In 1980s, Donaldson discovered his famous invariant of 4-manifolds which was subsequently proved to be an integral on the moduli space of semistable sheaves when the 4-manifold is an algebraic surface. In 1994, the Seiberg-Witten invariant was discovered and conjectured to be equivalent to the Donaldson invariant (still open). In late 1990s, Taubes  proved that the Seiberg-Witten invariant also counts pseudo-holomorphic curves.

The Donaldson-Thomas invariant of a Calabi-Yau 3-fold Y (complex projective manifold of dimension 3 with nowhere vanishing holomorphic 3-form) can be thought of as a generalization of the Donaldson invariant. It was defined by virtual integrals on the moduli space of stable sheaves on Y and expected to count algebraic curves in Y. The categorification conjecture due to Kontsevich-Soibelman, Joyce-Song, Behrend-Bryan-Szendroi and others claims that there should be a cohomology theory on the moduli space of stable sheaves whose Euler number coincides with the Donaldson-Thomas invariant.

I will talk about recent progress about the categorification conjecture by using perverse sheaves. Locally the moduli space is the critical locus of a holomorphic function on a complex manifold called a Chern-Simons chart and we have the perverse sheaf of vanishing cycles on the critical locus. By constructing suitable Chern-Simons charts and homotopies using gauge theory, it is possible to glue the perverse sheaves of vanishing cycles to obtain a globally defined perverse sheaf whose hypercohomology is the desired categorified Donaldson-Thomas invariant. As an application, we can provide a mathematical theory of the Gopakumar-Vafa (BPS) invariant. I will also discuss wall crossing formulas for these invariants.

It is known that a compact complex manifold of dimension 2 with the same Betti numbers as the complex projective plane is projective. Such a manifold is called a fake projective plane if it is not isomorphic to the complex projective plane.

A fake projective plane has betti numbers, $b_1=0$, $b_2=1$, hence $q=p_g=0$, $c_2=3$ and by Noether formula $c_1^2=9$. Either its canonical class or its anti-canonical class is ample. The latter case yields the complex projective plane. So a fake projective plane is exactly a smooth surface of general type with $p_g=0$ and $c_1^2=3c_2=9$. By Aubin and Yau, its universal cover is the unit complex 2-ball and hence its fundamental group $\pi_1(X)$ is infinite. More precisely, $\pi_1(X)$ is a discrete torsion-free cocompact subgroup $\Pi$ of $PU(2,1)$ having minimal Betti numbers and finite abelianization. By Mostow's  rigidity theorem, such a ball quotient is strongly rigid, i.e., $\Pi$ determines a fake projective plane up to holomorphic or anti-holomorphic isomorphism. By Kharlamov and Kulikov, no fake projective plane can be anti-holomorphic to itself. Thus the moduli space of fake projective planes consists of a finite number of points, and the number is the double of the number of distinct fundamental groups $\Pi$. By Hirzebruch's proportionality principle, $\Pi$ has covolume 1 in $PU(2,1)$. Furthermore, Klingler proved that the discrete torsion-free cocompact subgroups of $PU(2,1)$ having minimal Betti numbers are arithmetic. With these informations, Prasad and Yeung recently have carried out a classification of fundamental groups of fake projective planes. In this talk I will focus on fake projective planes which cover a (2,3)-Dolgachev surface, and discuss their derived categories.

A toric variety of complex dimension $n$ is a normal algebraic variety over $\mathbb C$ with an effective algebraic action of $(\mathbb C^\ast)^n$ having an open dense orbit. A compact non-singular toric variety is called a toric manifold. Little is known about the diffeomorphism or homeomorphism classification of toric manifolds, but the following problem, called the cohomological rigidity problem for toric manifolds, is proposed by Masuda and Suh in 2008; are two toric manifolds diffeomorphic if their integral cohomology rings are isomorphic as graded rings? This problem is still open, but only partial results have been known.

A Bott tower of height $n$ is a sequence of projective bundles $$     B_n \stackrel{\pi_n}\longrightarrow B_{n-1} \stackrel{\pi_{n-1}}\longrightarrow \cdots \stackrel{\pi_2}\longrightarrow B_1 \stackrel{\pi_1}\longrightarrow B_0 = \{\text{a point}\}, $$ where each $\pi_i$ is the projectivization of a Whitney sum of two complex line bundles. We call $B_n$ an $n$-stage Bott manifold. It is known that a Bott manifold is a toric manifold. A one-stage Bott manifold is a complex projective space, and a two-stage Bott manifold is known as a Hirzebruch surface. In this talk, we survey the current progress on the cohomological rigidity problem, and we shall present many affirmative evidences of that any cohomology ring isomorphism between two Bott manifolds is indeed realizable by a certain diffeomorphism.

  세상에 대한 기존 지식을 인식하는 방법들을 분류하여 삶의 지혜를 효율적으로 깨닫는 방법을 찾고자 한다. 삶에 대한 탐구는 가시적인 일상영역으로부터 보이지 않는 극대, 극소세계로 끝없이 넓혀나가야 한다. 내부관점으로 살아가기에 삶의 탐구 방법은 생각하는 힘을 발휘하여 외부관점을 확보하면서 근사한 모델 세계를 창으로 하여 통찰하는 것이 보다 효율적이다.

  대부분의 연구가 선행연구를 언급하고 관계되는 인용을 거쳐 과학적으로 지식의 세계를 확장하고 있다. 그러나 내부관점에서의 연구는 사안의 본질과 나아갈 목표를 외면하고 지엽적으로 흘러갈 경우에 그 가치가 떨어진다. 반면 외부에서 전체 사안을 살피는 거시적 연구는 모델세계가 사안에 근사하게 설정될수록 핵심과 나아갈 방향을 잘 찾을 수 있게 해준다. 확장되는 지식들을 창으로 하여 삶에 대한 지혜를 하루 빨리 깨닫게 하는 것이야말로 많은 연구들이 추구해야 할 방향이며 목표다.

  위상수학, 수학교육, 교양강좌를 강의하면서 행복한 삶으로 이끄는 수학의 큰 역할들을 찾아 왔으며, 인문학자와는 다른 방법으로 삶에 대하여 연구하고 있다. 삶의 지혜를 효율적으로 깨닫게 하는 수학적 사고법을 통해서 수학교육이 어느 교과보다 인생길을 가는 데에 꼭 필요함을 소개하고자 한다.

[Mathematics for Information Sciences]  김종락  Kim, Jon-Lark

• 서강대학교 수학과 교수
   Professor, Department of Mathematics, Sogang University

Title: Codes, Lattices, and Rings

Codes over rings have been used in the construction of interesting Euclidean or Hermitian lattices. We begin with some basic definitions of codes and lattices. Given a prime $p$, B. Fine proved that there are exactly three commutative rings with unity of order $p^2$ and characteristic $p$. Using C. Bachoc's results, we describe that how these rings can be related to certain quotient rings of the ring of algebraic integers of an imaginary quadratic number field. Then we construct Hermitian lattices from codes over these rings. Shaska, et. al. have studied the theta functions of these Hermitian lattices. We generalize the results of Bachoc, Shaska, et. al. Some parts of this talk are based on the joint work with Yoonjin Lee and Steven Dougherty.

[Cryptography]  윤아람  Yun, Aaram

• 울산과학기술대학교 전기전자컴퓨터공학부 교수
   Professor, School of Electrical and Computer Engineering,
   Ulsan National Institute of Science and Technology (UNIST)

Title: On Homomorphic Authenticated Encryption

Recently, Gentry proposed fully homomorphic encryption where a third party can homomorphically evaluate a function using only ciphertexts, therefore providing useful services and at the same time preserving privacy.  Yet, the authenticity of data is as important as privacy, but homomorphic encryption alone cannot provide authenticity of outsourced data.  Homomorphic authenticated encryption is a cryptographic primitive which protects both privacy and authenticity of outsourced data, while allowing homomorphic function evaluation.  We examine various security definitions of homomorphic authenticated encryption and their relationship with each other.  Also, we construct an example of homomorphic authenticated encryption scheme supporting arithmetic circuits, which is chosen-ciphertext secure both for privacy and authenticity.