Welcome to Geometry and Topology Lab at Sungshin Women's University, Seoul, South Korea. Our lab is focused on a classification problem of smooth and symplectic 4-manifolds.

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School of Mathematics, Statistics and Data Sciences

Sungshin Women’s University

2 Bomun-ro 34 Da-gil, Seongbuk-gu

Seoul 02844, Republic of Korea

Phone: +82-2-920-7534(Office)

FAX: +82-2-920-2046

- September 1994 - September 2000, Ph.D.(Mathematics)

University of California at Irvine.

Ph.D Thesis: Alexander polynomial of Plat Representation.

Academic Advisor: Prof. Ronald John Stern.

Mathematical Genealogy Project - March 1989 - February 1991, M.S.(Mathematics)

Seoul National University

Master Thesis: Descriptions of the Poincaré Homology 3-sphere

Academic Advisor: Prof. HyunKoo Lee - March 1985 - February 1989, B.S.(Mathematics)

Seoul National University

- Assistant, Associate and full Professor: March 2008 - current, Department of Mathematics, Sungshin Women’s University, Seoul, Korea
- Research fellow: August 2007 -Feburary 2008, Department of mathematical sciences, Seoul National University, Seoul, Korea.
- Research Fellow: July, 2003 - June, 2007, ABRL, Department of Mathematics, Konkuk University, Seoul, Korea
- Research Fellow: July 23, 2001 - July 22, 2003, Korea Institute for Advanced Study (KIAS), Seoul, Korea.
- Postgraduate Researcher: October 1, 2000 - June 30, 2001, University of California, Irvine, USA.

- Director of Sungshin R&DB Foundation, July 2020 - June 2022

- August, 2022 -- August, 2023, Korea Institute for Advanced Study (KIAS)
- January, 2015– February, 2015 & June, 2015– July, 2016, Alfred Renyi institute of Mathematics, Budapest, Hungary
- March, 2003–June, 2003, Institute for Pure and Applied Mathematics(IPAM), UCLA, USA

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- Mathscinet Reviewer
- Zentralblatt MATH Reviewer
- Editor: Bulletin of the Korean Mathematical Society
- Korean Mathematical Society, Vice President(General Affairs), January, 2021 - December. 2022
- Korean mathematical society, Secretary General, January, 2019 - December, 2020
- Seoul ICM (International Congress of Mathematicians) 2014 Organizing committee member
- Review Board member, Division of Natural Science, National Research Foundation of Korea (September, 2014-August, 2016)

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- Low dimensional Topology and Geometry, especially Lefschetz fibration structures on symplectic 4-manifold
- Classification of smooth 4-manifolds and symplectic 4-manifolds

- September 1994 – August 1997, The Korean Government Scholarship Program for Study Overseas (KGSPSO)
- May 2009 – April 2012, Basic science research program, National Research Foundation of Korea
- May 2012 – April 2015, Basic science research program, National Research Foundation of Korea
- August 2014– December 2015, Korea-EU(ERC) Researcher exchange program, National Research Foundation of Korea
- November 2015 – October 2018, Basic science research program, National Research Foundation of Korea
- December 2016 – November 2021, Samsung Science and Technology Foundation(co-PI)
- March 2019 - August 2022, Basic science research program, National Research Foundation of Korea

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Topology and its Applications 153 (2006), pp. 1994-2012

In this article we show that the signature of a Lefschetz fibration coming from a special involution as a product of right handed Dehn twists depends only on the number of genus on the involution axis. We investigate the geography of such Lefschetz fibrations and we identify it with a blow up of a ruled surface. We also get a geography of the Lefschetz fibration coming from a finite order element of mapping class group as a composition of two special involutions.

Journal of the knot theory and its ramifications Vol. 16, No. 4 (2007) pp. 499-521

In the article we study the multi-variable Alexander polynomial of a link in a plat form or in a closed braid form. By using the method, we find an algorithm how to compute the multi-variable Alexander polynomial of the 2a-fold dihedral cover and the a-fold irregular cover of a two bridge knot K(a,b)

Communications in contemporary Mathematics, Vol. 9, No. 5 (2007) pp. 681-690 (Joint work with Jongil Park)

We present a simple way to construct an infinite family of simply connected, nonspin, smooth 4-manifolds with one basic class which do not admit a symplectic structure with either orientation.

Transactions of the AMS Vol. 360 No.11(2008)pp. 5853-5868

In the article, we study the Fintushel-Stern’s knot surgery four manifold E(n)K and their monodromy factorization. For fibered knots we provide a smooth classification of knot surgery 4-manifolds up to twisted fiber sum. We then show that other constructions of 4-manifolds with the same Seiberg-Witten invariants are in fact diffeomorphic.

Mathematische Annalen Vol. 345(2009) pp. 581-597 (Joint work with Jongil Park)

In this article we construct an infinite family of simply connected minimal symplectic 4-manifolds, each of which admits at least two nonisomorphic Lefschetz fibration structures with the same generic fiber. We obtain such examples by performing knot surgery on an elliptic surface E(n) using a special type of 2-bridge knots.

Bulletin of the Korean Mathematical Society Vol.47(2010) No.5 pp. 961-971 (Joint work with Jongil Park)

As an application of ‘reverse engineering’ technique introduced by R. Fintushel, D. Park and R. Stern, we present a simple way to construct an infinite family of exotic \((2n+2l-1) \mathbf{CP}^2\sharp (2n+4l-1)\overline{\mathbf{CP}^2}\)’s for all \(n \ge 0\),\(l \ge 1\).

Michigan Mathematical Journal Vol. 60(2011) pp. 525-544 (Joint work with Jongil Park)

In this article we study Lefschetz fibration structures on knot surgery 4-manifolds obtained from an elliptic surface E(2) using Kanenobu knots K. As a result, we get an infinite family of simply connected mutually diffeomorphic 4-manifolds coming from a pair of inequivalent Kanenobu knots. We also obtain an infinite family of simply connected symplectic 4-manifolds, each of which admits more than one inequivalent Lefschetz fibration structures of the same generic fiber.

Journal of Symplectic Geometry 13(2015) No.2 pp. 279–303 (Joint work with Jongil Park)

In this article, we show that, at least for non-simply connected case, there exist an infinite family of nondiffeomorphic symplectic 4-manifolds with the same Seiberg-Witten invariants. The main techniques are knot surgery and a covering method developed in Fintushel and Stern’s paper.

Kyungpook Mathematical Journal, Vol. 53 No. 4(2013), pp. 603 –614

In the article we show that nondiffeomorphic symplectic 4-manifolds which admit marked Lefschetz fibrations can share the same monodromy group. Explicitly we prove that, for each integer g > 0, every knot surgery 4-manifold in a family \(\{ E(2)_K | K \text{ is a fibered 2-bridge knot of genus } g \text{ in } S^3\}\) admits a marked Lefschetz fibration structure which has the same monodromy group.

arXiv:1503.06272 (Joint work with Jongil Park), Michigan Mathematical Journal 66 (2017), no. 3, pp. 481-498

In this article we construct a family of knot surgery 4-manifolds admitting arbitrarily many nonisomorphic Lefschetz fibration structures with the same genus fiber. We obtain such families by performing knot surgery on an elliptic surface \(E(2)\) using connected sums of fibered knots obtained by Stallings twist from a slice knot \(3_1\sharp 3_1^*\). By comparing their monodromy groups induced from the corresponding monodromy factorizations, we show that they admit mutually nonisomorphic Lefschetz fibration structures.

Proc. Amer. Math. Soc. 145 (2017), pp. 3607-3616.(Joint work with András I. Stipsicz)

We show that the minimal number of singular fibers \(N(g,1)\) in a genus-\(g\) Lefschetz fibration over the torus is at least \(3\). As an application, we show that \(N(g,1) = \{3,4\}\) for \(g \ge 5\), \(N(g,1)\in \{3,4,5\}\) for \(g = 3,4\) and \(N(2,1) = 7\).

Joint work with Heesang Park, Jongil Park, Dongsoo Shin, Communications of the KMS 32(2017) no.3, pp.479-494

We construct various examples of simply connected minimal complex surfaces of general type with \(p_g=0\) and \(K^2=1,2\) using \(\mathbb{Q}\)-Gorenstein smoothing method.

Joint work with Hakho Choi and Jongil Park, Asian Journal of Mathematics 23(2019), no.5, pp. 735-748

In this article we show that all knot surgery 4-manifolds \(E(n)_K\) are mutually diffeomorphic after a connected sum with \(\mathbb{CP}^2\). Hence, by combining a known fact that every simply connected elliptic surface is almost completely decomposable, we conclude that every knot surgery \(4\)-manifold \(E(n)_K\) is also almost completely decomposable.

- 황지영(중등교사, 경기도 공립)
- 김도영(중등교사, 경기도 공립)
- 손지민(메가스터디)
- 손은아(통계청)
- 정혜원(중등교사, 경기도 공립)
- 김민서(중등교사, 경기도 공립)
- 교육대학원: 황지영, 채혜정, 김주영, 김예슬, 손은아, 고은희

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