Sungkyung Kang

Upcoming Talk/Travel

• Sep 09-29, Travelling and family reunion before moving to my new job

• Oct 03, London Geometry and Topology Seminar at Imperial College London

• Oct 06, Topology Seminar at Southampton

• Oct 22, Geometry & Topology Seminar at Cambridge

• Jan 03-08, International Congress of Chinese Mathematicians at Shanghai

• Jun 22-26, Geometry and Topology in Low Dimensions: Lasting Trends and Emerging Directions at Budapest

• Jul 06-10, Low-Dimensional Topology at Mathematisches Forschungsinstitut Oberwolfach

Academic positions

• 2023-present, Titchmarsh Research Fellow at University of Oxford

• 2020-2023, Postdoctoral Fellow at IBS Center for Geometry and Physics

• 2019-2020, Postdoctoral Fellow at the Chinese University of Hong Kong

• Trinity 2016, Non-stipediary Lecturer, Keble College, University of Oxford

Awards

• 2025 Frontiers of Science Award, awarded to the paper The (2,1)-cable of the figure-eight knot is not smoothly slice

Research

I'm interested in using Heegaard Floer theory (and other related techniques) to solve low-dimensional topological problems. 

My favorite tool is involutive Heegaard Floer homology. But recently I'm also doing gauge theory.

My papers:

• Smooth concordance of cables of the figure-eight knot, joint with Junghwan Park and Masaki Taniguchi

• Exotic Dehn twists and homotopy coherent group actions, joint with Junghwan Park and Masaki Taniguchi

• Cables of the figure-eight knot via real Frøyshov invariants, joint with Junghwan Park and Masaki Taniguchi, to appear in Geometry & Topology

• Invariant splitting principles for the Lipshitz-Ozsvath-Thurston correspondence, joint with Gary Guth

• Central extensions of higher groups: Green-Schwarz mechanism and 2-connections, joint with Monica Jinwoo Kang

• Doubled disks and satellite surfaces, joint with Gary Guth,  Kyle Hayden, and Junghwan Park

• Topologically and rationally slice knots, joint with Jennifer Hom and Junghwan Park, to appear in Selecta Mathematica

• One stabilization is not enough for closed knotted surfaces, joint with Kyle Hayden and Anubhav Mukherjee

• One stabilization is not enough for contractible 4-manifolds

• The (2,1)-cable of the figure-eight knot is not smoothly slice, joint with Irving Dai, Junghwan Park, Abhishek Mallick, and Matthew Stoffregen, Inventiones mathematicae (2024): 1-20.

  • This preprint was featured in Quanta Magazine. See "outreach" section

• Torsion in the knot concordance group and cabling, joint with JungHwan Park, Journal of the European Mathematical Society (2024).

• Involutive knot Floer homology and bordered modules, to appear in Quantum Topology

• Stabilization and satellite construction of doubly slice links, joint with Seungwon Kim and Hongtaek Jung

• On the nonorientable four-ball genus of torus knots, joint with Fraser Binns, Jonathan Simone, and Paula Truöl, to appear in Algebraic & Geometric Topology

• Linear independence of rationally slice knots, joint with Jennifer Hom, Junghwan Park, and Matthew Stoffregen, Geometry & Topology 26.7 (2023): 3143-3172.

• Concordance invariants and the Turaev genus, joint with Seungwon Kim and Hongtaek Jung, International Mathematics Research Notices 2022.19 (2022): 15410-15420.

• Ribbon knots, cablings, and handle decompositions, joint with Jennifer Hom and Junghwan Park, Mathematical Research Letters, Vol. 28, No. 5 (2021), pp. 1441-1457.

• Link homology theories and ribbon concordances, Quantum Topology 13.1 (2022): 183-205.

• Connected Floer homology of covering involutions, joint with Antonio Alfieri and Andras Stipsicz, Mathematische Annalen, 2020, 377.3: 1427-1452 

• Z2-equivariant Heegaard Floer cohomology of knots in S3 as a strong Heegaard invariant

• A transverse knot invariant from Z2-equivariant Heegaard Floer cohomology

• Spectral order for contact manifolds with convex boundary, joint with Andras Juhasz, Algebr. Geom. Topol. 18 (2018) 3315-3338  

Works in progress:

• Exotic diffeomorphisms of contractible 4-manifolds and stabilizations, joint with JungHwan Park and Masaki Taniguchi

• A concordance-satellite formula via relative Lipshitz-Ozsvath-Thurston correspondence, joint with Gary Guth

• Some work going on, joint with my sister, Monica Jinwoo Kang, and Johan Asplund

Outreach

• Mathematicians Eliminate Long-Standing Threat to Knot Conjecture, Quanta Magazine, Feb 2023

An interesting open question

• Is there a Z/2Z-homology 3-sphere which is not Z/2Z-homology cobordant to branched double covers of S^3 along a knot? 

I've been investigating this problem for almost two years, but still have no clue. Maybe equivariant Heegaard Floer homology of Hendricks-Lipshitz-Sarkar can help?

Programming

Given a quasi-alternating knot, its branched double cover Σ(K) is an L-space. Thus the induced homotopy involution, induced by the deck transformation, on the Heegaard Floer chain complex of Σ(K) is nulhomotopic. Now, if we consider the knot filtration, we get the knot Floer theory of (Σ(K),K), and the deck transformation action here might be nontrivial. 

Question (originally by Stipsicz): Can we find an example of such a knot K?

I was working on this question earlier this year. Given a n-by-n grid diagram of K, one can explicitly compute the Z/2-action on the hat-flavored HFK of (Σ(K),K), with space complexity and time complexity O((n!)^2), which is doable when n is at most 9. But for all examples of K that I tested had trivial action on HFK. 

If somebody can find an example that works, please let me know!

Here's the C++ code that I wrote to solve this.

Teaching

• Tutor - Topology and Groups, University of Oxford, Michaelmas 2024

• Tutor - Algebraic Topology, University of Oxford, Michaelmas 2024

• Tutor - Prelim Analysis, Queen's College, University of Oxford, Michaelmas 2024

• Tutor - Prelim Geometry, Queen's College, University of Oxford, Michaelmas 2024

• Tutor - Metric Spaces, Keble College, University of Oxford, Michaelmas 2024

• Tutor - Group Theory, Keble College, University of Oxford, Trinity 2024

• Tutor - Low-dimensional Topology, University of Oxford, Hilary 2024

• Tutor - Rings and Modules, Keble College, University of Oxford, Michaelmas 2023

• Tutor - Algebraic Topology, University of Oxford, Michaelmas 2023

• Tutor -  Linear Algebra(Prelims), Keble College, University of Oxford, Michaelmas 2023

The list of teaching experiences I had before the current employment is as follows. Note that the Korean law prevented me from teaching classes while doing an alternative military service (i.e. my previous position at IBS). 

• Lecturer - Linear Algebra, The Chinese University of Hong Kong, Fall 2019

• Tutor - Group Theory, Keble College, University of Oxford, Trinity 2019 

• Teaching Assistant - Geometric Group Theory, University of Oxford, Hilary 2018

• Teaching Assistant - Algebraic Topology, University of Oxford, Michaelmas 2017 

• Class Tutor, Keble College, University of Oxford, Hilary 2017 - Trinity 2017

• Teaching Assistant - Algebraic Topology, University of Oxford, Michaelmas 2016 

•  Teaching Assistant - Algebraic Number Theory, University of Oxford, Hilary 2016

• Teaching Assistant - Algebraic Topology, University of Oxford, Michaelmas 2015 

• Teaching Assistant - Logic and Set Theory, KAIST, Fall 2014 

Past Plenary Talks

•  East Asian Conference on Geometric Topology, Soochow University, China (via Zoom), Feb 2023

Past Invited Talks

• Categorification in Low Dimensional Topology, Ruhr-Universtat Bochum, Germany, July 2025

• International Congress of Basic Sciences 2025, Beijing Institute of Mathematical Sciences and Applications, China, July 2025

• Trisectors Workshop 2025: Connections with Diffeomorphism Groups, UT Austin, United States, June 2025

• Conference on Modern Developments in Low-Dimensional Topology, ICTP Trieste, Italy, June 2025

• Topology Seminar, KAIST, South Korea, April 2025

• UK TILDE 2025, Newcastle Universty, United Kingdom, April 2025

• London Low-dimensional Topology Seminar, King's College London, United Kingdom, March 2025

• Topology Seminar, Stanford University, United States, January 2025

• Geometry and Topology Seminar, University of Glasgow, United Kingdom, December 2024

• Michigan Topology Intercity Symposium (MITIS), University of Michigan, United States, November 2024

• Topology Seminar, Michigan State University, United States, November 2024

• Integrability, geometry and QFTs: 1st MaPhyAg workshop, Spain, October 2024

• Topology Seminar, Princeton University, United States, October 2024

• Low-dimensional Topology Seminar, Max-Planck-Institut für Mathematik, Germany, September 2024

• JNU/SNU/KAIST Geometric Topology Workshop 2024, KAIST, South Korea, August 2024

• Topologie, Oberwolfach, Germany, July 2024

• 2024 MATRIX-IBSCGP Workshop on Symplectic and Low-dimensional Topology, IBS Center for Geometry and Physics, Pohang, South Korea, June 2024

• Georgia topology conference, University of Georgia, United States, May 2024

• Topology Seminar, KAIST, South Korea, April 2024

• Differential Topology '24, Tsukuba University, Japan, March 2024

• Seminar in Simons senester, IMPAN Warsaw, March 2024

• Colloquium, Center for Mathematical Physics Hamburg, January 2024

• Topology Seminar, University of Oxford, January 2024

• Virtual seminar on geometry and topology, KIAS, South Korea, August 2023

• Special session in low-dimensional topology, KMS Spring Meeting, South Korea, April 2023

• Mini-workshop on low-dimensional topology, POSTECH, South Korea, April 2023

• Gauge Theory in Kyoto, Kyoto University, Japan, March 2023

• Informal seminar during a focus week, Alfréd Rényi Institute of Mathematics, Hungary, March 2023

• Interactions of 3- & 4-dimensional Topology, China (via Zoom), March 2023

• Algebraic Geometry and Differential Topology seminar, Alfréd Rényi Institute of Mathematics, Hungary, March 2023

• Knot Online Seminar K-OS, Feb 2023

• Morphisms in Low Dimensions, Oberwolfach, Germany, Jan 2023

• Moscow-Beijing Topoolgy Seminar, Tsinghua University, China (via Zoom), Jan 2023

• Taubes' secret seminar, United States (via Zoom), December 2022

• Algebraic Geometry and Differential Topology Seminar, Alfréd Rényi Institute of Mathematics, Hungary (via Zoom), December 2022

• Taubes' secret seminar, United States (via Zoom), November 2022

• Floer homotopical methods in low dimensional and symplectic topology, MSRI, United States, November 2022

• PK^2 workshop, Busan,  South Korea, November 2022

• Rookies Pitch seminar, Seoul National University, South Korea, November 2022

• Geometry and Topology seminar, Caltech, United States, October 2022

• IBS-CGP 10th anniversary workshop, Pohang, South Korea, October 2022

• Topology Seminar, Princeton University, United States, September 2022

• Topology Seminar, Tokyo University, Japan (via Zoom), July 2022

• Topology Seminar, Michigan State University, United States, June 2022

• Geometry and Topology Seminar, Rutgers University (via Zoom), April 2022

• Workshop for young mathematicians in Korea, South Korea (via Zoom), January 2022

• Pohang Mathematical Workshop, Busan, South Korea, December 2021

• Topology Seminar, KAIST, South Korea, October 2021

• Recent techniques in Floer and Khovanov homology, NCNGT, June 2021

• Special Session on Floer theory, KMS Spring Meeting, South Korea (via Zoom), April 2021

• Special Session on Low-dimensional Topology, KMS Fall Meeting(via Zoom), October 2020

• Moscow-Beijing Topology Seminar, Tsinghua University, China (via Zoom), July 2020

• Topology Seminar, IMS, CUHK, Hong Kong (via Zoom), July 2020

• Mini-Symposium : Knot Theory in Okinawa, OIST, Japan, February 2020

• Topology Seminar, IBS-Center for Geometry and Physics, South Korea, December 2019

• Topology Seminar, KIAS, South Korea, November 2019

• Symplectic Geometry Seminar, Stony Brook University, United States, November 2019

• Topology Seminar, Boston College, United States, October 2019

• Topology Seminar, Georgia Tech, United States, October 2019

• Topology Seminar, Institute of Mathematical Sciences, The Chinese University of Hong Kong, September 2019

• Topology Seminar, UCLA, United States, July 2019

• Topology Seminar, University of Oregon, United States, July 2019

• Topology Seminar, KIAS, South Korea, June 2019

• Poincare Seminar, University of Oxford, United Kingdom, May 2019

• Chekanov-Eliashberg algebra and knot invariants, Alfréd Rényi Institute of Mathematics, Hungary, March 2019

• Geometry and Topology Seminar, Seoul National University, South Korea, July 2018

• Topology Seminar, University of Oxford, United Kingdom, January 2018

Past Contributed Talks

• Lightning talk, Summer school - New structures in low-dimensional topology, Alfréd Rényi Institute of Mathematics, Hungary, July 2024

• East Asian Conference on Geometric Topology, RIMS, Japan, February 2024

• Lightning talk, Morphisms in Low Dimensions, Oberwolfach, Germany, January 2023

• Workshop for Young Mathematicians in Korea, KAIST, South Korea, December 2022

• Lightning Talk, New Perspectives in Four Dimensions, University of Victoria, June 2022

• East Asian Conference on Geometric Topology, KIAS, Korea (via Zoom), January 2022

• East Asian Conference on Geometric Topology, University of Tokyo, Japan (via Zoom), January 2021

• East Asian Conference on Geometric Topology, RIMS, Japan, February 2020

• Short Talk, Floer Homotopy 2019, University of Oregon, United States, July 2019 

• Early Career Topology Conference, University of Sheffield, United Kingdom, June 2019

• Young Researchers in Mathematics 2018, University of Southampton, United Kingdom, June 2018 

• British and Irish Geometry Meeting 2018, Queen’s University Belfast, United Kingdom, March 2018

References

• Professor Andras Juhasz, Mathematical Institute, University of Oxford

• Professor Ciprian Manolescu, Department of Mathematics, Stanford University

• Professor Jennifer Hom, School of Mathematics, Georgian Institute of Technology

• Professor Yong-Geun Oh, IBS Center for Geometry and Physics / Pohang University of Science and Technology (POSTECH)

• Professor JungHwan Park, Department of Mathematical Sciences, Korea Advanced Institute of Science and Technology (KAIST)

• (Regards Teaching) Dr Richard David Wade, Mathematical Institute, University of Oxford