Robert Lipshitz

Papers

Preprints

• "Diagonals and A-infinity Tensor Products" with Peter Ozsváth and Dylan Thurston. arXiv:2009.05222. Last updated September 13, 2023.
• "Bordered HF- for three-manifolds with torus boundary" with Peter Ozsváth and Dylan Thurston. arXiv:2305.07754. Last updated May 12, 2023.
• "Local equivalence and refinements of Rasmussen's s-invariant" with Nathan Dunfield and Dirk Schütz. arXiv:2312.09114. Last updated December 14, 2023.
• "Bordered Floer homology, handlebody detection, and compressing diffeomorphisms" with Akram Alishahi. arXiv:2408.06619. Last updated August 21, 2024.

Accepted or published

Roughly, my work so far falls into four categories: bordered Heegaard Floer homology, an extension of Heegaard Floer homology to 3-manifolds with boundary; equivariant Lagrangian intersection Floer homology and its applications to Heegaard Floer homology and symplectic Khovanov homology; other topics in Heegaard Floer homology; and Khovanov homology and its stable homotopy refinement. (There is some overlap.) Papers in each category are in approximately the order they were written. A link to each paper on the arXiv and its review on MathSciNet is at the end of each entry. I have not read any of these reviews.

Bordered Heegaard Floer homology

• Bordered Heegaard Floer homology  with Peter Ozsváth and Dylan Thurston. Memoirs of the American Mathematical Society, Vol. 254, No. 1216 (2018). math/0810.0687. MR3827056.

• "Bimodules in bordered Heegaard Floer homology" with Peter Ozsváth and Dylan Thurston. Geometry & Topology, Vol. 19 (2015), no. 2, 525-724. arXiv:1003.0598. MR3336273.

• "Computing HF^ by factoring mapping classes" with Peter Ozsváth and Dylan Thurston. Geometry & Topology, Vol. 18 (2014), no. 5, 2547-2681. arXiv:1010.2550. MR3285222.

• "Bordered Floer homology and the spectral sequence of a branched double cover I" with Peter Ozsváth and Dylan Thurston. Journal of Topology, Vol. 7 (2014), no. 4, 1155-1199. arXiv:1011.0499. MR3286900.

• "Bordered Floer homology and the spectral sequence of a branched double cover II: the spectral sequences agree" with Peter Ozsváth and Dylan Thurston. Journal of Topology, Vol. 9 (2016) no. 2, 607-686. arXiv:1404.2894. MR3509974.
  • Some of the figures in the last section have some beta-circles reflected from how they should be. Corrected versions of those figures are here.

• "A faithful linear-categorical action of the mapping class group of a surface with boundary" with Peter Ozsváth and Dylan Thurston. Journal of the European Mathematical Society, Vol. 15 (2013), no. 4, pp. 1279-1307. arXiv:1012.1032. MR3055762.

• "Heegaard Floer homology as morphism spaces" with Peter Ozsváth and Dylan Thurston. Quantum Topology, Vol. 2 (2011), no. 4, pp. 381-449. arXiv:1005.1248. MR2844535.

• "Relative Q-gradings from bordered Floer theory" with Peter Ozsváth and Dylan Thurston. Michigan Mathematical Journal, Vol. 67 (2018) No. 4, 827-838. arXiv:1211.6990. MR3877439.

• "Noncommutative Hodge-to-de Rham spectral sequence and the Heegaard Floer homology of double covers" with David Treumann. Journal of the European Mathematical Society, Vol. 18 (2016), no. 2, 281-325. arXiv:1203.2963. MR3459952

• Cornered Heegaard Floer homology with Christopher Douglas and Ciprian Manolescu. Memoirs of the American Mathematical Society, Vol. 262, No. 1266 (2019). arXiv:1309.0155. MR4045965.

• "Involutive bordered Floer homology" with Kristen Hendricks. Transactions of the American Mathematical Society, Vol. 372 (2019) No. 1, 389-424. arXiv:1706.06557. MR3968773.

• "Bordered Floer homology and incompressible surfaces" with Akram Alishahi. Annales de l'Institut Fourier, Vol. 69 (2019), no. 4, 1525-1573. arXiv:1708.05121. MR4010864.

• "A bordered HF^- algebra for the torus" with Peter Ozsváth and Dylan Thurston. arXiv:2108.12488. Journal of Symplectic Geometry, accepted.

Expository

• "Slicing planar grid diagrams: a gentle introduction to bordered Heegaard Floer homology" with Peter Ozsváth and Dylan Thurston. Proceedings of 15th Gökova Geometry-Topology Conference, 91-119. math/0810.0695. MR2500575.

• "A tour of bordered Floer theory" with Peter Ozsváth and Dylan Thurston. Proceedings of the National Academy of Science, Vol. 108, no. 20, pp. 8085-8092. May 17, 2011. arXiv:1107.5621. MR2806643.

• "Floer homology beyond borders" with Peter Ozsváth and Dylan Thurston. arXiv:2307.16330. Accepted for publication in the Proceedings of the International Congress of Basic Science (2023).

"A faithful linear-categorical action of the mapping class group of a surface with boundary," above, is also partly expository.

Equivariant Floer homology

• "A flexible construction of equivariant Floer homology and applications" with Kristen Hendricks and Sucharit Sarkar. Journal of Topology, Vol. 9 (2016) No. 4, 1153-1236. arXiv:1510.02449. MR3620455.
  • See also the erratum below.

• "Corrigendum: 'A flexible construction of equivariant Floer homology and applications'" with Kristen Hendricks and Sucharit Sarkar. Journal of Topology, Vol. 13 (2020) No. 3, 1317-1331. arXiv:1810.03049. [MR4125758]{https://mathscinet.ams.org/mathscinet/article?mr=4125758}

• "A simplicial construction of G-equivariant Floer homology" with Kristen Hendricks and Sucharit Sarkar. Proceedings of the London Mathematical Society, Vol. 121 (2020) No. 6, 1798-1866. arXiv:1609.09132. MR4201124

• "Rank inequalities for the Heegaard Floer homology of branched covers" with Kristen Hendricks and Tye Lidman. Documenta Mathematica, Vol. 27 (2022), 581-612. arXiv:2007.03023. MR4424030

See also "Noncommutative Hodge-to-de Rham spectral sequence and the Heegaard Floer homology of double covers", above.

Contact homology

• "Torsion in linearized contact homology for Legendrian knots" with Lenhard Ng. arXiv:2308.13482. Michigan Mathematical Journal, accepted.

Other topics in Heegaard Floer homology

• "A cylindrical reformulation of Heegaard Floer homology". Geometry and Topology 10 (2006), pp. 955-1097. math/0502494. MR2240908.
  • See errata below.
  • Further minor corrections and explanations. Last updated February 2021.

• "Errata to 'A cylindrical reformulation of Heegaard Floer homology'". Geometry & Topology, Vol. 18 (2014) 17-30. arXiv:1301.4919. MR3158771.

• "Covering spaces and Q-gradings on Heegaard Floer homology" with Dan Lee. Journal of Symplectic Geometry, Vol. 6 (2008), no. 1, 33-59. math/0608001. Also available here. MR2417439.

• "A Heegaard-Floer invariant of bordered 3-manifolds." Ph.D. thesis, Stanford University, 2006. Available here.

• "Heegaard Floer homology, double points and nice diagrams." Proceedings of the Conference "New Perspectives and Challenges in Symplectic Field Theory, Stanford, June 2007." CRM Proceedings & Lecture Notes, Vol. 49 (2009), 327–342. Available here. MR2555944.

• "Combinatorial cobordism maps in hat Heegaard Floer theory" with Ciprian Manolescu and Jiajun Wang. Duke Mathematical Journal, Vol. 145 (2008), no 2., 207-247. math/0611927. MR2449946.

Expository

• "Heegaard Floer Homologies" in Lectures on quantum topology in dimension three, Panoramas et synthèses 48 (2016), Société Mathématique de France, 131-174. arXiv:1411.4540. MR3617634.

Khovanov homology and homotopy

• "A Khovanov stable homotopy type" with Sucharit Sarkar. Journal of the American Mathematical Society, Vol. 27 (2014), 983-1042. arXiv:1112.3932. MR3230817.
  • Former titles: "A Khovanov homotopy type or two"; using an idea of Oleg Viro we can now show the two variants agree. "A Khovanov homotopy type"; the referee suggested adding the word "stable" to avoid confusion.

• "Khovanov homotopy types and the Dold-Thom functor" with Brent Everitt, Sucharit Sarkar and Paul Turner. Homology, Homotopy and Applications, Vol. 18 (2016), no. 2, 177-181. arXiv:1202.1856. MR3547241.

• "A Steenrod square on Khovanov homology" with Sucharit Sarkar. Journal of Topology, 7 (2014), no. 3, 817-848. arXiv:1204.5776. MR3252965.
  • A minor correction to this paper, Formula (3.1).

• "A refinement of Rasmussen's s-invariant" with Sucharit Sarkar. Duke Mathematical Journal, Vol. 163 (2014), no. 5, 923-952. arXiv:1206.3632. MR3189434.

• "On transverse invariants from Khovanov homology" with Lenhard Ng and Sucharit Sarkar. Quantum Topology, Vol. 6 (2015), no. 3, 475-513. arXiv:1303.6371. MR3392962.

• "Khovanov homotopy type, Burnside category, and products" with Tyler Lawson and Sucharit Sarkar. Geometry & Topology, Vol. 24 (2020), no. 2, 623-745. arXiv:1505.00213. MR4153651

• "Khovanov spectra for tangles" with Tyler Lawson and Sucharit Sarkar. Journal of the Institute of Mathematics of Jussieu, accepted. arXiv:1706.02346.

• "Chen-Khovanov spectra for tangles" with Tyler Lawson and Sucharit Sarkar. Michigan Mathematical Journal, Vol. 71 (2022), No. 2, 401-443. arXiv:1909.12994. MR4484244

• "Khovanov homology detects split links" with Sucharit Sarkar. American Journal of Mathematics, Vol. 144 (2022) No. 6., 1745-1781. arXiv:1910.04246. MR4521052

• "Homotopy functoriality for Khovanov spectra" with Tyler Lawson and Sucharit Sarkar. Journal of Topology, Vol. 15 (2022), No. 4, 2426-2471. arXiv:2104.12907. MR4584595

• "A mixed invariant of non-orientable surfaces in equivariant Khovanov homology" with Sucharit Sarkar. Transactions of the American Mathematical Society, Vol 375 (2022) No. 12, 8807-8849. arXiv:2109.09018. MR4504654

• "Khovanov homology of strongly invertible knots and their quotients" with Sucharit Sarkar. arXiv:2203.13895. \emph{Frontiers in Geometry and Topology}, American Mathematical Society Proceedings of Symposia in Pure Mathematics, Vol. 109 (2024), 157--182.

Expository

• "The cube and the Burnside category" with Tyler Lawson and Sucharit Sarkar. Categorification in Algebra, Geometry and Physics, Contemporary Mathematics, American Mathematics Society. arXiv:1505.00512. MR3611723.

• "Spatial refinements and Khovanov homology" with Sucharit Sarkar. Proceedings of the International Congress of Mathematicians (ICM 2018), pp. 1153-1173 (2019). arXiv:1709.03602.
  • This expository article was not peer reviewed.

• "A remark on quantum Hochschild homology." Proceedings of the Conference "Interactions of gauge theory with contact and symplectic geometry in dimensions 3 and 4", accepted. arXiv:2008.03155.

• • "Categorical lifting of the Jones polynomial: a survey" with Mikhail Khovanov. Bulletin of the American Mathematical Society arXiv:2202.02473.

Funding

My research has been supported by a National Science Foundation (NSF) Graduate Research Fellowship, an NSF Mathematical Sciences Postdoctoral Research Fellowship, a Sloan Research Fellowship, and NSF Grants DMS-0905796, DMS-1149800, DMS-1560783, DMS-1642067, and DMS-1810893. The support specific to each paper is listed in that paper.

Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.