는 이러한 이유로 파생결합증권의 발행이 증권회사의 건전성에 미치는 영향에 대한 실증분석을 수행하는 것을 목적으로 한다. 파생결합증권을 발행한 증권회사의 손실 메카니즘을 분석하고 2015년 및 2016년 헤지손실을 기록했던 증권회사의 사례를 소개한다. 이를 바탕으로 ELS의 만기가 도래하는 2018년 증권사의 유동성 위기에 대한 가능성을 예상한다. 다음으로 증권회사의 건전성 규제에 대해 소개하고 파생결합증권 발행에 따른 건전성 규제에 미치는 영향을 살펴본다.
The purpose of this work is to mathematically justify the phenomena of the plasma sheath formation near the surface of a ball-shaped material immersed in a bulk plasma, and to get some qualitative information of such a boundary sheath layer. To this end, we employ the Euler-Poisson equations (both the stationary and nonstationary models) in the three dimensional annular domain to investigate the existence, time-asymptotic behavior and quasi-neutral limit of the boundary layer solutions. This is a joint work with C.-Y. Jung (UNIST) and M. Suzuki (Nogoya Inst. Tech.).
Stochastic partial differential equations (SPDEs) are differential equations which include the effects of random forces and environments. The theory of SPDE was started in 1970s, and L_p-theory of SPDE was first introduced by Krylov in 1994.
Since then, L_p-theory has become one of main approaches to study the regularity of solutions to SPDEs.
In this talk, I will give a short description on SPDEs and introduce classical L_p-theory of second-order SPDEs.
I will also introduce recent results on SPDEs having non-local operators.
Abstract
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.
Black holes are perhaps the most celebrated predictions of general relativity. Miraculously, these complicated spacetimes arise as explicit (i.e., exact expression can be written down!) solutions to the vacuum Einstein equation. Looking these explicit black hole solutions leads to an intriguing observation: While the black hole exterior look qualitatively similar for every realistic black hole, the structure of the interior, in particular the nature of the `singularity' inside the black hole, changes drastically depending on whether the black hole is spinning (Kerr) or not (Schwarzschild).
A proposed picture for what happens in general is the so-called strong cosmic censorship conjecture of R. Penrose, which is a central conjecture in general relativity. In this colloquium, I will give a short introduction to general relativity and explain what this conjecture is. Time permitting, I will present some recent progress (joint work with J. Luk at Stanford) on related topics, using tools from nonlinear hyperbolic PDEs.
VOD 보기
We introduce graded manifold as an ordinary smooth manifold with Z/2Z - graded functions, that are nothing but the sections of the exterior bundle of a vector bundle. We extend ordinary differential calculus to this setting, and introduce a graded extended version of the variational bicomplex and Lagrangian field theory, that can incorporate the idea of fermions in physics.
In the inhomogeneous heat equation
ut(t; x) = u(t; x) + f(t; x);
The term f models the interruptions to the heat di usion along time and on space locations.
Especially, the e ect of f in time direction is more troublesome and the regularity of u is subject
to the regularity of f. In this talk we discuss a type of modeling of f in the form Nt(!)g(t; x),
where N is a random noise process which can be whiter than the white noise. We also discuss
a regularity relation between Ng and u.
Intelligent systems with deep learning have emerged as a key technique for a wide range of different applications including vision processing, autonomous driving and robot navigation. SoC implementations in deep learning-based intelligent systems give us higher performance and low-power operations in many applications.
VOD 보기
The pharmacometrics (PM) is originated from roots of pharmacokinetics and pharmacodynamics through population and physiological based pharmacokinetics to pharmacometrics. PM is developing science that how to apply and use mathematical and statistical methods to understand a drug's behavior in body, quantify uncertainty of information, and make rational for data-driven decision in the drug discovery and pharmacotherapy.
Improvements in drug discovery is necessarily require to enhance translational research from pre-clinical to clinical stages. According to this stream, PM are occurring to powerful and efficient skill to unify divided knowledge. Nowadays, it is hard to imagine a more efficient, powerful and informative drug development process without the expansion of the role of PM. Pharmacotherapy is also in great need of improved dosing strategy selection for the avoidance of adverse events and the improvement in efficacy. This will get through the development of pragmatic PM models that provide knowledge about drug behavior and how the drug can be optimally used. As more pragmatic PM models are developed, optimal dosing strategies based can be implemented.
In recent, paradigm of PM is going to be more wider than now to cover biological process, as system pharmacology. Therefore, the importance of PM is not able to be overemphasized in whole process for drug discovery and clinical application.
Keywords
Pharmacometrics, Drug discovery, Clinical application
- Speaker: Prof. David Eisenbud (UC Berkeley)
- Title: How to make homogeneous prime ideals of high regularity
- Date: Friday 23 September, 2016
- Time: 16:00pm.
- Place: Rm. 1409, Natural Sciences B/D E6-1, KAIST
- Language: English
- Organizer: Sijong Kwak
Hilbert syzygy theorem says that any finitely generated graded module $M$ over the standard graded polynomial ring $S=K[x_1,ldots,x_n]$ has a finite free resolution
$$
0 leftarrow M leftarrow F_0 leftarrow F_1 leftarrow ldots leftarrow F_c leftarrow 0
$$
with $F_i = oplus_j S(-i)^{beta_{ij}}$ a free module with $beta_{ij}$ generators
in degree $j$. Hilbert proved his syzygy theorem to exhibit the polynomial nature of the Hilbert series:
$$
H_M(t) = sum_k dim M_k t^k = frac{sum_i (-1)^i sum_j beta_{ij}t^j}{(1-t)^n}
$$
In the talk I will report on the question, what kind of more information about $M$
is encoded in the graded Betti numbers $beta_{ij}(M)$, what are the possible values
of these numbers, and, what can be said about extremal cases.
VOD 보기
The ideas of Lagrangian and Hamiltonian field theories from Physics can be precisely formulated using the formalisms of infinite jet bundles and polysymplectic manifolds, respectively. Moreover, the classical stages of their BRST quantization procedures can also be described on simple extensions of them involving Grassmann variables. We plan to follow chapters 1, 2, 9, 3, 4 of "Advanced classical field theory" by Giachetta-Mangiarotti-Sardanashvily (http://www.worldscientific.com/worldscibooks/10.1142/7189) in four lectures, in the order as indicated. We assume some experience with differential forms on smooth manifolds. In the first lecture, we cover chapter 1, that reviews geometry of fiber bundles and introduces infinite jet formalism.
Suppose that a rational map between two projective spaces over a field is defined by a set of homogeneous polynomials of the same degree. It is interesting and important to study if such a map is birational onto its image. In this talk, we present an algebraic characterization under some assumptions on the ideal. Our result is obtained by analyzing the defining ideal of the special fiber ring. This is joint work with Vivek Mukundan.
E6, 1409
Discrete Math
Ringi Kim (University of Waterloo)
Unavoidable subtournaments in tournaments with large chromatic number
E6, 1409
Discrete Math
Unavoidable subtournaments in tournaments with large chromatic number
Ringi Kim (김린기)
University of Waterloo, Waterloo, Ontario, Canada
University of Waterloo, Waterloo, Ontario, Canada
2016/9/9 Fri 4PM-5PM
For a tournament T, the chromatic number of T is the minimum number of transitive sets with union V(T). We say a set H of tournaments is heroic if there exists c such that every tournament excluding all members of H has chromatic number at most c. Berger et al. explicitly characterized all heroic sets of size one. In this talk, we study heroic sets of size two. This is a joint work with Maria Chudnovsky, Ilhee Kim, and Paul Seymour.
Theoretical Computer Science provides the sound foundation
and rigorous concepts underlying contemporary algorithm
design and software development -- for discrete problems:
Problems in the continuous realm commonly considered in Numerical Engineering are largely treated by 'recipes' and 'methods'
whose correctness and efficiency is usually shown empirically.
We extend and apply the theory of computation over discrete structures
to continuous domains: It turns out that famous complexity classes like
P, NP, #P, and PSPACE naturally re-emerge in the setting of real numbers,
sequences, continuous functions, operators, and Euclidean subsets
(including a reformulation of a Millennium Prize Problem as a numerical one).
We currently work towards a rigorous computability and complexity
classification for partial differential equations, namely over
Sobolev spaces that their solutions naturally 'live' in.
VOD 보기
In this talk, I will review energy momentum fields and conformal weights of vertex operator algebras (VOAs). I will show conformal weight decompositions of VOAs with some examples. If time allows, I will introduce relations between vertex algebras and Poisson vertex algebras.