Knots and Surfaces in four-manifolds

Smooth four-manifolds – 4D topological manifolds with a smooth structure – are not well characterised. Funded by the European Research Council, the KnotSurf4d project aims to address this gap in the knowledge of high-dimensional manifolds by leveraging the genus function and its enhanced version that takes knots and their slice surfaces into account. This novel approach, which emphasises knots and their slice properties in various four-manifolds, could ultimately provide a candidate for an invariant that is a smooth generalisation of the intersection form and characterises smooth four-manifolds. The team will also study divisibility and torsion questions in the concordance group via knot Floer homology, as well as potential counterexamples for the famous Slice-Ribbon conjecture.

Publications

1. M. Beke, O. Plamenevskaya, L. Starkston. An Unexpected Rational Blowdown. arXiv:2510.00115
2. M. Eper, Sz. Szabo. Rank three representations of Painleve systems: III. Dolbeault structure, spectral correspondence. arXiv:2509.00418
3. M. Eper, Sz. Szabo. Rank three representations of Painlevé systems: I. Wild character varieties. arXiv:2505.21186
4. M. Beke. Minimal rational graphs admitting a QHD smoothing. arXiv:2504.06929
5. Gyenge Ádám , Szabó Szilárd: Blow-ups and the quantum spectrum of surfaces, ADVANCES IN MATHEMATICS (0001-8708 1090-2082): 479 Paper 110432. 39 p. (2025)
6. Mochizuki Takuro , Szabó Szilárd:  Asymptotic behaviour of large-scale solutions of Hitchin’s equations in higher rank, Moduli (2949-7647 2977-1382): 2 Paper e6. (2025)
7. Szabó Szilárd: Exponential decay of harmonic 1-forms for wild harmonic bundles on curves, COMPTES RENDUS MATHEMATIQUE (1631-073X 1778-3569): 362 pp 1015-1021 (2024)
8. J. Hanselman, M. Marengon, B. Wong. Correction terms of double branched covers and symmetries of immersed curves. arXiv:2408.02857
9. M. Eper, Sz. Szabo. Asymptotic geometry of non-abelian Hodge theory and Riemann--Hilbert correspondence, rank three E˜6 case. arXiv:2407.20638
10. R. I. Baykur, A. I. Stipsicz, Z. Szabo. Smooth structures on four-manifolds with finite cyclic fundamental groups. arXiv:2406.09007
11. Sz. Szabó, F. Živanović. Floer-theoretic filtration on Painlevé Hitchin systems. arXiv:2405.14850

Talks

1. Nonabelian Hodge Theory, Riemann--Hilbert Correspondence and Exponential Integrals, 23 October 2025, Modern Musings on Monopoles, Simons Center for Geometry and Physics, Stony Brook (Szabó Sz.)
2. Columbia Symplectic Geometry, Gauge Theory, and Categorification Seminar
   Sep. 19. 2025., invited speaker, title: “Legendrian Two-Bridge Knots”, (Földvári V.)
   Columbia University in the City of New York (New York)
3. Rank three representations of Painlevé systems, 2 September 2025, Workshop In Algebraic Geometry and Physics, Geometry And Physics of Higgs Moduli, Razlog (Bulgaria)
   5. Wild nonabelian Hodge theory and Gromov--Witten theory , 25 September 2025, POSTECH-ELTE Joint Workshop, Pohang (Republic of Korea) (Szabó Sz.)
4. Algebra and Geometry Seminar of BME
   Feb. 18. 2025, invited speaker, title: “Convex Surface Theory for Legendrian Classification”, (Földvári V.)
   Budapest University of Technology and Economics (Budapest)
5. Moduli of Atoms, 18 December 2024, Theory of Atoms Educational workshop, Institute of Mathematics and Informatics at the Bulgarian Academy of Sciences, Sofia (Bulgaria) (Szabó Sz.)
6. IberoSing International Workshop
   Nov. 29. 2024, plenary speaker, title: “Convex Surface Theory for Legendrian Classification”
   Universidad Politécnica de Madrid (Madrid) (Földvári V.)
7. Asymptotic geometry of non-abelian Hodge theory and Riemann--Hilbert correspondence in a rank three parabolic case, 7 October 2024, MaPhyAG Workshop: Integrability, Geometry and QFT, Universidad Complutense de Madrid (Szabó Sz.)
8. Asymptotic Geometry of non-abelian Hodge theory and Riemann-Hilbert correspondence in a rank 3 case, 26 August 2024, RIMS workshop on Moduli spaces of connections, Higgs Bundles and Riemann-Hilbert correspondences, Kyoto (Szabó Sz.)

Conferences

1. Deformation of Complex Singularities and Related Topics 2025.01.12-17.

Invited Speakers

1. Aceto Paolo
2. Santoro Diego
3. Hedden Matthew
4. Stefan Friedl
5. Brendan Owens
6. Gironella Fabio
7. Antonio Alfieri