Bruno Buchberger

• Professor, Computer Mathematics, Johannes Kepler University, Linz, Austria
• Research Institute for Symbolic Computation
   
Title: Mathematics and Computer: The Story Only Starts   
The computer is a deeply mathematical invention. A decade before the first computer was built (around 1940), its simple but ingenious principle was invented by mathematicians (in fact, “meta”-mathematicians who studied the foundations of mathematics by mathematical methods). And a decade before the first computer was built, one of these meta-mathematicians (Gödel 1931) already showed that mathematical thinking can be automated to an arbitrary degree of sophistication but, at the same time, every level of sophistication leaves potential mathematical insights open that cannot be discovered and proved automatically by the current level of sophistication.

We will report on the current state of automated discovery and proof in mathematics by discussing progress made in the speaker’s Theorema Project: By recent advances in automated proving and algorithm synthesis we were able to automatically synthesize a Gröbner bases algorithm automatically from the problem specification. This seems to be epistemologically interesting since the invention of the first Gröbner bases algorithm in 1965 solved a problem that was 65 years open.

We will discuss some perspectives of the increasing power of automated reasoning methods on the future of mathematical research, teaching, and applications. In particular, we will report on a task force that aims at building up a “Global Math Formal Knowledge Base”. This global program was initiated at the International Congress of Mathematicians 2014 in Seoul, and so, the Meeting of the KMS seems to be an appropriate occasion for discussing the current state of the program.

박용문  Park, Yong Moon

• 연세대학교 이과대학 수학과 명예교수
   Emeritus Professor, Department of Mathematics, Yonsei University
   
Title: 수학과 물리학으로의 산책  
본 강연은 두 부분으로 이루어졌다. 먼저 1915년 연희전문대학교가 설립되고 수물과(수학 및 물리학과)가 개설되면서 서구의 근대수학이 도입된 후 한국수학이 어떻게 발전하여 왔는가를 살펴보고자 한다. 특히 1970년대 초까지 걸음마 수준이었던 한국수학의 연구 활동이 어떻게 선진국 수준에 근접할 수 있게 되었는가를 본인의 연구경험을 토대로 하여 설명하고자 한다. 다음으로 수학이 어떤 학문인가를 설명하기 위하여 수학의 특징과 구조 그리고 공리체계에 대하여 개략적으로 설명하고자 한다. 그리고 수학과 가장 가까운 학문분야인 물리학이 수학과 어떻게 상호보완 하면서 발전하여 왔는가를 간략하게 언급하려고 한다.

Jin Akiyama
 
• Professor, Tokyo University of Science, Japan
• Director, Research Center for Math & Science Education, Japan
 
Title: Math Spectacle Show  
It is demonstrated how powerful mathematical theories can be when applied to real life situations. Everywhere you look has mathematics. Keep your eyes open and you can see math used everywhere around you.

Gunnar Carlsson
 
• Co-founder of AYASDI, USA
   (Former professor, Standford University, USA)
 
Title: Topology for Data Analysis  
Topology (the mathematical subdiscipline that concerns itself with the study of shape) is an old subject in mathematics. The first paper was published in 1736, and the subject has developed within pure mathematics over the last few centuries, with particular acceleration in the 20th century. Since the beginning of the 21st century, mathematicians have been working on applying the methods to a new domain, namely point clouds, or data sets. The new methods include extensions of techniques for "measuring" shape, called homology, as well as novel ways of representing data. We will discuss these ideas, with numerous examples.

Chieh-Yu Chang  

• Professor, Department of Mathematics, National Tsing Hua University
• The Mathematical Society of Republic of China
   
Title: Linear relations among double zeta values in positive characteristic  
This talk is motivated by the conjecture of Zagier on the explicit dimension formula for the space of the same weight double zeta values in terms of the dimension of elliptic cusp forms for the full modular group. We will present an effective criterion for computing the dimension of the same weight double zeta values over a rational function field in positive characteristic. Contrary to the Zagier's conjecture, the analogue of Zagier's conjectural dimension provides a lower bound for the dimension of the double zeta values when the weight is 'even'.

[위상수학 Topology]  이대웅  Lee, Dae-Woong

• 전북대학교 자연과학대학 수학과
   Professor, Department of Mathematics, Chonbuk National University
    
Title: Strong homology groups, phantom maps, comultiplications and same $n$-types   
In this talk, the following topics in algebraic topology will be briefly outlined:
(1) strong (co)homology groups,
(2) phantom maps,
(3) comultiplication structures for a wedge of spheres, and
(4) the same n-types of CW-complexes.  

A subordinate Brownian motion is a Lévy process which can be obtained by replacing the time of the Brownian motion by an independent subordinator. The infinitesimal generator of a subordinate Brownian motion is $-\phi(-\Delta)$, where $\phi$ is the Laplace exponent of the subordinator. In this talk we discuss  the sharp two-sided estimates on the Green functions of  a large class of subordinate Brownian motions without diffusion component in any bounded $\kappa$-fat open set $D$. When $D$ is a bounded $C^{1,1}$ open set, such estimates have an explicit form in terms of the distance to the boundary and $\phi$. We also discuss consequences of such sharp Green function estimates such as boundary Harnack principle in $C^{1,1}$ open sets with explicit rate of decay, Poisson Kernel estimates, Fatou type theorem.

This talk is based on several joint papers with Jaehoon Kang, Yunju Lee, Ante Mimica, Renming Song, Zoran Vondraček.

[응용수학 Applied Mathematics]  임미경  Lim, Mikyoung

• 카이스트 자연과학대학 수리과학과 교수
   Professor, Department of Mathematical Sciences, KAIST (Korea Advanced Institute of Science and Technology)
    
Title: Asymptotic analysis for the electric field enhancement in between two nearly touching conductors with extreme conductivities

Two nearly touching inclusions with extreme conductivities can cause the enhancement of the electric field. In an external electric field of long wavelength compared to the size of conductors, the presence of nearly touching conductors induces a very large electric field confined in the narrow gap region between the conductors. In this talk, we analyze this field enhancement based on the quasi-static approximation and the related conductivity equation for the electric scalar potential. The generic blow-up rate of the gradient of the solution to the conductivity problem in presence of two perfect conductors is $|\epsilon\ln\epsilon|^{-1}$ in three dimensions and $\epsilon^{-1/2}$ in two dimensions, where $\epsilon$ is the distance between the two conductors. I derive asymptotic formulas for the solution to the conductivity problem, which characterize the gradient blow-up of the solution.
 

[암호학 Cryptography]  이주영  Lee, Jooyoung

• 세종대학교 자연과학대학 수학통계학부 교수
   Professor, Mathematics-Applied Statistics, Sejong University
    
Title: Partition and Mix: Generalizing the Swap-or-Not Shuffle

Recently, card shuffle algorithms has begun to attract renewed interest from a cryptographic point of view with applications to format preserving encryption. In this work, we naturally extend the swap-or-not shuffle by replacing a perfect matching used in each round of the swap-or-not shuffle by a certain uniform keyed partition. By using such a random set partition of blocks of size D, we prove that the number of rounds can be reduced by 1/log D for the same level of security compared to the swap-or-not shuffle, improving the efficiency of the resulting encryption scheme~(in terms of the number of rounds). We also give a couple of examples of the uniform keyed partitions that allow efficient implementation in practice.