Slides for "happy hour" talk on "Weyl laws and dense periodic orbits", June 18, 2020
FAQ for prospective graduate students, March 16, 2020

Research

• Papers 2020-
  • (with J. Chaidez) Computing Reeb dynamics on four-dimensional convex polytopes, J. Computational Dynamics 8 (2021), 403-445, almost final version here.
  • (with J. Gutt and V. Ramos) Examples around the strong Viterbo conjecture, J. Fixed Point Theory Appl. 24 (2022), no. 2. Paper No. 41, 22 pp., almost final version here.
  • (with D. Cristofaro-Gardiner, U. Hryniewicz, and H. Liu) Contact three-manifolds with exactly two simple Reeb orbits, Geometry and Topology 27 (2023), no. 9. 3801-3831, almost final version here.
  • (with O. Edtmair) PFH spectral invariants and C-infinity closing lemmas, arXiv:2110.02463.
  • An elementary alternative to ECH capacities, PNAS 119 (35) e2203090119, almost final version here.
  • Elementary spectral invariants and quantitative closing lemmas for contact three-manifolds, J. Modern Dynamics 20 (2024), 635-662, almost final version here.
  • Braid stability for periodic orbits of area-preserving surface diffeomorphisms, IMRN Volume 2025, Issue 16, rnaf207, almost final version here.
  • (with D. Cristofaro-Gardiner, U. Hryniewicz, and H. Liu) Proof of Hofer-Wysocki-Zehnder's two or infinity conjecture, arXiv:2310.07636.
  • (with U. Hryniewicz and V. Ramos) Hopf orbits and the first ECH capacity, arXiv:2312.11830, to appear in J. Topology.
  • Zeta functions of dynamically tame Liouville domains, arXiv:2402.07003.
  • (with A. Roy, M. Weiler, and Y. Yao) Anchored symplectic embeddings, arXiv:2407.08512.
  • Reeb orbits frequently intersecting a symplectic surface, arXiv:2504.19332.
• Older papers
  • Embedded contact homology and applications
  • Other kinds of contact and Floer homology
  • Reidemeister torsion in circle-valued Morse theory
  • Soap bubbles and isoperimetric problems
  • Other research papers
  • Survey articles
• Blog, 2011-18 (mostly about research)

Teaching

• Courses
  • Math H53, Honors Multivariable Calculus, Fall 2025
  • Math 214, Differential Topology, Fall 2025
  • Past courses.
• Lecture notes etc. (Please use freely.)
  • A very basic introduction to mathematical proofs (pdf, 27 pages). These notes are rather ancient, but maybe still helpful.
  • Lecture notes on Morse homology (with an eye towards Floer theory and pseudoholomorphic curves), Fall 2002 (postscript, 63 pages). Despite the mistakes in these notes, some people have found them useful as an introduction, so I will keep them up here while hoping to revise them sometime.
  • Slides for a Berkeley Math Circle colloquium talk on Rubik's cube, 2022. (The actual lecture did not cover all of the slides.)
  • Multivariable calculus videos.
• PhD students and theses
  • Theodore Coyne, Galen Liang, Audrey Rosevear, Nancy Eagles, Gabriel Beiner (in progress)
  • Oliver Edtmair, Some quantitative results in symplectic geometry, 2024
  • Ziwen Zhao, Algebraic structures of fixed point Floer homology of Dehn twists, 2024
  • Yuan Yao, Morse-Bott embedded contact homology, 2023
  • Luya Wang, New structures on embedded contact homology and applications to low-dimensional topology, 2023
  • Julian Chaidez, Convexity in contact geometry and Reeb dynamics, 2021. (Preprints here, here, and here.)
  • Morgan Weiler, Mean action of periodic orbits of area-preserving annulus diffeomorphisms, 2019. (Preprint here.)
  • Mihai Munteanu, Nontrivial tori in spaces of symplectic embeddings, 2019. (Preprints here and here.)
  • Chris Gerig, Seiberg-Witten and Gromov invariants for self-dual harmonic 2-forms, 2018. (Preprints here and here.)
  • Keon Choi, The embedded contact homology of toric contact manifolds, 2013. (Preprint here.)
  • Dan Cristofaro-Gardiner, Some results involving embedded contact homology, 2013. (Preprints here, here, and here.)
  • Vinicius Gripp Barros Ramos, The asymptotics of ECH capacities and absolute gradings on Floer homologies, 2013. (Preprints here, here, and here.)
  • David Farris, The embedded contact homology of nontrivial circle bundles over Riemann surfaces, 2011.
  • Andrew Cotton-Clay, Symplectic Floer homology of area-preserving surface diffeomorphisms and sharp fixed point bounds, 2009.
  • Eli Lebow, Embedded contact homology of 2-torus bundles over the circle, 2007.
  • Tamas Kalman, Contact homology and one parameter families of Legendrian knots, 2004.

Personal

• Who I am
• Who I am not

Contact

Office: 923 Evans
Postal address: Mathematics Dept, 970 Evans Hall, University of California, Berkeley CA 94720
Phone: (510) 642-4329 Department fax: (510) 642-8204
Email: [My last name with the last letter deleted]@math.berkeley.edu

Berkeley Mathematics Department.

This material is based upon work supported by the National Science Foundation. 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 (NSF).