Click here for an annotated list of my papers and preprints.
Research interests
I am interested in geometric lowdimensional (usually 4, but sometimes 3 or 5) topology. Many of my previous projects have involved embeddings of surfaces in 4manifolds, 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 4manifolds can be used to create exotic 4manifolds or understand hcobordisms of 4manifolds; they are a natural object of study to 4dimensional 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 B^{4} with no local maxima with respect to radial height) are arguably the simplest possible surfaces and live in perhaps the simplest 4manifold (B^{4}), 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 π_{0} 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 4dimensional topology.
Papers and preprints
Click a title to be redirected to the arXiv entry.
 K. Hayden, S. Kim, M. Miller, JH. Park, and I. Sundberg, Seifert surfaces in the 4ball, arXiv:2205:15283 [math.GT], May 2022.
 J. Joseph, J. Meier, M. Miller, A. Zupan, Bridge trisections and classical knotted surface theory, arXiv:2112.11557 [math.GT], Dec. 2021. To appear in Pac. J. Math.
 M. C. Hughes, S. Kim, M. Miller, Knotted handlebodies in the 4sphere and 5ball, arXiv:2111.13255 [math.GT], Nov. 2021. Submitted.
 M. C. Hughes, S. Kim, and M. Miller, Band diagrams of immersed surfaces in 4manifolds, arXiv: 2108.12794 [math.GT], Aug. 2021. Submitted.
 K. Hayden, A. Kjuchukova, S. Krishna, M. Miller, M. Powell, and N. Sunukjian, Brunnian exotic surface links in the 4ball, arXiv:2106.13776 [math.GT], June 2021. Submitted.
 M. Miller and B. Ozbagci, Lefschetz fibrations on nonorientable 4manifolds, Pac. J. Math. 312(1):177–202, 2021.
 M. Miller and P. Naylor, Trisections of nonorientable 4manifolds, arXiv:2010.07433 [math.GT], Oct. 2020. To appear in Mich. Math. J.

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

P. Aceto, J. Meier, A. N. Miller, M. Miller, JH. Park, and A. I. Stipsicz, Branched covers bounding rational homology balls, arXiv:2002.10324 [math.GT], Feb. 2020. To appear in Algebr. Geom. Topol.

A. Juhász, M. Miller, and I. Zemke, Transverse invariants and exotic surfaces in the 4ball, arXiv:2001.07191 [math.GT], Jan. 2020. To appear in Geom Topol.

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

N. A. Castro, G. Islambouli, M. Miller, and M. Tomova, The relative Linvariant of a compact 4manifold, arXiv:1908.05371 [math.GT], Aug. 2019. To appear in Pac. J. Math.

I. Dai and M. Miller, The 0concordance monoid is infinitely generated, arXiv:1907.07166 [math.GT], Jul. 2019. To appear in Proc. Amer. Math. Soc.

M. Miller, The effect of link Dehn surgery on the Thurston norm, arXiv:1906.08458 [math.GT], Jun. 2019. Submitted.

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

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

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

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

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

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

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

M. Miller, Concordances from the standard surface in S^{2} × S^{2}, J. Knot Theory Ramifications 29(9):1950–57, 2019.
