Maggie Miller


Click here for an annotated list of my papers and preprints.

Research interests

I am interested in geometric low-dimensional (usually 4, but sometimes 3 or 5) topology. Many of my previous projects have involved embeddings of surfaces in 4-manifolds, e.g. constructing/obstructing isotopies and concordances between surfaces, finding interesting structures on the complements of surfaces, or understanding surfaces from a diagrammatic point of view. Surfaces in 4-manifolds can be used to create exotic 4-manifolds or understand h-cobordisms of 4-manifolds; they are a natural object of study to 4-dimensional topologists.

My favorite open problem is to decide whether the unknot bounds a unique ribbon disk up to isotopy rel boundary. I find this problem interesting because ribbon disks (disks in B4 with no local maxima with respect to radial height) are arguably the simplest possible surfaces and live in perhaps the simplest 4-manifold (B4), yet even if we restrict to the simplest possible boundary (the unknot), we still do not understand the class of all such objects. (In fact, we do not even understand π0 of the space of all ribbon disks bounded by the unknot, let alone the whole space.) I think this illustrates some of the complexity of 4-dimensional topology.

Papers and preprints

Click a title to be redirected to the arXiv entry.

  1. A. Conway, I. Dai, M. Miller, Z-disks in CP2, arXiv:2403.10080 [math.GT], Mar. 2024.

  2. M. Hughes, S. Kim, M. Miller, Branched covers of twist-roll spun knots, arXiv:2402.11706 [math.GT], Feb. 2024.

  3. A. J. Lobb, M. Miller, A. Ray, Morphisms in Low Dimensions, Oberwolfach Rep. 20(1):215–259, 2023.

  4. M. Hughes, S. Kim, M. Miller, Non-isotopic splitting spheres for a split link in S4, arXiv:2307.12140 [math.GT], July 2023. Submitted.

  5. P. Aceto, N. A. Castro, M. Miller, JH. Park, A. Stipsicz, Slice obstructions from genus bounds in definite 4-manifolds, arXiv:2303.10587 [math.GT], Mar. 2023. Submitted.

  6. M. Miller, Explicitly describing fibered 3-manifolds through families of singularly fibered surfaces, arXiv:2306.13081 [math.GT], June 2023. To appear in Proc. Sympos. Pure Math. "Frontiers in Geometry and Topology."

  7. M. Klug and M. Miller, Concordance of spheres in 4-manifolds with an immersed dual sphere, arXiv:2211.07177 [math.GT], Nov. 2022. Submitted.

  8. J. Joseph, J. Meier, M. Miller, A. Zupan, Bridge trisections and Seifert solids, arXiv:2210.09669 [math.GT], Oct. 2022. To appear in Algebr. Geom. Topol.

  9. K. Hayden, S. Kim, M. Miller, JH. Park, I. Sundberg, Seifert surfaces in the 4-ball, arXiv:2205.15283 [math.GT], May 2022. Submitted.

  10. J. Joseph, J. Meier, M. Miller, A. Zupan, Bridge trisections and classical knotted surface theory, Pac. J. Math. 319(2):343–369, 2022.

  11. M. C. Hughes, S. Kim, M. Miller, Knotted handlebodies in the 4-sphere and 5-ball, arXiv:2111.13255 [math.GT], Nov. 2021. To appear in J. Eur. Math. Soc.

  12. M. C. Hughes, S. Kim, M. Miller, Band diagrams of immersed surfaces in 4-manifolds, arXiv: 2108.12794 [math.GT], Aug. 2021. Submitted.

  13. K. Hayden, A. Kjuchukova, S. Krishna, M. Miller, M. Powell, N. Sunukjian, Brunnian exotic surface links in the 4-ball, arXiv:2106.13776 [math.GT], June 2021. To appear in Mich. Math. J.

  14. M. Miller and B. Ozbagci, Lefschetz fibrations on nonorientable 4-manifolds, Pac. J. Math. 312(1):177–202, 2021.

  15. M. Miller and P. Naylor, Trisections of non-orientable 4-manifolds, arXiv:2010.07433 [math.GT], Oct. 2020. To appear in Mich. Math. J.

  16. M. Miller and A. Zupan, Equivalent characterizations of handle-ribbon knots, arXiv:2005.11243 [math.GT], May 2020. To appear in Commun. Anal. Geom.

  17. P. Aceto, J. Meier, A. N. Miller, M. Miller, JH. Park, A. I. Stipsicz, Branched covers bounding rational homology balls, Algebr. Geom. Topol. 21(7):3569–3599, 2021.

  18. A. Juhász, M. Miller, I. Zemke, Transverse invariants and exotic surfaces in the 4-ball, Geom. Topol. 25(6):2963–3012, 2021.

  19. M. R. Klug and M. Miller, Concordance of surfaces and the Freedman-Quinn invariant, J. Topol. 14(2):560–586, 2021.

  20. N. A. Castro, G. Islambouli, M. Miller, M. Tomova, The relative L-invariant of a compact 4-manifold, Pac. J. Math. 315(2):305–346, 2021.

  21. I. Dai and M. Miller, The 0-concordance monoid is infinitely generated, Proc. Amer. Math. Soc. 151(8):3601–3609, 2023.

  22. M. Miller, The effect of link Dehn surgery on the Thurston norm, Int. Math. Res. Not. 2023(22):19069–19114, 2023.

  23. A. Juhász, M. Miller, I. Zemke, Knot cobordisms, torsion, and Floer homology, J. Topol. 13(4):1701–1724, 2020.

  24. P. Lambert-Cole and M. Miller, Trisections of 5-manifolds, 2019 MATRIX Annals, MATRIX Book Ser., Springer, pp. 117–134, 2021.

  25. M. Miller and I. Zemke, Knot Floer homology and strongly homotopy-ribbon concordances, Math. Res. Lett. 28(3):849–861, 2021.

  26. M. Miller, A Concordance Analogue of the 4D Light Bulb Theorem, Int. Math. Res. Not. IMRN 2021(4):2565–2587, 2021.

  27. M. Hughes, S. Kim, M. Miller, Isotopies of surfaces in 4-manifolds via banded unlink diagrams, Geom. Topol. 24(3):1519–1569, 2020.

  28. M. Miller, Extending fibrations on knot complements to ribbon disk complements, Geom. Topol. 25(3):1479–1550, 2021.

  29. S. Kim and M. Miller, Trisecting surface complements and Price twist surgery, Algebr. Geom. Topol. 20(1):343–373, 2020.

  30. M. Miller, Concordances from the standard surface in S2 × S2, J. Knot Theory Ramifications 29(9):1950–1957, 2019.