# Geometry and Topology Lab:

## Personal Information

Department of Mathematics,
Sungshin Women’s University
Seoul 02844, Korea
• Phone: +82-2-920-7534(Office)
• FAX: +82-2-920-2046

## Education

• September 1994 - September 2000, Ph.D.
University of California at Irvine.
Ph.D Thesis: Alexander polynomial of Plat Representation.
• March 1989 - February 1991, M.S.
Seoul National University
Master Thesis: Descriptions of the Poincaré Homology 3-sphere
• March 1985 - February 1989, B.S.
Seoul National University

## Work Experiences

• Assistant Professor and Associate 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.

## Long term visit

• 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

## Research Interests

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

## Research Grant

• September 1994 – August 1997,
Study Abroad Scholarship supported by Korean government
• 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)

## Mathematical Services

• Mathscinet Reviewer
• Zentralblatt MATH Reviewer
• Seoul ICM (International Congress of Mathematicians) 2014 Organizing committee member
• Editor: Bulletin of the Korean Mathematical Society
• Review Board member, Division of Natural Science, National Research Foundation of Korea (September, 2014-August, 2016)

## Publications and Abstracts

• On the signature of a Lefschetz fibration coming from an involution,

• 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.

• Multi-variable Alexander polynomial of a plat,

• 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)

• Rational blow-downs and nonsymplectic 4-manifolds with one basic class,

• 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.

• Twisted fiber sums of Fintushel-Stern’s knot surgery 4-manifolds,

• 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.

• Nonisomorphic Lefschetz fibrations on a knot-surgered 4-manifold,

• 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.

• Exotic smooth structures on $$(2n+2l-1) \mathbf{CP}^2\sharp (2n+4l-1)\overline{\mathbf{CP}^2}$$,

• 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 $$n \ge 0$$,$$l \ge 1$$.

• Lefschetz fibration structures on knot surgery 4-manifolds,

• 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.

• Families of non-diffeomorphic 4-manifolds with the same Seiberg-Witten invariants,

• 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.

• Monodromy groups on knot surgery 4-manifolds,

• 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.

• Lefschetz fibrations on knot surgery 4-manifolds via Stallings twist,

• arXiv:1503.06272 (Joint work with Jongil Park), to appear in Michigan Mathematical Journal

• 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.

• Minimal number of singular fibers in Lefschetz fibrations over the torus,

• arXiv:1604.04877 (Joint work with András I. Stipsicz), Proceedings of the AMS (published online)

• 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 $$N(2,1) = 7$$.

• Simply connected complex surfaces of general type with $$p_g=0$$ and $$K^2=1,2$$

• Joint work with Heesang Park, Jongil Park, Dongsoo Shin, Communications of the KMS (published online)

• 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.

• On dissolving knot surgery 4-manifolds under a $$\mathbf{CP}^2$$-connected sum.

• arXiv:1704.02181(joint work with Hakho Choi and Jongil Park)

• 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.